From Wikipedia, the free encyclopedia
Een numerieke differentiatie formule is een formule voor de numerieke benadering van een afgeleide als gewogen som van functiewaarden in equidistante punten. De basisgedachte daarbij is de functie te benaderen door een polynoom en het benaderende polynoom exact te differentiëren.
Gegeven is de op het interval [a,b] differentieerbare functie f . Het interval wordt opgedeeld in n deelintervallen van gelijke lengte:
a
=
x
0
<
x
1
<
…
<
x
n
=
b
{\displaystyle a=x_{0}<x_{1}<\ldots <x_{n}=b}
de stapgrootte:
h
=
b
−
a
n
=
x
i
−
x
i
−
1
,
i
=
1
,
…
,
n
{\displaystyle h={\frac {b-a}{n}}=x_{i}-x_{i-1},\quad i=1,\ldots ,n}
en de functiewaarden:
f
i
=
f
(
x
i
)
=
f
(
a
+
i
h
)
,
i
=
0
,
1
,
…
,
n
{\displaystyle f_{i}=f(x_{i})=f(a+ih),\quad i=0,1,\ldots ,n}
2-punts 1e afgeleide:
h
f
(
1
)
(
x
0
)
=
−
f
0
+
f
1
−
1
2
h
2
f
(
2
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{0})=-f_{0}+f_{1}-{\frac {1}{2}}h^{2}f^{(2)}(\xi )}
h
f
(
1
)
(
x
1
)
=
−
f
0
+
f
1
+
1
2
h
2
f
(
2
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{1})=-f_{0}+f_{1}+{\frac {1}{2}}h^{2}f^{(2)}(\xi )}
3-punts 1e afgeleide:
h
f
(
1
)
(
x
0
)
=
1
2
(
−
3
f
0
+
4
f
1
−
f
2
)
+
1
3
h
3
f
(
3
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{0})={\frac {1}{2}}(-3f_{0}+4f_{1}-f_{2})+{\frac {1}{3}}h^{3}f^{(3)}(\xi )}
h
f
(
1
)
(
x
1
)
=
1
2
(
−
f
0
+
f
2
)
−
1
6
h
3
f
(
3
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{1})={\frac {1}{2}}(-f_{0}+f_{2})-{\frac {1}{6}}h^{3}f^{(3)}(\xi )}
h
f
(
1
)
(
x
2
)
=
1
2
(
f
0
−
4
f
1
+
3
f
2
)
+
1
3
h
3
f
(
3
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{2})={\frac {1}{2}}(f_{0}-4f_{1}+3f_{2})+{\frac {1}{3}}h^{3}f^{(3)}(\xi )}
3-punts 2e afgeleide:
h
2
f
(
2
)
(
x
0
)
=
f
0
−
2
f
1
+
f
2
−
h
3
f
(
3
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{0})=f_{0}-2f_{1}+f_{2}-h^{3}f^{(3)}(\xi )}
h
2
f
(
2
)
(
x
1
)
=
f
0
−
2
f
1
+
f
2
−
1
12
h
4
f
(
4
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{1})=f_{0}-2f_{1}+f_{2}-{\frac {1}{12}}h^{4}f^{(4)}(\xi )}
h
2
f
(
2
)
(
x
2
)
=
f
0
−
2
f
1
+
f
2
+
h
3
f
(
3
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{2})=f_{0}-2f_{1}+f_{2}+h^{3}f^{(3)}(\xi )}
4-punts 1e afgeleide:
h
f
(
1
)
(
x
0
)
=
1
6
(
−
11
f
0
+
18
f
1
−
9
f
2
+
2
f
3
)
−
1
4
h
4
f
(
4
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{0})={\frac {1}{6}}(-11f_{0}+18f_{1}-9f_{2}+2f_{3})-{\frac {1}{4}}h^{4}f^{(4)}(\xi )}
h
f
(
1
)
(
x
1
)
=
1
6
(
−
2
f
0
−
3
f
1
+
6
f
2
−
f
3
)
+
1
12
h
4
f
(
4
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{1})={\frac {1}{6}}(-2f_{0}-3f_{1}+6f_{2}-f_{3})+{\frac {1}{12}}h^{4}f^{(4)}(\xi )}
h
f
(
1
)
(
x
2
)
=
1
6
(
f
0
−
6
f
1
+
3
f
2
+
2
f
3
)
−
1
12
h
4
f
(
4
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{2})={\frac {1}{6}}(f_{0}-6f_{1}+3f_{2}+2f_{3})-{\frac {1}{12}}h^{4}f^{(4)}(\xi )}
h
f
(
1
)
(
x
3
)
=
1
6
(
−
2
f
0
+
9
f
1
−
18
f
2
+
11
f
3
)
+
1
4
h
4
f
(
4
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{3})={\frac {1}{6}}(-2f_{0}+9f_{1}-18f_{2}+11f_{3})+{\frac {1}{4}}h^{4}f^{(4)}(\xi )}
4-punts 2e afgeleide:
h
2
f
(
2
)
(
x
0
)
=
2
f
0
−
5
f
1
+
4
f
2
−
f
3
+
11
12
h
4
f
(
4
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{0})=2f_{0}-5f_{1}+4f_{2}-f_{3}+{\frac {11}{12}}h^{4}f^{(4)}(\xi )}
h
2
f
(
2
)
(
x
1
)
=
f
0
−
2
f
1
+
f
2
−
1
12
h
4
f
(
4
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{1})=f_{0}-2f_{1}+f_{2}-{\frac {1}{12}}h^{4}f^{(4)}(\xi )}
h
2
f
(
2
)
(
x
2
)
=
f
1
−
2
f
2
+
f
3
−
1
12
h
4
f
(
4
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{2})=f_{1}-2f_{2}+f_{3}-{\frac {1}{12}}h^{4}f^{(4)}(\xi )}
h
2
f
(
2
)
(
x
3
)
=
−
f
0
+
4
f
1
−
5
f
2
+
2
f
3
+
11
12
h
4
f
(
4
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{3})=-f_{0}+4f_{1}-5f_{2}+2f_{3}+{\frac {11}{12}}h^{4}f^{(4)}(\xi )}
4-punts 3e afgeleide:
h
3
f
(
3
)
(
x
0
)
=
−
f
0
+
3
f
1
−
3
f
2
+
f
3
−
3
2
h
4
f
(
4
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{0})=-f_{0}+3f_{1}-3f_{2}+f_{3}-{\frac {3}{2}}h^{4}f^{(4)}(\xi )}
h
3
f
(
3
)
(
x
1
)
=
−
f
0
+
3
f
1
−
3
f
2
+
f
3
−
1
2
h
4
f
(
4
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{1})=-f_{0}+3f_{1}-3f_{2}+f_{3}-{\frac {1}{2}}h^{4}f^{(4)}(\xi )}
h
3
f
(
3
)
(
x
2
)
=
−
f
0
+
3
f
1
−
3
f
2
+
f
3
+
1
2
h
4
f
(
4
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{2})=-f_{0}+3f_{1}-3f_{2}+f_{3}+{\frac {1}{2}}h^{4}f^{(4)}(\xi )}
h
3
f
(
3
)
(
x
3
)
=
−
f
0
+
3
f
1
−
3
f
2
+
f
3
+
3
2
h
4
f
(
4
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{3})=-f_{0}+3f_{1}-3f_{2}+f_{3}+{\frac {3}{2}}h^{4}f^{(4)}(\xi )}
5-punts 1e afgeleide:
h
f
(
1
)
(
x
0
)
=
1
12
(
−
25
f
0
+
48
f
1
−
36
f
2
+
16
f
3
−
3
f
4
)
+
1
5
h
5
f
(
5
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{0})={\frac {1}{12}}(-25f_{0}+48f_{1}-36f_{2}+16f_{3}-3f_{4})+{\frac {1}{5}}h^{5}f^{(5)}(\xi )}
h
f
(
1
)
(
x
1
)
=
1
12
(
−
3
f
0
−
10
f
1
+
18
f
2
−
6
f
3
+
f
4
)
−
1
20
h
5
f
(
5
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{1})={\frac {1}{12}}(-3f_{0}-10f_{1}+18f_{2}-6f_{3}+f_{4})-{\frac {1}{20}}h^{5}f^{(5)}(\xi )}
h
f
(
1
)
(
x
2
)
=
1
12
(
f
0
−
8
f
1
+
8
f
3
−
f
4
)
+
1
30
h
5
f
(
5
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{2})={\frac {1}{12}}(f_{0}-8f_{1}+8f_{3}-f_{4})+{\frac {1}{30}}h^{5}f^{(5)}(\xi )}
h
f
(
1
)
(
x
3
)
=
1
12
(
−
f
0
+
6
f
1
−
18
f
2
+
10
f
3
+
3
f
4
)
−
1
20
h
5
f
(
5
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{3})={\frac {1}{12}}(-f_{0}+6f_{1}-18f_{2}+10f_{3}+3f_{4})-{\frac {1}{20}}h^{5}f^{(5)}(\xi )}
h
f
(
1
)
(
x
4
)
=
1
12
(
3
f
0
−
16
f
1
+
36
f
2
−
48
f
3
+
25
f
4
)
+
1
5
h
5
f
(
5
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{4})={\frac {1}{12}}(3f_{0}-16f_{1}+36f_{2}-48f_{3}+25f_{4})+{\frac {1}{5}}h^{5}f^{(5)}(\xi )}
5-punts 2e afgeleide:
h
2
f
(
2
)
(
x
0
)
=
1
12
(
35
f
0
−
104
f
1
+
114
f
2
−
56
f
3
+
11
f
4
)
−
5
6
h
5
f
(
5
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{0})={\frac {1}{12}}(35f_{0}-104f_{1}+114f_{2}-56f_{3}+11f_{4})-{\frac {5}{6}}h^{5}f^{(5)}(\xi )}
h
2
f
(
2
)
(
x
1
)
=
1
12
(
11
f
0
−
20
f
1
+
6
f
2
+
4
f
3
−
f
4
)
+
1
12
h
5
f
(
5
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{1})={\frac {1}{12}}(11f_{0}-20f_{1}+6f_{2}+4f_{3}-f_{4})+{\frac {1}{12}}h^{5}f^{(5)}(\xi )}
h
2
f
(
2
)
(
x
2
)
=
1
12
(
−
f
0
+
16
f
1
−
30
f
2
+
16
f
3
−
f
4
)
+
1
90
h
6
f
(
6
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{2})={\frac {1}{12}}(-f_{0}+16f_{1}-30f_{2}+16f_{3}-f_{4})+{\frac {1}{90}}h^{6}f^{(6)}(\xi )}
h
2
f
(
2
)
(
x
3
)
=
1
12
(
−
f
0
+
4
f
1
+
6
f
2
−
20
f
3
+
11
f
4
)
−
1
12
h
5
f
(
5
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{3})={\frac {1}{12}}(-f_{0}+4f_{1}+6f_{2}-20f_{3}+11f_{4})-{\frac {1}{12}}h^{5}f^{(5)}(\xi )}
h
2
f
(
2
)
(
x
4
)
=
1
12
(
11
f
0
−
56
f
1
+
114
f
2
−
104
f
3
+
35
f
4
)
+
5
6
h
5
f
(
5
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{4})={\frac {1}{12}}(11f_{0}-56f_{1}+114f_{2}-104f_{3}+35f_{4})+{\frac {5}{6}}h^{5}f^{(5)}(\xi )}
5-punts 3e afgeleide:
h
3
f
(
3
)
(
x
0
)
=
1
2
(
−
5
f
0
+
18
f
1
−
24
f
2
+
14
f
3
−
3
f
4
)
+
7
4
h
5
f
(
5
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{0})={\frac {1}{2}}(-5f_{0}+18f_{1}-24f_{2}+14f_{3}-3f_{4})+{\frac {7}{4}}h^{5}f^{(5)}(\xi )}
h
3
f
(
3
)
(
x
1
)
=
1
2
(
−
3
f
0
+
10
f
1
−
12
f
2
+
6
f
3
−
f
4
)
+
1
4
h
5
f
(
5
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{1})={\frac {1}{2}}(-3f_{0}+10f_{1}-12f_{2}+6f_{3}-f_{4})+{\frac {1}{4}}h^{5}f^{(5)}(\xi )}
h
3
f
(
3
)
(
x
2
)
=
1
2
(
−
f
0
+
2
f
1
−
2
f
3
+
f
4
)
−
1
4
h
5
f
(
5
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{2})={\frac {1}{2}}(-f_{0}+2f_{1}-2f_{3}+f_{4})-{\frac {1}{4}}h^{5}f^{(5)}(\xi )}
h
3
f
(
3
)
(
x
3
)
=
1
2
(
f
0
−
6
f
1
+
12
f
2
−
10
f
3
+
3
f
4
)
+
1
4
h
5
f
(
5
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{3})={\frac {1}{2}}(f_{0}-6f_{1}+12f_{2}-10f_{3}+3f_{4})+{\frac {1}{4}}h^{5}f^{(5)}(\xi )}
h
3
f
(
3
)
(
x
4
)
=
1
2
(
3
f
0
−
14
f
1
+
24
f
2
−
18
f
3
+
5
f
4
)
+
7
4
h
5
f
(
5
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{4})={\frac {1}{2}}(3f_{0}-14f_{1}+24f_{2}-18f_{3}+5f_{4})+{\frac {7}{4}}h^{5}f^{(5)}(\xi )}
5-punts 4e afgeleide:
h
4
f
(
4
)
(
x
0
)
=
f
0
−
4
f
1
+
6
f
2
−
4
f
3
+
f
4
−
2
h
5
f
(
5
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{0})=f_{0}-4f_{1}+6f_{2}-4f_{3}+f_{4}-2h^{5}f^{(5)}(\xi )}
h
4
f
(
4
)
(
x
1
)
=
f
0
−
4
f
1
+
6
f
2
−
4
f
3
+
f
4
−
h
5
f
(
5
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{1})=f_{0}-4f_{1}+6f_{2}-4f_{3}+f_{4}-h^{5}f^{(5)}(\xi )}
h
4
f
(
4
)
(
x
2
)
=
f
0
−
4
f
1
+
6
f
2
−
4
f
3
+
f
4
−
1
6
h
6
f
(
6
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{2})=f_{0}-4f_{1}+6f_{2}-4f_{3}+f_{4}-{\frac {1}{6}}h^{6}f^{(6)}(\xi )}
h
4
f
(
4
)
(
x
3
)
=
f
0
−
4
f
1
+
6
f
2
−
4
f
3
+
f
4
+
h
5
f
(
5
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{3})=f_{0}-4f_{1}+6f_{2}-4f_{3}+f_{4}+h^{5}f^{(5)}(\xi )}
h
4
f
(
4
)
(
x
4
)
=
f
0
−
4
f
1
+
6
f
2
−
4
f
3
+
f
4
+
2
h
5
f
(
5
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{4})=f_{0}-4f_{1}+6f_{2}-4f_{3}+f_{4}+2h^{5}f^{(5)}(\xi )}
6-punts 1e afgeleide:
h
f
(
1
)
(
x
0
)
=
1
60
(
−
137
f
0
+
300
f
1
−
300
f
2
+
200
f
3
−
75
f
4
+
12
f
5
)
−
1
6
h
6
f
(
6
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{0})={\frac {1}{60}}(-137f_{0}+300f_{1}-300f_{2}+200f_{3}-75f_{4}+12f_{5})-{\frac {1}{6}}h^{6}f^{(6)}(\xi )}
h
f
(
1
)
(
x
1
)
=
1
60
(
−
12
f
0
−
65
f
1
+
120
f
2
−
60
f
3
+
20
f
4
−
3
f
5
)
+
1
30
h
6
f
(
6
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{1})={\frac {1}{60}}(-12f_{0}-65f_{1}+120f_{2}-60f_{3}+20f_{4}-3f_{5})+{\frac {1}{30}}h^{6}f^{(6)}(\xi )}
h
f
(
1
)
(
x
2
)
=
1
60
(
3
f
0
−
30
f
1
−
20
f
2
+
60
f
3
−
15
f
4
+
2
f
5
)
−
1
60
h
6
f
(
6
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{2})={\frac {1}{60}}(3f_{0}-30f_{1}-20f_{2}+60f_{3}-15f_{4}+2f_{5})-{\frac {1}{60}}h^{6}f^{(6)}(\xi )}
h
f
(
1
)
(
x
3
)
=
1
60
(
−
2
f
0
+
15
f
1
−
60
f
2
+
20
f
3
+
30
f
4
−
3
f
5
)
+
1
60
h
6
f
(
6
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{3})={\frac {1}{60}}(-2f_{0}+15f_{1}-60f_{2}+20f_{3}+30f_{4}-3f_{5})+{\frac {1}{60}}h^{6}f^{(6)}(\xi )}
h
f
(
1
)
(
x
4
)
=
1
60
(
3
f
0
−
20
f
1
+
60
f
2
−
120
f
3
+
65
f
4
+
12
f
5
)
−
1
30
h
6
f
(
6
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{4})={\frac {1}{60}}(3f_{0}-20f_{1}+60f_{2}-120f_{3}+65f_{4}+12f_{5})-{\frac {1}{30}}h^{6}f^{(6)}(\xi )}
h
f
(
1
)
(
x
5
)
=
1
60
(
−
12
f
0
+
75
f
1
−
200
f
2
+
300
f
3
−
300
f
4
+
137
f
5
)
+
1
6
h
6
f
(
6
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{5})={\frac {1}{60}}(-12f_{0}+75f_{1}-200f_{2}+300f_{3}-300f_{4}+137f_{5})+{\frac {1}{6}}h^{6}f^{(6)}(\xi )}
6-punts 2e afgeleide:
h
2
f
(
2
)
(
x
0
)
=
1
12
(
45
f
0
−
154
f
1
+
214
f
2
−
156
f
3
+
61
f
4
−
10
f
5
)
+
137
180
h
6
f
(
6
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{0})={\frac {1}{12}}(45f_{0}-154f_{1}+214f_{2}-156f_{3}+61f_{4}-10f_{5})+{\frac {137}{180}}h^{6}f^{(6)}(\xi )}
h
2
f
(
2
)
(
x
1
)
=
1
12
(
10
f
0
−
15
f
1
−
4
f
2
+
14
f
3
−
6
f
4
+
f
5
)
−
13
180
h
6
f
(
6
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{1})={\frac {1}{12}}(10f_{0}-15f_{1}-4f_{2}+14f_{3}-6f_{4}+f_{5})-{\frac {13}{180}}h^{6}f^{(6)}(\xi )}
h
2
f
(
2
)
(
x
2
)
=
1
12
(
−
f
0
+
16
f
1
−
30
f
2
+
16
f
3
−
f
4
)
+
1
90
h
6
f
(
6
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{2})={\frac {1}{12}}(-f_{0}+16f_{1}-30f_{2}+16f_{3}-f_{4})+{\frac {1}{90}}h^{6}f^{(6)}(\xi )}
h
2
f
(
2
)
(
x
3
)
=
1
12
(
−
f
1
+
16
f
2
−
30
f
3
+
16
f
4
−
f
5
)
+
1
90
h
6
f
(
6
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{3})={\frac {1}{12}}(-f_{1}+16f_{2}-30f_{3}+16f_{4}-f_{5})+{\frac {1}{90}}h^{6}f^{(6)}(\xi )}
h
2
f
(
2
)
(
x
4
)
=
1
12
(
f
0
−
6
f
1
+
14
f
2
−
4
f
3
−
15
f
4
+
10
f
5
)
−
13
180
h
6
f
(
6
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{4})={\frac {1}{12}}(f_{0}-6f_{1}+14f_{2}-4f_{3}-15f_{4}+10f_{5})-{\frac {13}{180}}h^{6}f^{(6)}(\xi )}
h
2
f
(
2
)
(
x
5
)
=
1
12
(
−
10
f
0
+
61
f
1
−
156
f
2
+
214
f
3
−
154
f
4
+
45
f
5
)
+
137
180
h
6
f
(
6
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{5})={\frac {1}{12}}(-10f_{0}+61f_{1}-156f_{2}+214f_{3}-154f_{4}+45f_{5})+{\frac {137}{180}}h^{6}f^{(6)}(\xi )}
6-punts 3e afgeleide:
h
3
f
(
3
)
(
x
0
)
=
1
4
(
−
17
f
0
+
71
f
1
−
118
f
2
+
98
f
3
−
41
f
4
+
7
f
5
)
−
15
8
h
6
f
(
6
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{0})={\frac {1}{4}}(-17f_{0}+71f_{1}-118f_{2}+98f_{3}-41f_{4}+7f_{5})-{\frac {15}{8}}h^{6}f^{(6)}(\xi )}
h
3
f
(
3
)
(
x
1
)
=
1
4
(
−
7
f
0
+
25
f
1
−
34
f
2
+
22
f
3
−
7
f
4
+
f
5
)
−
1
8
h
6
f
(
6
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{1})={\frac {1}{4}}(-7f_{0}+25f_{1}-34f_{2}+22f_{3}-7f_{4}+f_{5})-{\frac {1}{8}}h^{6}f^{(6)}(\xi )}
h
3
f
(
3
)
(
x
2
)
=
1
4
(
−
f
0
−
f
1
+
10
f
2
−
14
f
3
+
7
f
4
−
f
5
)
+
1
8
h
6
f
(
6
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{2})={\frac {1}{4}}(-f_{0}-f_{1}+10f_{2}-14f_{3}+7f_{4}-f_{5})+{\frac {1}{8}}h^{6}f^{(6)}(\xi )}
h
3
f
(
3
)
(
x
3
)
=
1
4
(
f
0
−
7
f
1
+
14
f
2
−
10
f
3
+
f
4
+
f
5
)
−
1
8
h
6
f
(
6
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{3})={\frac {1}{4}}(f_{0}-7f_{1}+14f_{2}-10f_{3}+f_{4}+f_{5})-{\frac {1}{8}}h^{6}f^{(6)}(\xi )}
h
3
f
(
3
)
(
x
4
)
=
1
4
(
−
f
0
+
7
f
1
−
22
f
2
+
34
f
3
−
25
f
4
+
7
f
5
)
+
1
8
h
6
f
(
6
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{4})={\frac {1}{4}}(-f_{0}+7f_{1}-22f_{2}+34f_{3}-25f_{4}+7f_{5})+{\frac {1}{8}}h^{6}f^{(6)}(\xi )}
h
3
f
(
3
)
(
x
5
)
=
1
4
(
−
7
f
0
+
41
f
1
−
98
f
2
+
118
f
3
−
71
f
4
+
17
f
5
)
+
15
8
h
6
f
(
6
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{5})={\frac {1}{4}}(-7f_{0}+41f_{1}-98f_{2}+118f_{3}-71f_{4}+17f_{5})+{\frac {15}{8}}h^{6}f^{(6)}(\xi )}
6-punts 4e afgeleide:
h
4
f
(
4
)
(
x
0
)
=
3
f
0
−
14
f
1
+
26
f
2
−
24
f
3
+
11
f
4
−
2
f
5
+
17
6
h
6
f
(
6
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{0})=3f_{0}-14f_{1}+26f_{2}-24f_{3}+11f_{4}-2f_{5}+{\frac {17}{6}}h^{6}f^{(6)}(\xi )}
h
4
f
(
4
)
(
x
1
)
=
2
f
0
−
9
f
1
+
16
f
2
−
14
f
3
+
6
f
4
−
f
5
+
5
6
h
6
f
(
6
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{1})=2f_{0}-9f_{1}+16f_{2}-14f_{3}+6f_{4}-f_{5}+{\frac {5}{6}}h^{6}f^{(6)}(\xi )}
h
4
f
(
4
)
(
x
2
)
=
f
0
−
4
f
1
+
6
f
2
−
4
f
3
+
f
4
−
1
6
h
6
f
(
6
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{2})=f_{0}-4f_{1}+6f_{2}-4f_{3}+f_{4}-{\frac {1}{6}}h^{6}f^{(6)}(\xi )}
h
4
f
(
4
)
(
x
3
)
=
f
1
−
4
f
2
+
6
f
3
−
4
f
4
+
f
5
−
1
6
h
6
f
(
6
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{3})=f_{1}-4f_{2}+6f_{3}-4f_{4}+f_{5}-{\frac {1}{6}}h^{6}f^{(6)}(\xi )}
h
4
f
(
4
)
(
x
4
)
=
−
f
0
+
6
f
1
−
14
f
2
+
16
f
3
−
9
f
4
+
2
f
5
+
5
6
h
6
f
(
6
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{4})=-f_{0}+6f_{1}-14f_{2}+16f_{3}-9f_{4}+2f_{5}+{\frac {5}{6}}h^{6}f^{(6)}(\xi )}
h
4
f
(
4
)
(
x
5
)
=
−
2
f
0
+
11
f
1
−
24
f
2
+
26
f
3
−
14
f
4
+
3
f
5
+
17
6
h
6
f
(
6
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{5})=-2f_{0}+11f_{1}-24f_{2}+26f_{3}-14f_{4}+3f_{5}+{\frac {17}{6}}h^{6}f^{(6)}(\xi )}
6-punts 5e afgeleide:
h
5
f
(
5
)
(
x
0
)
=
−
f
0
+
5
f
1
−
10
f
2
+
10
f
3
−
5
f
4
+
f
5
−
5
2
h
6
f
(
6
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{0})=-f_{0}+5f_{1}-10f_{2}+10f_{3}-5f_{4}+f_{5}-{\frac {5}{2}}h^{6}f^{(6)}(\xi )}
h
5
f
(
5
)
(
x
1
)
=
−
f
0
+
5
f
1
−
10
f
2
+
10
f
3
−
5
f
4
+
f
5
−
3
2
h
6
f
(
6
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{1})=-f_{0}+5f_{1}-10f_{2}+10f_{3}-5f_{4}+f_{5}-{\frac {3}{2}}h^{6}f^{(6)}(\xi )}
h
5
f
(
5
)
(
x
2
)
=
−
f
0
+
5
f
1
−
10
f
2
+
10
f
3
−
5
f
4
+
f
5
−
1
2
h
6
f
(
6
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{2})=-f_{0}+5f_{1}-10f_{2}+10f_{3}-5f_{4}+f_{5}-{\frac {1}{2}}h^{6}f^{(6)}(\xi )}
h
5
f
(
5
)
(
x
3
)
=
−
f
0
+
5
f
1
−
10
f
2
+
10
f
3
−
5
f
4
+
f
5
+
1
2
h
6
f
(
6
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{3})=-f_{0}+5f_{1}-10f_{2}+10f_{3}-5f_{4}+f_{5}+{\frac {1}{2}}h^{6}f^{(6)}(\xi )}
h
5
f
(
5
)
(
x
4
)
=
−
f
0
+
5
f
1
−
10
f
2
+
10
f
3
−
5
f
4
+
f
5
+
3
2
h
6
f
(
6
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{4})=-f_{0}+5f_{1}-10f_{2}+10f_{3}-5f_{4}+f_{5}+{\frac {3}{2}}h^{6}f^{(6)}(\xi )}
h
5
f
(
5
)
(
x
5
)
=
−
f
0
+
5
f
1
−
10
f
2
+
10
f
3
−
5
f
4
+
f
5
+
5
2
h
6
f
(
6
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{5})=-f_{0}+5f_{1}-10f_{2}+10f_{3}-5f_{4}+f_{5}+{\frac {5}{2}}h^{6}f^{(6)}(\xi )}
7-punts 1e afgeleide:
h
f
(
1
)
(
x
0
)
=
1
60
(
−
147
f
0
+
360
f
1
−
450
f
2
+
400
f
3
−
225
f
4
+
72
f
5
−
10
f
6
)
+
1
7
h
7
f
(
7
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{0})={\frac {1}{60}}(-147f_{0}+360f_{1}-450f_{2}+400f_{3}-225f_{4}+72f_{5}-10f_{6})+{\frac {1}{7}}h^{7}f^{(7)}(\xi )}
h
f
(
1
)
(
x
1
)
=
1
60
(
−
10
f
0
−
77
f
1
+
150
f
2
−
100
f
3
+
50
f
4
−
15
f
5
+
2
f
6
)
−
1
42
h
7
f
(
7
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{1})={\frac {1}{60}}(-10f_{0}-77f_{1}+150f_{2}-100f_{3}+50f_{4}-15f_{5}+2f_{6})-{\frac {1}{42}}h^{7}f^{(7)}(\xi )}
h
f
(
1
)
(
x
2
)
=
1
60
(
2
f
0
−
24
f
1
−
35
f
2
+
80
f
3
−
30
f
4
+
8
f
5
−
f
6
)
+
1
105
h
7
f
(
7
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{2})={\frac {1}{60}}(2f_{0}-24f_{1}-35f_{2}+80f_{3}-30f_{4}+8f_{5}-f_{6})+{\frac {1}{105}}h^{7}f^{(7)}(\xi )}
h
f
(
1
)
(
x
3
)
=
1
60
(
−
f
0
+
9
f
1
−
45
f
2
+
45
f
4
−
9
f
5
+
f
6
)
−
1
140
h
7
f
(
7
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{3})={\frac {1}{60}}(-f_{0}+9f_{1}-45f_{2}+45f_{4}-9f_{5}+f_{6})-{\frac {1}{140}}h^{7}f^{(7)}(\xi )}
h
f
(
1
)
(
x
4
)
=
1
60
(
f
0
−
8
f
1
+
30
f
2
−
80
f
3
+
35
f
4
+
24
f
5
−
2
f
6
)
+
1
105
h
7
f
(
7
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{4})={\frac {1}{60}}(f_{0}-8f_{1}+30f_{2}-80f_{3}+35f_{4}+24f_{5}-2f_{6})+{\frac {1}{105}}h^{7}f^{(7)}(\xi )}
h
f
(
1
)
(
x
5
)
=
1
60
(
−
2
f
0
+
15
f
1
−
50
f
2
+
100
f
3
−
150
f
4
+
77
f
5
+
10
f
6
)
−
1
42
h
7
f
(
7
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{5})={\frac {1}{60}}(-2f_{0}+15f_{1}-50f_{2}+100f_{3}-150f_{4}+77f_{5}+10f_{6})-{\frac {1}{42}}h^{7}f^{(7)}(\xi )}
h
f
(
1
)
(
x
6
)
=
1
60
(
10
f
0
−
72
f
1
+
225
f
2
−
400
f
3
+
450
f
4
−
360
f
5
+
147
f
6
)
+
1
7
h
7
f
(
7
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{6})={\frac {1}{60}}(10f_{0}-72f_{1}+225f_{2}-400f_{3}+450f_{4}-360f_{5}+147f_{6})+{\frac {1}{7}}h^{7}f^{(7)}(\xi )}
7-punts 2e afgeleide:
h
2
f
(
2
)
(
x
0
)
=
1
180
(
812
f
0
−
3132
f
1
+
5265
f
2
−
5080
f
3
+
2970
f
4
−
972
f
5
+
137
f
6
)
−
7
10
h
7
f
(
7
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{0})={\frac {1}{180}}(812f_{0}-3132f_{1}+5265f_{2}-5080f_{3}+2970f_{4}-972f_{5}+137f_{6})-{\frac {7}{10}}h^{7}f^{(7)}(\xi )}
h
2
f
(
2
)
(
x
1
)
=
1
180
(
137
f
0
−
147
f
1
−
255
f
2
+
470
f
3
−
285
f
4
+
93
f
5
−
13
f
6
)
+
11
180
h
7
f
(
7
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{1})={\frac {1}{180}}(137f_{0}-147f_{1}-255f_{2}+470f_{3}-285f_{4}+93f_{5}-13f_{6})+{\frac {11}{180}}h^{7}f^{(7)}(\xi )}
h
2
f
(
2
)
(
x
2
)
=
1
180
(
−
13
f
0
+
228
f
1
−
420
f
2
+
200
f
3
+
15
f
4
−
12
f
5
+
2
f
6
)
−
1
90
h
7
f
(
7
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{2})={\frac {1}{180}}(-13f_{0}+228f_{1}-420f_{2}+200f_{3}+15f_{4}-12f_{5}+2f_{6})-{\frac {1}{90}}h^{7}f^{(7)}(\xi )}
h
2
f
(
2
)
(
x
3
)
=
1
180
(
2
f
0
−
27
f
1
+
270
f
2
−
490
f
3
+
270
f
4
−
27
f
5
+
2
f
6
)
−
1
560
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{3})={\frac {1}{180}}(2f_{0}-27f_{1}+270f_{2}-490f_{3}+270f_{4}-27f_{5}+2f_{6})-{\frac {1}{560}}h^{8}f^{(8)}(\xi )}
h
2
f
(
2
)
(
x
4
)
=
1
180
(
2
f
0
−
12
f
1
+
15
f
2
+
200
f
3
−
420
f
4
+
228
f
5
−
13
f
6
)
+
1
90
h
7
f
(
7
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{4})={\frac {1}{180}}(2f_{0}-12f_{1}+15f_{2}+200f_{3}-420f_{4}+228f_{5}-13f_{6})+{\frac {1}{90}}h^{7}f^{(7)}(\xi )}
h
2
f
(
2
)
(
x
5
)
=
1
180
(
−
13
f
0
+
93
f
1
−
285
f
2
+
470
f
3
−
255
f
4
−
147
f
5
+
137
f
6
)
−
11
180
h
7
f
(
7
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{5})={\frac {1}{180}}(-13f_{0}+93f_{1}-285f_{2}+470f_{3}-255f_{4}-147f_{5}+137f_{6})-{\frac {11}{180}}h^{7}f^{(7)}(\xi )}
h
2
f
(
2
)
(
x
6
)
=
1
180
(
137
f
0
−
972
f
1
+
2970
f
2
−
5080
f
3
+
5265
f
4
−
3132
f
5
+
812
f
6
)
+
7
10
h
7
f
(
7
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{6})={\frac {1}{180}}(137f_{0}-972f_{1}+2970f_{2}-5080f_{3}+5265f_{4}-3132f_{5}+812f_{6})+{\frac {7}{10}}h^{7}f^{(7)}(\xi )}
7-punts 3e afgeleide:
h
3
f
(
3
)
(
x
0
)
=
1
8
(
−
49
f
0
+
232
f
1
−
461
f
2
+
496
f
3
−
307
f
4
+
104
f
5
−
15
f
6
)
+
29
15
h
7
f
(
7
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{0})={\frac {1}{8}}(-49f_{0}+232f_{1}-461f_{2}+496f_{3}-307f_{4}+104f_{5}-15f_{6})+{\frac {29}{15}}h^{7}f^{(7)}(\xi )}
h
3
f
(
3
)
(
x
1
)
=
1
8
(
−
15
f
0
+
56
f
1
−
83
f
2
+
64
f
3
−
29
f
4
+
8
f
5
−
f
6
)
+
7
120
h
7
f
(
7
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{1})={\frac {1}{8}}(-15f_{0}+56f_{1}-83f_{2}+64f_{3}-29f_{4}+8f_{5}-f_{6})+{\frac {7}{120}}h^{7}f^{(7)}(\xi )}
h
3
f
(
3
)
(
x
2
)
=
1
8
(
−
f
0
−
8
f
1
+
35
f
2
−
48
f
3
+
29
f
4
−
8
f
5
+
f
6
)
−
1
15
h
7
f
(
7
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{2})={\frac {1}{8}}(-f_{0}-8f_{1}+35f_{2}-48f_{3}+29f_{4}-8f_{5}+f_{6})-{\frac {1}{15}}h^{7}f^{(7)}(\xi )}
h
3
f
(
3
)
(
x
3
)
=
1
8
(
f
0
−
8
f
1
+
13
f
2
−
13
f
4
+
8
f
5
−
f
6
)
+
7
120
h
7
f
(
7
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{3})={\frac {1}{8}}(f_{0}-8f_{1}+13f_{2}-13f_{4}+8f_{5}-f_{6})+{\frac {7}{120}}h^{7}f^{(7)}(\xi )}
h
3
f
(
3
)
(
x
4
)
=
1
8
(
−
f
0
+
8
f
1
−
29
f
2
+
48
f
3
−
35
f
4
+
8
f
5
+
f
6
)
−
1
15
h
7
f
(
7
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{4})={\frac {1}{8}}(-f_{0}+8f_{1}-29f_{2}+48f_{3}-35f_{4}+8f_{5}+f_{6})-{\frac {1}{15}}h^{7}f^{(7)}(\xi )}
h
3
f
(
3
)
(
x
5
)
=
1
8
(
f
0
−
8
f
1
+
29
f
2
−
64
f
3
+
83
f
4
−
56
f
5
+
15
f
6
)
+
7
120
h
7
f
(
7
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{5})={\frac {1}{8}}(f_{0}-8f_{1}+29f_{2}-64f_{3}+83f_{4}-56f_{5}+15f_{6})+{\frac {7}{120}}h^{7}f^{(7)}(\xi )}
h
3
f
(
3
)
(
x
6
)
=
1
8
(
15
f
0
−
104
f
1
+
307
f
2
−
496
f
3
+
461
f
4
−
232
f
5
+
49
f
6
)
+
29
15
h
7
f
(
7
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{6})={\frac {1}{8}}(15f_{0}-104f_{1}+307f_{2}-496f_{3}+461f_{4}-232f_{5}+49f_{6})+{\frac {29}{15}}h^{7}f^{(7)}(\xi )}
7-punts 4e afgeleide:
h
4
f
(
4
)
(
x
0
)
=
1
6
(
35
f
0
−
186
f
1
+
411
f
2
−
484
f
3
+
321
f
4
−
114
f
5
+
17
f
6
)
−
7
2
h
7
f
(
7
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{0})={\frac {1}{6}}(35f_{0}-186f_{1}+411f_{2}-484f_{3}+321f_{4}-114f_{5}+17f_{6})-{\frac {7}{2}}h^{7}f^{(7)}(\xi )}
h
4
f
(
4
)
(
x
1
)
=
1
6
(
17
f
0
−
84
f
1
+
171
f
2
−
184
f
3
+
111
f
4
−
36
f
5
+
5
f
6
)
−
2
3
h
7
f
(
7
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{1})={\frac {1}{6}}(17f_{0}-84f_{1}+171f_{2}-184f_{3}+111f_{4}-36f_{5}+5f_{6})-{\frac {2}{3}}h^{7}f^{(7)}(\xi )}
h
4
f
(
4
)
(
x
2
)
=
1
6
(
5
f
0
−
18
f
1
+
21
f
2
−
4
f
3
−
9
f
4
+
6
f
5
−
f
6
)
+
1
6
h
7
f
(
7
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{2})={\frac {1}{6}}(5f_{0}-18f_{1}+21f_{2}-4f_{3}-9f_{4}+6f_{5}-f_{6})+{\frac {1}{6}}h^{7}f^{(7)}(\xi )}
h
4
f
(
4
)
(
x
3
)
=
1
6
(
−
f
0
+
12
f
1
−
39
f
2
+
56
f
3
−
39
f
4
+
12
f
5
−
f
6
)
+
7
240
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{3})={\frac {1}{6}}(-f_{0}+12f_{1}-39f_{2}+56f_{3}-39f_{4}+12f_{5}-f_{6})+{\frac {7}{240}}h^{8}f^{(8)}(\xi )}
h
4
f
(
4
)
(
x
4
)
=
1
6
(
−
f
0
+
6
f
1
−
9
f
2
−
4
f
3
+
21
f
4
−
18
f
5
+
5
f
6
)
−
1
6
h
7
f
(
7
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{4})={\frac {1}{6}}(-f_{0}+6f_{1}-9f_{2}-4f_{3}+21f_{4}-18f_{5}+5f_{6})-{\frac {1}{6}}h^{7}f^{(7)}(\xi )}
h
4
f
(
4
)
(
x
5
)
=
1
6
(
5
f
0
−
36
f
1
+
111
f
2
−
184
f
3
+
171
f
4
−
84
f
5
+
17
f
6
)
+
2
3
h
7
f
(
7
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{5})={\frac {1}{6}}(5f_{0}-36f_{1}+111f_{2}-184f_{3}+171f_{4}-84f_{5}+17f_{6})+{\frac {2}{3}}h^{7}f^{(7)}(\xi )}
h
4
f
(
4
)
(
x
6
)
=
1
6
(
17
f
0
−
114
f
1
+
321
f
2
−
484
f
3
+
411
f
4
−
186
f
5
+
35
f
6
)
+
7
2
h
7
f
(
7
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{6})={\frac {1}{6}}(17f_{0}-114f_{1}+321f_{2}-484f_{3}+411f_{4}-186f_{5}+35f_{6})+{\frac {7}{2}}h^{7}f^{(7)}(\xi )}
7-punts 5e afgeleide:
h
5
f
(
5
)
(
x
0
)
=
1
2
(
−
7
f
0
+
40
f
1
−
95
f
2
+
120
f
3
−
85
f
4
+
32
f
5
−
5
f
6
)
+
25
6
h
7
f
(
7
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{0})={\frac {1}{2}}(-7f_{0}+40f_{1}-95f_{2}+120f_{3}-85f_{4}+32f_{5}-5f_{6})+{\frac {25}{6}}h^{7}f^{(7)}(\xi )}
h
5
f
(
5
)
(
x
1
)
=
1
2
(
−
5
f
0
+
28
f
1
−
65
f
2
+
80
f
3
−
55
f
4
+
20
f
5
−
3
f
6
)
+
5
3
h
7
f
(
7
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{1})={\frac {1}{2}}(-5f_{0}+28f_{1}-65f_{2}+80f_{3}-55f_{4}+20f_{5}-3f_{6})+{\frac {5}{3}}h^{7}f^{(7)}(\xi )}
h
5
f
(
5
)
(
x
2
)
=
1
2
(
−
3
f
0
+
16
f
1
−
35
f
2
+
40
f
3
−
25
f
4
+
8
f
5
−
f
6
)
+
1
6
h
7
f
(
7
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{2})={\frac {1}{2}}(-3f_{0}+16f_{1}-35f_{2}+40f_{3}-25f_{4}+8f_{5}-f_{6})+{\frac {1}{6}}h^{7}f^{(7)}(\xi )}
h
5
f
(
5
)
(
x
3
)
=
1
2
(
−
f
0
+
4
f
1
−
5
f
2
+
5
f
4
−
4
f
5
+
f
6
)
−
1
3
h
7
f
(
7
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{3})={\frac {1}{2}}(-f_{0}+4f_{1}-5f_{2}+5f_{4}-4f_{5}+f_{6})-{\frac {1}{3}}h^{7}f^{(7)}(\xi )}
h
5
f
(
5
)
(
x
4
)
=
1
2
(
f
0
−
8
f
1
+
25
f
2
−
40
f
3
+
35
f
4
−
16
f
5
+
3
f
6
)
+
1
6
h
7
f
(
7
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{4})={\frac {1}{2}}(f_{0}-8f_{1}+25f_{2}-40f_{3}+35f_{4}-16f_{5}+3f_{6})+{\frac {1}{6}}h^{7}f^{(7)}(\xi )}
h
5
f
(
5
)
(
x
5
)
=
1
2
(
3
f
0
−
20
f
1
+
55
f
2
−
80
f
3
+
65
f
4
−
28
f
5
+
5
f
6
)
+
5
3
h
7
f
(
7
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{5})={\frac {1}{2}}(3f_{0}-20f_{1}+55f_{2}-80f_{3}+65f_{4}-28f_{5}+5f_{6})+{\frac {5}{3}}h^{7}f^{(7)}(\xi )}
h
5
f
(
5
)
(
x
6
)
=
1
2
(
5
f
0
−
32
f
1
+
85
f
2
−
120
f
3
+
95
f
4
−
40
f
5
+
7
f
6
)
+
25
6
h
7
f
(
7
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{6})={\frac {1}{2}}(5f_{0}-32f_{1}+85f_{2}-120f_{3}+95f_{4}-40f_{5}+7f_{6})+{\frac {25}{6}}h^{7}f^{(7)}(\xi )}
7-punts 6e afgeleide:
h
6
f
(
6
)
(
x
0
)
=
f
0
−
6
f
1
+
15
f
2
−
20
f
3
+
15
f
4
−
6
f
5
+
f
6
−
3
h
7
f
(
7
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{0})=f_{0}-6f_{1}+15f_{2}-20f_{3}+15f_{4}-6f_{5}+f_{6}-3h^{7}f^{(7)}(\xi )}
h
6
f
(
6
)
(
x
1
)
=
f
0
−
6
f
1
+
15
f
2
−
20
f
3
+
15
f
4
−
6
f
5
+
f
6
−
2
h
7
f
(
7
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{1})=f_{0}-6f_{1}+15f_{2}-20f_{3}+15f_{4}-6f_{5}+f_{6}-2h^{7}f^{(7)}(\xi )}
h
6
f
(
6
)
(
x
2
)
=
f
0
−
6
f
1
+
15
f
2
−
20
f
3
+
15
f
4
−
6
f
5
+
f
6
−
h
7
f
(
7
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{2})=f_{0}-6f_{1}+15f_{2}-20f_{3}+15f_{4}-6f_{5}+f_{6}-h^{7}f^{(7)}(\xi )}
h
6
f
(
6
)
(
x
3
)
=
f
0
−
6
f
1
+
15
f
2
−
20
f
3
+
15
f
4
−
6
f
5
+
f
6
−
1
4
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{3})=f_{0}-6f_{1}+15f_{2}-20f_{3}+15f_{4}-6f_{5}+f_{6}-{\frac {1}{4}}h^{8}f^{(8)}(\xi )}
h
6
f
(
6
)
(
x
4
)
=
f
0
−
6
f
1
+
15
f
2
−
20
f
3
+
15
f
4
−
6
f
5
+
f
6
+
h
7
f
(
7
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{4})=f_{0}-6f_{1}+15f_{2}-20f_{3}+15f_{4}-6f_{5}+f_{6}+h^{7}f^{(7)}(\xi )}
h
6
f
(
6
)
(
x
5
)
=
f
0
−
6
f
1
+
15
f
2
−
20
f
3
+
15
f
4
−
6
f
5
+
f
6
+
2
h
7
f
(
7
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{5})=f_{0}-6f_{1}+15f_{2}-20f_{3}+15f_{4}-6f_{5}+f_{6}+2h^{7}f^{(7)}(\xi )}
h
6
f
(
6
)
(
x
6
)
=
f
0
−
6
f
1
+
15
f
2
−
20
f
3
+
15
f
4
−
6
f
5
+
f
6
+
3
h
7
f
(
7
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{6})=f_{0}-6f_{1}+15f_{2}-20f_{3}+15f_{4}-6f_{5}+f_{6}+3h^{7}f^{(7)}(\xi )}
8-punts 1e afgeleide:
h
f
(
1
)
(
x
0
)
=
1
420
(
−
1089
f
0
+
2940
f
1
−
4410
f
2
+
4900
f
3
−
3675
f
4
+
1764
f
5
−
490
f
6
+
60
f
7
)
−
1
8
h
8
f
(
8
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{0})={\frac {1}{420}}(-1089f_{0}+2940f_{1}-4410f_{2}+4900f_{3}-3675f_{4}+1764f_{5}-490f_{6}+60f_{7})-{\frac {1}{8}}h^{8}f^{(8)}(\xi )}
h
f
(
1
)
(
x
1
)
=
1
420
(
−
60
f
0
−
609
f
1
+
1260
f
2
−
1050
f
3
+
700
f
4
−
315
f
5
+
84
f
6
−
10
f
7
)
+
1
56
h
8
f
(
8
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{1})={\frac {1}{420}}(-60f_{0}-609f_{1}+1260f_{2}-1050f_{3}+700f_{4}-315f_{5}+84f_{6}-10f_{7})+{\frac {1}{56}}h^{8}f^{(8)}(\xi )}
h
f
(
1
)
(
x
2
)
=
1
420
(
10
f
0
−
140
f
1
−
329
f
2
+
700
f
3
−
350
f
4
+
140
f
5
−
35
f
6
+
4
f
7
)
−
1
168
h
8
f
(
8
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{2})={\frac {1}{420}}(10f_{0}-140f_{1}-329f_{2}+700f_{3}-350f_{4}+140f_{5}-35f_{6}+4f_{7})-{\frac {1}{168}}h^{8}f^{(8)}(\xi )}
h
f
(
1
)
(
x
3
)
=
1
420
(
−
4
f
0
+
42
f
1
−
252
f
2
−
105
f
3
+
420
f
4
−
126
f
5
+
28
f
6
−
3
f
7
)
+
1
280
h
8
f
(
8
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{3})={\frac {1}{420}}(-4f_{0}+42f_{1}-252f_{2}-105f_{3}+420f_{4}-126f_{5}+28f_{6}-3f_{7})+{\frac {1}{280}}h^{8}f^{(8)}(\xi )}
h
f
(
1
)
(
x
4
)
=
1
420
(
3
f
0
−
28
f
1
+
126
f
2
−
420
f
3
+
105
f
4
+
252
f
5
−
42
f
6
+
4
f
7
)
−
1
280
h
8
f
(
8
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{4})={\frac {1}{420}}(3f_{0}-28f_{1}+126f_{2}-420f_{3}+105f_{4}+252f_{5}-42f_{6}+4f_{7})-{\frac {1}{280}}h^{8}f^{(8)}(\xi )}
h
f
(
1
)
(
x
5
)
=
1
420
(
−
4
f
0
+
35
f
1
−
140
f
2
+
350
f
3
−
700
f
4
+
329
f
5
+
140
f
6
−
10
f
7
)
+
1
168
h
8
f
(
8
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{5})={\frac {1}{420}}(-4f_{0}+35f_{1}-140f_{2}+350f_{3}-700f_{4}+329f_{5}+140f_{6}-10f_{7})+{\frac {1}{168}}h^{8}f^{(8)}(\xi )}
h
f
(
1
)
(
x
6
)
=
1
420
(
10
f
0
−
84
f
1
+
315
f
2
−
700
f
3
+
1050
f
4
−
1260
f
5
+
609
f
6
+
60
f
7
)
−
1
56
h
8
f
(
8
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{6})={\frac {1}{420}}(10f_{0}-84f_{1}+315f_{2}-700f_{3}+1050f_{4}-1260f_{5}+609f_{6}+60f_{7})-{\frac {1}{56}}h^{8}f^{(8)}(\xi )}
h
f
(
1
)
(
x
7
)
=
1
420
(
−
60
f
0
+
490
f
1
−
1764
f
2
+
3675
f
3
−
4900
f
4
+
4410
f
5
−
2940
f
6
+
1089
f
7
)
+
1
8
h
8
f
(
8
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{7})={\frac {1}{420}}(-60f_{0}+490f_{1}-1764f_{2}+3675f_{3}-4900f_{4}+4410f_{5}-2940f_{6}+1089f_{7})+{\frac {1}{8}}h^{8}f^{(8)}(\xi )}
8-punts 2e afgeleide:
h
2
f
(
2
)
(
x
0
)
=
1
180
(
938
f
0
−
4014
f
1
+
7911
f
2
−
9490
f
3
+
7380
f
4
−
3618
f
5
+
1019
f
6
−
126
f
7
)
+
363
560
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{0})={\frac {1}{180}}(938f_{0}-4014f_{1}+7911f_{2}-9490f_{3}+7380f_{4}-3618f_{5}+1019f_{6}-126f_{7})+{\frac {363}{560}}h^{8}f^{(8)}(\xi )}
h
2
f
(
2
)
(
x
1
)
=
1
180
(
126
f
0
−
70
f
1
−
486
f
2
+
855
f
3
−
670
f
4
+
324
f
5
−
90
f
6
+
11
f
7
)
−
29
560
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{1})={\frac {1}{180}}(126f_{0}-70f_{1}-486f_{2}+855f_{3}-670f_{4}+324f_{5}-90f_{6}+11f_{7})-{\frac {29}{560}}h^{8}f^{(8)}(\xi )}
h
2
f
(
2
)
(
x
2
)
=
1
180
(
−
11
f
0
+
214
f
1
−
378
f
2
+
130
f
3
+
85
f
4
−
54
f
5
+
16
f
6
−
2
f
7
)
+
47
5040
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{2})={\frac {1}{180}}(-11f_{0}+214f_{1}-378f_{2}+130f_{3}+85f_{4}-54f_{5}+16f_{6}-2f_{7})+{\frac {47}{5040}}h^{8}f^{(8)}(\xi )}
h
2
f
(
2
)
(
x
3
)
=
1
180
(
2
f
0
−
27
f
1
+
270
f
2
−
490
f
3
+
270
f
4
−
27
f
5
+
2
f
6
)
−
1
560
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{3})={\frac {1}{180}}(2f_{0}-27f_{1}+270f_{2}-490f_{3}+270f_{4}-27f_{5}+2f_{6})-{\frac {1}{560}}h^{8}f^{(8)}(\xi )}
h
2
f
(
2
)
(
x
4
)
=
1
180
(
2
f
1
−
27
f
2
+
270
f
3
−
490
f
4
+
270
f
5
−
27
f
6
+
2
f
7
)
−
1
560
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{4})={\frac {1}{180}}(2f_{1}-27f_{2}+270f_{3}-490f_{4}+270f_{5}-27f_{6}+2f_{7})-{\frac {1}{560}}h^{8}f^{(8)}(\xi )}
h
2
f
(
2
)
(
x
5
)
=
1
180
(
−
2
f
0
+
16
f
1
−
54
f
2
+
85
f
3
+
130
f
4
−
378
f
5
+
214
f
6
−
11
f
7
)
+
47
5040
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{5})={\frac {1}{180}}(-2f_{0}+16f_{1}-54f_{2}+85f_{3}+130f_{4}-378f_{5}+214f_{6}-11f_{7})+{\frac {47}{5040}}h^{8}f^{(8)}(\xi )}
h
2
f
(
2
)
(
x
6
)
=
1
180
(
11
f
0
−
90
f
1
+
324
f
2
−
670
f
3
+
855
f
4
−
486
f
5
−
70
f
6
+
126
f
7
)
−
29
560
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{6})={\frac {1}{180}}(11f_{0}-90f_{1}+324f_{2}-670f_{3}+855f_{4}-486f_{5}-70f_{6}+126f_{7})-{\frac {29}{560}}h^{8}f^{(8)}(\xi )}
h
2
f
(
2
)
(
x
7
)
=
1
180
(
−
126
f
0
+
1019
f
1
−
3618
f
2
+
7380
f
3
−
9490
f
4
+
7911
f
5
−
4014
f
6
+
938
f
7
)
+
363
560
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{7})={\frac {1}{180}}(-126f_{0}+1019f_{1}-3618f_{2}+7380f_{3}-9490f_{4}+7911f_{5}-4014f_{6}+938f_{7})+{\frac {363}{560}}h^{8}f^{(8)}(\xi )}
8-punts 3e afgeleide:
h
3
f
(
3
)
(
x
0
)
=
1
120
(
−
967
f
0
+
5104
f
1
−
11787
f
2
+
15560
f
3
−
12725
f
4
+
6432
f
5
−
1849
f
6
+
232
f
7
)
−
469
240
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{0})={\frac {1}{120}}(-967f_{0}+5104f_{1}-11787f_{2}+15560f_{3}-12725f_{4}+6432f_{5}-1849f_{6}+232f_{7})-{\frac {469}{240}}h^{8}f^{(8)}(\xi )}
h
3
f
(
3
)
(
x
1
)
=
1
120
(
−
232
f
0
+
889
f
1
−
1392
f
2
+
1205
f
3
−
680
f
4
+
267
f
5
−
64
f
6
+
7
f
7
)
−
1
48
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{1})={\frac {1}{120}}(-232f_{0}+889f_{1}-1392f_{2}+1205f_{3}-680f_{4}+267f_{5}-64f_{6}+7f_{7})-{\frac {1}{48}}h^{8}f^{(8)}(\xi )}
h
3
f
(
3
)
(
x
2
)
=
1
120
(
−
7
f
0
−
176
f
1
+
693
f
2
−
1000
f
3
+
715
f
4
−
288
f
5
+
71
f
6
−
8
f
7
)
+
3
80
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{2})={\frac {1}{120}}(-7f_{0}-176f_{1}+693f_{2}-1000f_{3}+715f_{4}-288f_{5}+71f_{6}-8f_{7})+{\frac {3}{80}}h^{8}f^{(8)}(\xi )}
h
3
f
(
3
)
(
x
3
)
=
1
120
(
8
f
0
−
71
f
1
+
48
f
2
+
245
f
3
−
440
f
4
+
267
f
5
−
64
f
6
+
7
f
7
)
−
7
240
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{3})={\frac {1}{120}}(8f_{0}-71f_{1}+48f_{2}+245f_{3}-440f_{4}+267f_{5}-64f_{6}+7f_{7})-{\frac {7}{240}}h^{8}f^{(8)}(\xi )}
h
3
f
(
3
)
(
x
4
)
=
1
120
(
−
7
f
0
+
64
f
1
−
267
f
2
+
440
f
3
−
245
f
4
−
48
f
5
+
71
f
6
−
8
f
7
)
+
7
240
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{4})={\frac {1}{120}}(-7f_{0}+64f_{1}-267f_{2}+440f_{3}-245f_{4}-48f_{5}+71f_{6}-8f_{7})+{\frac {7}{240}}h^{8}f^{(8)}(\xi )}
h
3
f
(
3
)
(
x
5
)
=
1
120
(
8
f
0
−
71
f
1
+
288
f
2
−
715
f
3
+
1000
f
4
−
693
f
5
+
176
f
6
+
7
f
7
)
−
3
80
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{5})={\frac {1}{120}}(8f_{0}-71f_{1}+288f_{2}-715f_{3}+1000f_{4}-693f_{5}+176f_{6}+7f_{7})-{\frac {3}{80}}h^{8}f^{(8)}(\xi )}
h
3
f
(
3
)
(
x
6
)
=
1
120
(
−
7
f
0
+
64
f
1
−
267
f
2
+
680
f
3
−
1205
f
4
+
1392
f
5
−
889
f
6
+
232
f
7
)
+
1
48
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{6})={\frac {1}{120}}(-7f_{0}+64f_{1}-267f_{2}+680f_{3}-1205f_{4}+1392f_{5}-889f_{6}+232f_{7})+{\frac {1}{48}}h^{8}f^{(8)}(\xi )}
h
3
f
(
3
)
(
x
7
)
=
1
120
(
−
232
f
0
+
1849
f
1
−
6432
f
2
+
12725
f
3
−
15560
f
4
+
11787
f
5
−
5104
f
6
+
967
f
7
)
+
469
240
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{7})={\frac {1}{120}}(-232f_{0}+1849f_{1}-6432f_{2}+12725f_{3}-15560f_{4}+11787f_{5}-5104f_{6}+967f_{7})+{\frac {469}{240}}h^{8}f^{(8)}(\xi )}
8-punts 4e afgeleide:
h
4
f
(
4
)
(
x
0
)
=
1
6
(
56
f
0
−
333
f
1
+
852
f
2
−
1219
f
3
+
1056
f
4
−
555
f
5
+
164
f
6
−
21
f
7
)
+
967
240
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{0})={\frac {1}{6}}(56f_{0}-333f_{1}+852f_{2}-1219f_{3}+1056f_{4}-555f_{5}+164f_{6}-21f_{7})+{\frac {967}{240}}h^{8}f^{(8)}(\xi )}
h
4
f
(
4
)
(
x
1
)
=
1
6
(
21
f
0
−
112
f
1
+
255
f
2
−
324
f
3
+
251
f
4
−
120
f
5
+
33
f
6
−
4
f
7
)
+
127
240
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{1})={\frac {1}{6}}(21f_{0}-112f_{1}+255f_{2}-324f_{3}+251f_{4}-120f_{5}+33f_{6}-4f_{7})+{\frac {127}{240}}h^{8}f^{(8)}(\xi )}
h
4
f
(
4
)
(
x
2
)
=
1
6
(
4
f
0
−
11
f
1
+
31
f
3
−
44
f
4
+
27
f
5
−
8
f
6
+
f
7
)
−
11
80
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{2})={\frac {1}{6}}(4f_{0}-11f_{1}+31f_{3}-44f_{4}+27f_{5}-8f_{6}+f_{7})-{\frac {11}{80}}h^{8}f^{(8)}(\xi )}
h
4
f
(
4
)
(
x
3
)
=
1
6
(
−
f
0
+
12
f
1
−
39
f
2
+
56
f
3
−
39
f
4
+
12
f
5
−
f
6
)
+
7
240
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{3})={\frac {1}{6}}(-f_{0}+12f_{1}-39f_{2}+56f_{3}-39f_{4}+12f_{5}-f_{6})+{\frac {7}{240}}h^{8}f^{(8)}(\xi )}
h
4
f
(
4
)
(
x
4
)
=
1
6
(
−
f
1
+
12
f
2
−
39
f
3
+
56
f
4
−
39
f
5
+
12
f
6
−
f
7
)
+
7
240
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{4})={\frac {1}{6}}(-f_{1}+12f_{2}-39f_{3}+56f_{4}-39f_{5}+12f_{6}-f_{7})+{\frac {7}{240}}h^{8}f^{(8)}(\xi )}
h
4
f
(
4
)
(
x
5
)
=
1
6
(
f
0
−
8
f
1
+
27
f
2
−
44
f
3
+
31
f
4
−
11
f
6
+
4
f
7
)
−
11
80
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{5})={\frac {1}{6}}(f_{0}-8f_{1}+27f_{2}-44f_{3}+31f_{4}-11f_{6}+4f_{7})-{\frac {11}{80}}h^{8}f^{(8)}(\xi )}
h
4
f
(
4
)
(
x
6
)
=
1
6
(
−
4
f
0
+
33
f
1
−
120
f
2
+
251
f
3
−
324
f
4
+
255
f
5
−
112
f
6
+
21
f
7
)
+
127
240
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{6})={\frac {1}{6}}(-4f_{0}+33f_{1}-120f_{2}+251f_{3}-324f_{4}+255f_{5}-112f_{6}+21f_{7})+{\frac {127}{240}}h^{8}f^{(8)}(\xi )}
h
4
f
(
4
)
(
x
7
)
=
1
6
(
−
21
f
0
+
164
f
1
−
555
f
2
+
1056
f
3
−
1219
f
4
+
852
f
5
−
333
f
6
+
56
f
7
)
+
967
240
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{7})={\frac {1}{6}}(-21f_{0}+164f_{1}-555f_{2}+1056f_{3}-1219f_{4}+852f_{5}-333f_{6}+56f_{7})+{\frac {967}{240}}h^{8}f^{(8)}(\xi )}
8-punts 5e afgeleide:
h
5
f
(
5
)
(
x
0
)
=
1
6
(
−
46
f
0
+
295
f
1
−
810
f
2
+
1235
f
3
−
1130
f
4
+
621
f
5
−
190
f
6
+
25
f
7
)
−
35
6
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{0})={\frac {1}{6}}(-46f_{0}+295f_{1}-810f_{2}+1235f_{3}-1130f_{4}+621f_{5}-190f_{6}+25f_{7})-{\frac {35}{6}}h^{8}f^{(8)}(\xi )}
h
5
f
(
5
)
(
x
1
)
=
1
6
(
−
25
f
0
+
154
f
1
−
405
f
2
+
590
f
3
−
515
f
4
+
270
f
5
−
79
f
6
+
10
f
7
)
−
5
3
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{1})={\frac {1}{6}}(-25f_{0}+154f_{1}-405f_{2}+590f_{3}-515f_{4}+270f_{5}-79f_{6}+10f_{7})-{\frac {5}{3}}h^{8}f^{(8)}(\xi )}
h
5
f
(
5
)
(
x
2
)
=
1
6
(
−
10
f
0
+
55
f
1
−
126
f
2
+
155
f
3
−
110
f
4
+
45
f
5
−
10
f
6
+
f
7
)
−
11
144
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{2})={\frac {1}{6}}(-10f_{0}+55f_{1}-126f_{2}+155f_{3}-110f_{4}+45f_{5}-10f_{6}+f_{7})-{\frac {11}{144}}h^{9}f^{(9)}(\xi )}
h
5
f
(
5
)
(
x
3
)
=
1
6
(
−
f
0
−
2
f
1
+
27
f
2
−
70
f
3
+
85
f
4
−
54
f
5
+
17
f
6
−
2
f
7
)
+
1
6
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{3})={\frac {1}{6}}(-f_{0}-2f_{1}+27f_{2}-70f_{3}+85f_{4}-54f_{5}+17f_{6}-2f_{7})+{\frac {1}{6}}h^{8}f^{(8)}(\xi )}
h
5
f
(
5
)
(
x
4
)
=
1
6
(
2
f
0
−
17
f
1
+
54
f
2
−
85
f
3
+
70
f
4
−
27
f
5
+
2
f
6
+
f
7
)
−
1
6
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{4})={\frac {1}{6}}(2f_{0}-17f_{1}+54f_{2}-85f_{3}+70f_{4}-27f_{5}+2f_{6}+f_{7})-{\frac {1}{6}}h^{8}f^{(8)}(\xi )}
h
5
f
(
5
)
(
x
5
)
=
1
6
(
−
f
0
+
10
f
1
−
45
f
2
+
110
f
3
−
155
f
4
+
126
f
5
−
55
f
6
+
10
f
7
)
−
11
144
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{5})={\frac {1}{6}}(-f_{0}+10f_{1}-45f_{2}+110f_{3}-155f_{4}+126f_{5}-55f_{6}+10f_{7})-{\frac {11}{144}}h^{9}f^{(9)}(\xi )}
h
5
f
(
5
)
(
x
6
)
=
1
6
(
−
10
f
0
+
79
f
1
−
270
f
2
+
515
f
3
−
590
f
4
+
405
f
5
−
154
f
6
+
25
f
7
)
+
5
3
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{6})={\frac {1}{6}}(-10f_{0}+79f_{1}-270f_{2}+515f_{3}-590f_{4}+405f_{5}-154f_{6}+25f_{7})+{\frac {5}{3}}h^{8}f^{(8)}(\xi )}
h
5
f
(
5
)
(
x
7
)
=
1
6
(
−
25
f
0
+
190
f
1
−
621
f
2
+
1130
f
3
−
1235
f
4
+
810
f
5
−
295
f
6
+
46
f
7
)
+
35
6
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{7})={\frac {1}{6}}(-25f_{0}+190f_{1}-621f_{2}+1130f_{3}-1235f_{4}+810f_{5}-295f_{6}+46f_{7})+{\frac {35}{6}}h^{8}f^{(8)}(\xi )}
8-punts 6e afgeleide:
h
6
f
(
6
)
(
x
0
)
=
4
f
0
−
27
f
1
+
78
f
2
−
125
f
3
+
120
f
4
−
69
f
5
+
22
f
6
−
3
f
7
+
23
4
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{0})=4f_{0}-27f_{1}+78f_{2}-125f_{3}+120f_{4}-69f_{5}+22f_{6}-3f_{7}+{\frac {23}{4}}h^{8}f^{(8)}(\xi )}
h
6
f
(
6
)
(
x
1
)
=
3
f
0
−
20
f
1
+
57
f
2
−
90
f
3
+
85
f
4
−
48
f
5
+
15
f
6
−
2
f
7
+
11
4
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{1})=3f_{0}-20f_{1}+57f_{2}-90f_{3}+85f_{4}-48f_{5}+15f_{6}-2f_{7}+{\frac {11}{4}}h^{8}f^{(8)}(\xi )}
h
6
f
(
6
)
(
x
2
)
=
2
f
0
−
13
f
1
+
36
f
2
−
55
f
3
+
50
f
4
−
27
f
5
+
8
f
6
−
f
7
+
3
4
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{2})=2f_{0}-13f_{1}+36f_{2}-55f_{3}+50f_{4}-27f_{5}+8f_{6}-f_{7}+{\frac {3}{4}}h^{8}f^{(8)}(\xi )}
h
6
f
(
6
)
(
x
3
)
=
f
0
−
6
f
1
+
15
f
2
−
20
f
3
+
15
f
4
−
6
f
5
+
f
6
−
1
4
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{3})=f_{0}-6f_{1}+15f_{2}-20f_{3}+15f_{4}-6f_{5}+f_{6}-{\frac {1}{4}}h^{8}f^{(8)}(\xi )}
h
6
f
(
6
)
(
x
4
)
=
f
1
−
6
f
2
+
15
f
3
−
20
f
4
+
15
f
5
−
6
f
6
+
f
7
−
1
4
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{4})=f_{1}-6f_{2}+15f_{3}-20f_{4}+15f_{5}-6f_{6}+f_{7}-{\frac {1}{4}}h^{8}f^{(8)}(\xi )}
h
6
f
(
6
)
(
x
5
)
=
−
f
0
+
8
f
1
−
27
f
2
+
50
f
3
−
55
f
4
+
36
f
5
−
13
f
6
+
2
f
7
+
3
4
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{5})=-f_{0}+8f_{1}-27f_{2}+50f_{3}-55f_{4}+36f_{5}-13f_{6}+2f_{7}+{\frac {3}{4}}h^{8}f^{(8)}(\xi )}
h
6
f
(
6
)
(
x
6
)
=
−
2
f
0
+
15
f
1
−
48
f
2
+
85
f
3
−
90
f
4
+
57
f
5
−
20
f
6
+
3
f
7
+
11
4
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{6})=-2f_{0}+15f_{1}-48f_{2}+85f_{3}-90f_{4}+57f_{5}-20f_{6}+3f_{7}+{\frac {11}{4}}h^{8}f^{(8)}(\xi )}
h
6
f
(
6
)
(
x
7
)
=
−
3
f
0
+
22
f
1
−
69
f
2
+
120
f
3
−
125
f
4
+
78
f
5
−
27
f
6
+
4
f
7
+
23
4
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{7})=-3f_{0}+22f_{1}-69f_{2}+120f_{3}-125f_{4}+78f_{5}-27f_{6}+4f_{7}+{\frac {23}{4}}h^{8}f^{(8)}(\xi )}
8-punts 7e afgeleide:
h
7
f
(
7
)
(
x
0
)
=
−
f
0
+
7
f
1
−
21
f
2
+
35
f
3
−
35
f
4
+
21
f
5
−
7
f
6
+
f
7
−
7
2
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{7}f^{(7)}(x_{0})=-f_{0}+7f_{1}-21f_{2}+35f_{3}-35f_{4}+21f_{5}-7f_{6}+f_{7}-{\frac {7}{2}}h^{8}f^{(8)}(\xi )}
h
7
f
(
7
)
(
x
1
)
=
−
f
0
+
7
f
1
−
21
f
2
+
35
f
3
−
35
f
4
+
21
f
5
−
7
f
6
+
f
7
−
5
2
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{7}f^{(7)}(x_{1})=-f_{0}+7f_{1}-21f_{2}+35f_{3}-35f_{4}+21f_{5}-7f_{6}+f_{7}-{\frac {5}{2}}h^{8}f^{(8)}(\xi )}
h
7
f
(
7
)
(
x
2
)
=
−
f
0
+
7
f
1
−
21
f
2
+
35
f
3
−
35
f
4
+
21
f
5
−
7
f
6
+
f
7
−
3
2
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{7}f^{(7)}(x_{2})=-f_{0}+7f_{1}-21f_{2}+35f_{3}-35f_{4}+21f_{5}-7f_{6}+f_{7}-{\frac {3}{2}}h^{8}f^{(8)}(\xi )}
h
7
f
(
7
)
(
x
3
)
=
−
f
0
+
7
f
1
−
21
f
2
+
35
f
3
−
35
f
4
+
21
f
5
−
7
f
6
+
f
7
−
1
2
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{7}f^{(7)}(x_{3})=-f_{0}+7f_{1}-21f_{2}+35f_{3}-35f_{4}+21f_{5}-7f_{6}+f_{7}-{\frac {1}{2}}h^{8}f^{(8)}(\xi )}
h
7
f
(
7
)
(
x
4
)
=
−
f
0
+
7
f
1
−
21
f
2
+
35
f
3
−
35
f
4
+
21
f
5
−
7
f
6
+
f
7
+
1
2
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{7}f^{(7)}(x_{4})=-f_{0}+7f_{1}-21f_{2}+35f_{3}-35f_{4}+21f_{5}-7f_{6}+f_{7}+{\frac {1}{2}}h^{8}f^{(8)}(\xi )}
h
7
f
(
7
)
(
x
5
)
=
−
f
0
+
7
f
1
−
21
f
2
+
35
f
3
−
35
f
4
+
21
f
5
−
7
f
6
+
f
7
+
3
2
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{7}f^{(7)}(x_{5})=-f_{0}+7f_{1}-21f_{2}+35f_{3}-35f_{4}+21f_{5}-7f_{6}+f_{7}+{\frac {3}{2}}h^{8}f^{(8)}(\xi )}
h
7
f
(
7
)
(
x
6
)
=
−
f
0
+
7
f
1
−
21
f
2
+
35
f
3
−
35
f
4
+
21
f
5
−
7
f
6
+
f
7
+
5
2
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{7}f^{(7)}(x_{6})=-f_{0}+7f_{1}-21f_{2}+35f_{3}-35f_{4}+21f_{5}-7f_{6}+f_{7}+{\frac {5}{2}}h^{8}f^{(8)}(\xi )}
h
7
f
(
7
)
(
x
7
)
=
−
f
0
+
7
f
1
−
21
f
2
+
35
f
3
−
35
f
4
+
21
f
5
−
7
f
6
+
f
7
+
7
2
h
8
f
(
8
)
(
ξ
)
{\displaystyle h^{7}f^{(7)}(x_{7})=-f_{0}+7f_{1}-21f_{2}+35f_{3}-35f_{4}+21f_{5}-7f_{6}+f_{7}+{\frac {7}{2}}h^{8}f^{(8)}(\xi )}
9-punts 1e afgeleide:
h
f
(
1
)
(
x
0
)
=
1
840
(
−
2283
f
0
+
6720
f
1
−
11760
f
2
+
15680
f
3
−
14700
f
4
+
9408
f
5
−
3920
f
6
+
960
f
7
−
105
f
8
)
+
1
9
h
9
f
(
9
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{0})={\frac {1}{840}}(-2283f_{0}+6720f_{1}-11760f_{2}+15680f_{3}-14700f_{4}+9408f_{5}-3920f_{6}+960f_{7}-105f_{8})+{\frac {1}{9}}h^{9}f^{(9)}(\xi )}
h
f
(
1
)
(
x
1
)
=
1
840
(
−
105
f
0
−
1338
f
1
+
2940
f
2
−
2940
f
3
+
2450
f
4
−
1470
f
5
+
588
f
6
−
140
f
7
+
15
f
8
)
−
1
72
h
9
f
(
9
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{1})={\frac {1}{840}}(-105f_{0}-1338f_{1}+2940f_{2}-2940f_{3}+2450f_{4}-1470f_{5}+588f_{6}-140f_{7}+15f_{8})-{\frac {1}{72}}h^{9}f^{(9)}(\xi )}
h
f
(
1
)
(
x
2
)
=
1
840
(
15
f
0
−
240
f
1
−
798
f
2
+
1680
f
3
−
1050
f
4
+
560
f
5
−
210
f
6
+
48
f
7
−
5
f
8
)
+
1
252
h
9
f
(
9
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{2})={\frac {1}{840}}(15f_{0}-240f_{1}-798f_{2}+1680f_{3}-1050f_{4}+560f_{5}-210f_{6}+48f_{7}-5f_{8})+{\frac {1}{252}}h^{9}f^{(9)}(\xi )}
h
f
(
1
)
(
x
3
)
=
1
840
(
−
5
f
0
+
60
f
1
−
420
f
2
−
378
f
3
+
1050
f
4
−
420
f
5
+
140
f
6
−
30
f
7
+
3
f
8
)
−
1
504
h
9
f
(
9
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{3})={\frac {1}{840}}(-5f_{0}+60f_{1}-420f_{2}-378f_{3}+1050f_{4}-420f_{5}+140f_{6}-30f_{7}+3f_{8})-{\frac {1}{504}}h^{9}f^{(9)}(\xi )}
h
f
(
1
)
(
x
4
)
=
1
840
(
3
f
0
−
32
f
1
+
168
f
2
−
672
f
3
+
672
f
5
−
168
f
6
+
32
f
7
−
3
f
8
)
+
1
630
h
9
f
(
9
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{4})={\frac {1}{840}}(3f_{0}-32f_{1}+168f_{2}-672f_{3}+672f_{5}-168f_{6}+32f_{7}-3f_{8})+{\frac {1}{630}}h^{9}f^{(9)}(\xi )}
h
f
(
1
)
(
x
5
)
=
1
840
(
−
3
f
0
+
30
f
1
−
140
f
2
+
420
f
3
−
1050
f
4
+
378
f
5
+
420
f
6
−
60
f
7
+
5
f
8
)
−
1
504
h
9
f
(
9
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{5})={\frac {1}{840}}(-3f_{0}+30f_{1}-140f_{2}+420f_{3}-1050f_{4}+378f_{5}+420f_{6}-60f_{7}+5f_{8})-{\frac {1}{504}}h^{9}f^{(9)}(\xi )}
h
f
(
1
)
(
x
6
)
=
1
840
(
5
f
0
−
48
f
1
+
210
f
2
−
560
f
3
+
1050
f
4
−
1680
f
5
+
798
f
6
+
240
f
7
−
15
f
8
)
+
1
252
h
9
f
(
9
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{6})={\frac {1}{840}}(5f_{0}-48f_{1}+210f_{2}-560f_{3}+1050f_{4}-1680f_{5}+798f_{6}+240f_{7}-15f_{8})+{\frac {1}{252}}h^{9}f^{(9)}(\xi )}
h
f
(
1
)
(
x
7
)
=
1
840
(
−
15
f
0
+
140
f
1
−
588
f
2
+
1470
f
3
−
2450
f
4
+
2940
f
5
−
2940
f
6
+
1338
f
7
+
105
f
8
)
−
1
72
h
9
f
(
9
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{7})={\frac {1}{840}}(-15f_{0}+140f_{1}-588f_{2}+1470f_{3}-2450f_{4}+2940f_{5}-2940f_{6}+1338f_{7}+105f_{8})-{\frac {1}{72}}h^{9}f^{(9)}(\xi )}
h
f
(
1
)
(
x
8
)
=
1
840
(
105
f
0
−
960
f
1
+
3920
f
2
−
9408
f
3
+
14700
f
4
−
15680
f
5
+
11760
f
6
−
6720
f
7
+
2283
f
8
)
+
1
9
h
9
f
(
9
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{8})={\frac {1}{840}}(105f_{0}-960f_{1}+3920f_{2}-9408f_{3}+14700f_{4}-15680f_{5}+11760f_{6}-6720f_{7}+2283f_{8})+{\frac {1}{9}}h^{9}f^{(9)}(\xi )}
9-punts 2e afgeleide:
h
2
f
(
2
)
(
x
0
)
=
1
5040
(
29531
f
0
−
138528
f
1
+
312984
f
2
−
448672
f
3
+
435330
f
4
−
284256
f
5
+
120008
f
6
−
29664
f
7
+
3267
f
8
)
−
761
1260
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{0})={\frac {1}{5040}}(29531f_{0}-138528f_{1}+312984f_{2}-448672f_{3}+435330f_{4}-284256f_{5}+120008f_{6}-29664f_{7}+3267f_{8})-{\frac {761}{1260}}h^{9}f^{(9)}(\xi )}
h
2
f
(
2
)
(
x
1
)
=
1
5040
(
3267
f
0
+
128
f
1
−
20916
f
2
+
38556
f
3
−
37030
f
4
+
23688
f
5
−
9828
f
6
+
2396
f
7
−
261
f
8
)
+
223
5040
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{1})={\frac {1}{5040}}(3267f_{0}+128f_{1}-20916f_{2}+38556f_{3}-37030f_{4}+23688f_{5}-9828f_{6}+2396f_{7}-261f_{8})+{\frac {223}{5040}}h^{9}f^{(9)}(\xi )}
h
2
f
(
2
)
(
x
2
)
=
1
5040
(
−
261
f
0
+
5616
f
1
−
9268
f
2
+
1008
f
3
+
5670
f
4
−
4144
f
5
+
1764
f
6
−
432
f
7
+
47
f
8
)
−
19
2520
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{2})={\frac {1}{5040}}(-261f_{0}+5616f_{1}-9268f_{2}+1008f_{3}+5670f_{4}-4144f_{5}+1764f_{6}-432f_{7}+47f_{8})-{\frac {19}{2520}}h^{9}f^{(9)}(\xi )}
h
2
f
(
2
)
(
x
3
)
=
1
5040
(
47
f
0
−
684
f
1
+
7308
f
2
−
13216
f
3
+
6930
f
4
−
252
f
5
−
196
f
6
+
72
f
7
−
9
f
8
)
+
1
560
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{3})={\frac {1}{5040}}(47f_{0}-684f_{1}+7308f_{2}-13216f_{3}+6930f_{4}-252f_{5}-196f_{6}+72f_{7}-9f_{8})+{\frac {1}{560}}h^{9}f^{(9)}(\xi )}
h
2
f
(
2
)
(
x
4
)
=
1
5040
(
−
9
f
0
+
128
f
1
−
1008
f
2
+
8064
f
3
−
14350
f
4
+
8064
f
5
−
1008
f
6
+
128
f
7
−
9
f
8
)
+
1
3150
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{4})={\frac {1}{5040}}(-9f_{0}+128f_{1}-1008f_{2}+8064f_{3}-14350f_{4}+8064f_{5}-1008f_{6}+128f_{7}-9f_{8})+{\frac {1}{3150}}h^{10}f^{(10)}(\xi )}
h
2
f
(
2
)
(
x
5
)
=
1
5040
(
−
9
f
0
+
72
f
1
−
196
f
2
−
252
f
3
+
6930
f
4
−
13216
f
5
+
7308
f
6
−
684
f
7
+
47
f
8
)
−
1
560
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{5})={\frac {1}{5040}}(-9f_{0}+72f_{1}-196f_{2}-252f_{3}+6930f_{4}-13216f_{5}+7308f_{6}-684f_{7}+47f_{8})-{\frac {1}{560}}h^{9}f^{(9)}(\xi )}
h
2
f
(
2
)
(
x
6
)
=
1
5040
(
47
f
0
−
432
f
1
+
1764
f
2
−
4144
f
3
+
5670
f
4
+
1008
f
5
−
9268
f
6
+
5616
f
7
−
261
f
8
)
+
19
2520
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{6})={\frac {1}{5040}}(47f_{0}-432f_{1}+1764f_{2}-4144f_{3}+5670f_{4}+1008f_{5}-9268f_{6}+5616f_{7}-261f_{8})+{\frac {19}{2520}}h^{9}f^{(9)}(\xi )}
h
2
f
(
2
)
(
x
7
)
=
1
5040
(
−
261
f
0
+
2396
f
1
−
9828
f
2
+
23688
f
3
−
37030
f
4
+
38556
f
5
−
20916
f
6
+
128
f
7
+
3267
f
8
)
−
223
5040
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{7})={\frac {1}{5040}}(-261f_{0}+2396f_{1}-9828f_{2}+23688f_{3}-37030f_{4}+38556f_{5}-20916f_{6}+128f_{7}+3267f_{8})-{\frac {223}{5040}}h^{9}f^{(9)}(\xi )}
h
2
f
(
2
)
(
x
8
)
=
1
5040
(
3267
f
0
−
29664
f
1
+
120008
f
2
−
284256
f
3
+
435330
f
4
−
448672
f
5
+
312984
f
6
−
138528
f
7
+
29531
f
8
)
+
761
1260
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{8})={\frac {1}{5040}}(3267f_{0}-29664f_{1}+120008f_{2}-284256f_{3}+435330f_{4}-448672f_{5}+312984f_{6}-138528f_{7}+29531f_{8})+{\frac {761}{1260}}h^{9}f^{(9)}(\xi )}
9-punts 3e afgeleide:
h
3
f
(
3
)
(
x
0
)
=
1
240
(
−
2403
f
0
+
13960
f
1
−
36706
f
2
+
57384
f
3
−
58280
f
4
+
39128
f
5
−
16830
f
6
+
4216
f
7
−
469
f
8
)
+
29531
15120
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{0})={\frac {1}{240}}(-2403f_{0}+13960f_{1}-36706f_{2}+57384f_{3}-58280f_{4}+39128f_{5}-16830f_{6}+4216f_{7}-469f_{8})+{\frac {29531}{15120}}h^{9}f^{(9)}(\xi )}
h
3
f
(
3
)
(
x
1
)
=
1
240
(
−
469
f
0
+
1818
f
1
−
2924
f
2
+
2690
f
3
−
1710
f
4
+
814
f
5
−
268
f
6
+
54
f
7
−
5
f
8
)
−
1
945
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{1})={\frac {1}{240}}(-469f_{0}+1818f_{1}-2924f_{2}+2690f_{3}-1710f_{4}+814f_{5}-268f_{6}+54f_{7}-5f_{8})-{\frac {1}{945}}h^{9}f^{(9)}(\xi )}
h
3
f
(
3
)
(
x
2
)
=
1
240
(
−
5
f
0
−
424
f
1
+
1638
f
2
−
2504
f
3
+
2060
f
4
−
1080
f
5
+
394
f
6
−
88
f
7
+
9
f
8
)
−
331
15120
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{2})={\frac {1}{240}}(-5f_{0}-424f_{1}+1638f_{2}-2504f_{3}+2060f_{4}-1080f_{5}+394f_{6}-88f_{7}+9f_{8})-{\frac {331}{15120}}h^{9}f^{(9)}(\xi )}
h
3
f
(
3
)
(
x
3
)
=
1
240
(
9
f
0
−
86
f
1
−
100
f
2
+
882
f
3
−
1370
f
4
+
926
f
5
−
324
f
6
+
70
f
7
−
7
f
8
)
+
59
3780
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{3})={\frac {1}{240}}(9f_{0}-86f_{1}-100f_{2}+882f_{3}-1370f_{4}+926f_{5}-324f_{6}+70f_{7}-7f_{8})+{\frac {59}{3780}}h^{9}f^{(9)}(\xi )}
h
3
f
(
3
)
(
x
4
)
=
1
240
(
−
7
f
0
+
72
f
1
−
338
f
2
+
488
f
3
−
488
f
5
+
338
f
6
−
72
f
7
+
7
f
8
)
−
41
3024
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{4})={\frac {1}{240}}(-7f_{0}+72f_{1}-338f_{2}+488f_{3}-488f_{5}+338f_{6}-72f_{7}+7f_{8})-{\frac {41}{3024}}h^{9}f^{(9)}(\xi )}
h
3
f
(
3
)
(
x
5
)
=
1
240
(
7
f
0
−
70
f
1
+
324
f
2
−
926
f
3
+
1370
f
4
−
882
f
5
+
100
f
6
+
86
f
7
−
9
f
8
)
+
59
3780
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{5})={\frac {1}{240}}(7f_{0}-70f_{1}+324f_{2}-926f_{3}+1370f_{4}-882f_{5}+100f_{6}+86f_{7}-9f_{8})+{\frac {59}{3780}}h^{9}f^{(9)}(\xi )}
h
3
f
(
3
)
(
x
6
)
=
1
240
(
−
9
f
0
+
88
f
1
−
394
f
2
+
1080
f
3
−
2060
f
4
+
2504
f
5
−
1638
f
6
+
424
f
7
+
5
f
8
)
−
331
15120
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{6})={\frac {1}{240}}(-9f_{0}+88f_{1}-394f_{2}+1080f_{3}-2060f_{4}+2504f_{5}-1638f_{6}+424f_{7}+5f_{8})-{\frac {331}{15120}}h^{9}f^{(9)}(\xi )}
h
3
f
(
3
)
(
x
7
)
=
1
240
(
5
f
0
−
54
f
1
+
268
f
2
−
814
f
3
+
1710
f
4
−
2690
f
5
+
2924
f
6
−
1818
f
7
+
469
f
8
)
−
1
945
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{7})={\frac {1}{240}}(5f_{0}-54f_{1}+268f_{2}-814f_{3}+1710f_{4}-2690f_{5}+2924f_{6}-1818f_{7}+469f_{8})-{\frac {1}{945}}h^{9}f^{(9)}(\xi )}
h
3
f
(
3
)
(
x
8
)
=
1
240
(
469
f
0
−
4216
f
1
+
16830
f
2
−
39128
f
3
+
58280
f
4
−
57384
f
5
+
36706
f
6
−
13960
f
7
+
2403
f
8
)
+
29531
15120
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{8})={\frac {1}{240}}(469f_{0}-4216f_{1}+16830f_{2}-39128f_{3}+58280f_{4}-57384f_{5}+36706f_{6}-13960f_{7}+2403f_{8})+{\frac {29531}{15120}}h^{9}f^{(9)}(\xi )}
9-punts 4e afgeleide:
h
4
f
(
4
)
(
x
0
)
=
1
240
(
3207
f
0
−
21056
f
1
+
61156
f
2
−
102912
f
3
+
109930
f
4
−
76352
f
5
+
33636
f
6
−
8576
f
7
+
967
f
8
)
−
89
20
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{0})={\frac {1}{240}}(3207f_{0}-21056f_{1}+61156f_{2}-102912f_{3}+109930f_{4}-76352f_{5}+33636f_{6}-8576f_{7}+967f_{8})-{\frac {89}{20}}h^{9}f^{(9)}(\xi )}
h
4
f
(
4
)
(
x
1
)
=
1
240
(
967
f
0
−
5496
f
1
+
13756
f
2
−
20072
f
3
+
18930
f
4
−
11912
f
5
+
4876
f
6
−
1176
f
7
+
127
f
8
)
−
101
240
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{1})={\frac {1}{240}}(967f_{0}-5496f_{1}+13756f_{2}-20072f_{3}+18930f_{4}-11912f_{5}+4876f_{6}-1176f_{7}+127f_{8})-{\frac {101}{240}}h^{9}f^{(9)}(\xi )}
h
4
f
(
4
)
(
x
2
)
=
1
240
(
127
f
0
−
176
f
1
−
924
f
2
+
3088
f
3
−
4070
f
4
+
2928
f
5
−
1244
f
6
+
304
f
7
−
33
f
8
)
+
13
120
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{2})={\frac {1}{240}}(127f_{0}-176f_{1}-924f_{2}+3088f_{3}-4070f_{4}+2928f_{5}-1244f_{6}+304f_{7}-33f_{8})+{\frac {13}{120}}h^{9}f^{(9)}(\xi )}
h
4
f
(
4
)
(
x
3
)
=
1
240
(
−
33
f
0
+
424
f
1
−
1364
f
2
+
1848
f
3
−
1070
f
4
+
88
f
5
+
156
f
6
−
56
f
7
+
7
f
8
)
−
7
240
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{3})={\frac {1}{240}}(-33f_{0}+424f_{1}-1364f_{2}+1848f_{3}-1070f_{4}+88f_{5}+156f_{6}-56f_{7}+7f_{8})-{\frac {7}{240}}h^{9}f^{(9)}(\xi )}
h
4
f
(
4
)
(
x
4
)
=
1
240
(
7
f
0
−
96
f
1
+
676
f
2
−
1952
f
3
+
2730
f
4
−
1952
f
5
+
676
f
6
−
96
f
7
+
7
f
8
)
−
41
7560
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{4})={\frac {1}{240}}(7f_{0}-96f_{1}+676f_{2}-1952f_{3}+2730f_{4}-1952f_{5}+676f_{6}-96f_{7}+7f_{8})-{\frac {41}{7560}}h^{10}f^{(10)}(\xi )}
h
4
f
(
4
)
(
x
5
)
=
1
240
(
7
f
0
−
56
f
1
+
156
f
2
+
88
f
3
−
1070
f
4
+
1848
f
5
−
1364
f
6
+
424
f
7
−
33
f
8
)
+
7
240
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{5})={\frac {1}{240}}(7f_{0}-56f_{1}+156f_{2}+88f_{3}-1070f_{4}+1848f_{5}-1364f_{6}+424f_{7}-33f_{8})+{\frac {7}{240}}h^{9}f^{(9)}(\xi )}
h
4
f
(
4
)
(
x
6
)
=
1
240
(
−
33
f
0
+
304
f
1
−
1244
f
2
+
2928
f
3
−
4070
f
4
+
3088
f
5
−
924
f
6
−
176
f
7
+
127
f
8
)
−
13
120
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{6})={\frac {1}{240}}(-33f_{0}+304f_{1}-1244f_{2}+2928f_{3}-4070f_{4}+3088f_{5}-924f_{6}-176f_{7}+127f_{8})-{\frac {13}{120}}h^{9}f^{(9)}(\xi )}
h
4
f
(
4
)
(
x
7
)
=
1
240
(
127
f
0
−
1176
f
1
+
4876
f
2
−
11912
f
3
+
18930
f
4
−
20072
f
5
+
13756
f
6
−
5496
f
7
+
967
f
8
)
+
101
240
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{7})={\frac {1}{240}}(127f_{0}-1176f_{1}+4876f_{2}-11912f_{3}+18930f_{4}-20072f_{5}+13756f_{6}-5496f_{7}+967f_{8})+{\frac {101}{240}}h^{9}f^{(9)}(\xi )}
h
4
f
(
4
)
(
x
8
)
=
1
240
(
967
f
0
−
8576
f
1
+
33636
f
2
−
76352
f
3
+
109930
f
4
−
102912
f
5
+
61156
f
6
−
21056
f
7
+
3207
f
8
)
+
89
20
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{8})={\frac {1}{240}}(967f_{0}-8576f_{1}+33636f_{2}-76352f_{3}+109930f_{4}-102912f_{5}+61156f_{6}-21056f_{7}+3207f_{8})+{\frac {89}{20}}h^{9}f^{(9)}(\xi )}
9-punts 5e afgeleide:
h
5
f
(
5
)
(
x
0
)
=
1
6
(
−
81
f
0
+
575
f
1
−
1790
f
2
+
3195
f
3
−
3580
f
4
+
2581
f
5
−
1170
f
6
+
305
f
7
−
35
f
8
)
+
1069
144
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{0})={\frac {1}{6}}(-81f_{0}+575f_{1}-1790f_{2}+3195f_{3}-3580f_{4}+2581f_{5}-1170f_{6}+305f_{7}-35f_{8})+{\frac {1069}{144}}h^{9}f^{(9)}(\xi )}
h
5
f
(
5
)
(
x
1
)
=
1
6
(
−
35
f
0
+
234
f
1
−
685
f
2
+
1150
f
3
−
1215
f
4
+
830
f
5
−
359
f
6
+
90
f
7
−
10
f
8
)
+
229
144
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{1})={\frac {1}{6}}(-35f_{0}+234f_{1}-685f_{2}+1150f_{3}-1215f_{4}+830f_{5}-359f_{6}+90f_{7}-10f_{8})+{\frac {229}{144}}h^{9}f^{(9)}(\xi )}
h
5
f
(
5
)
(
x
2
)
=
1
6
(
−
10
f
0
+
55
f
1
−
126
f
2
+
155
f
3
−
110
f
4
+
45
f
5
−
10
f
6
+
f
7
)
−
11
144
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{2})={\frac {1}{6}}(-10f_{0}+55f_{1}-126f_{2}+155f_{3}-110f_{4}+45f_{5}-10f_{6}+f_{7})-{\frac {11}{144}}h^{9}f^{(9)}(\xi )}
h
5
f
(
5
)
(
x
3
)
=
1
6
(
−
10
f
1
+
55
f
2
−
126
f
3
+
155
f
4
−
110
f
5
+
45
f
6
−
10
f
7
+
f
8
)
−
11
144
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{3})={\frac {1}{6}}(-10f_{1}+55f_{2}-126f_{3}+155f_{4}-110f_{5}+45f_{6}-10f_{7}+f_{8})-{\frac {11}{144}}h^{9}f^{(9)}(\xi )}
h
5
f
(
5
)
(
x
4
)
=
1
6
(
f
0
−
9
f
1
+
26
f
2
−
29
f
3
+
29
f
5
−
26
f
6
+
9
f
7
−
f
8
)
+
13
144
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{4})={\frac {1}{6}}(f_{0}-9f_{1}+26f_{2}-29f_{3}+29f_{5}-26f_{6}+9f_{7}-f_{8})+{\frac {13}{144}}h^{9}f^{(9)}(\xi )}
h
5
f
(
5
)
(
x
5
)
=
1
6
(
−
f
0
+
10
f
1
−
45
f
2
+
110
f
3
−
155
f
4
+
126
f
5
−
55
f
6
+
10
f
7
)
−
11
144
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{5})={\frac {1}{6}}(-f_{0}+10f_{1}-45f_{2}+110f_{3}-155f_{4}+126f_{5}-55f_{6}+10f_{7})-{\frac {11}{144}}h^{9}f^{(9)}(\xi )}
h
5
f
(
5
)
(
x
6
)
=
1
6
(
−
f
1
+
10
f
2
−
45
f
3
+
110
f
4
−
155
f
5
+
126
f
6
−
55
f
7
+
10
f
8
)
−
11
144
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{6})={\frac {1}{6}}(-f_{1}+10f_{2}-45f_{3}+110f_{4}-155f_{5}+126f_{6}-55f_{7}+10f_{8})-{\frac {11}{144}}h^{9}f^{(9)}(\xi )}
h
5
f
(
5
)
(
x
7
)
=
1
6
(
10
f
0
−
90
f
1
+
359
f
2
−
830
f
3
+
1215
f
4
−
1150
f
5
+
685
f
6
−
234
f
7
+
35
f
8
)
+
229
144
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{7})={\frac {1}{6}}(10f_{0}-90f_{1}+359f_{2}-830f_{3}+1215f_{4}-1150f_{5}+685f_{6}-234f_{7}+35f_{8})+{\frac {229}{144}}h^{9}f^{(9)}(\xi )}
h
5
f
(
5
)
(
x
8
)
=
1
6
(
35
f
0
−
305
f
1
+
1170
f
2
−
2581
f
3
+
3580
f
4
−
3195
f
5
+
1790
f
6
−
575
f
7
+
81
f
8
)
+
1069
144
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{8})={\frac {1}{6}}(35f_{0}-305f_{1}+1170f_{2}-2581f_{3}+3580f_{4}-3195f_{5}+1790f_{6}-575f_{7}+81f_{8})+{\frac {1069}{144}}h^{9}f^{(9)}(\xi )}
9-punts 6e afgeleide:
h
6
f
(
6
)
(
x
0
)
=
1
4
(
39
f
0
−
292
f
1
+
956
f
2
−
1788
f
3
+
2090
f
4
−
1564
f
5
+
732
f
6
−
196
f
7
+
23
f
8
)
−
9
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{0})={\frac {1}{4}}(39f_{0}-292f_{1}+956f_{2}-1788f_{3}+2090f_{4}-1564f_{5}+732f_{6}-196f_{7}+23f_{8})-9h^{9}f^{(9)}(\xi )}
h
6
f
(
6
)
(
x
1
)
=
1
4
(
23
f
0
−
168
f
1
+
536
f
2
−
976
f
3
+
1110
f
4
−
808
f
5
+
368
f
6
−
96
f
7
+
11
f
8
)
−
13
4
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{1})={\frac {1}{4}}(23f_{0}-168f_{1}+536f_{2}-976f_{3}+1110f_{4}-808f_{5}+368f_{6}-96f_{7}+11f_{8})-{\frac {13}{4}}h^{9}f^{(9)}(\xi )}
h
6
f
(
6
)
(
x
2
)
=
1
4
(
11
f
0
−
76
f
1
+
228
f
2
−
388
f
3
+
410
f
4
−
276
f
5
+
116
f
6
−
28
f
7
+
3
f
8
)
−
1
2
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{2})={\frac {1}{4}}(11f_{0}-76f_{1}+228f_{2}-388f_{3}+410f_{4}-276f_{5}+116f_{6}-28f_{7}+3f_{8})-{\frac {1}{2}}h^{9}f^{(9)}(\xi )}
h
6
f
(
6
)
(
x
3
)
=
1
4
(
3
f
0
−
16
f
1
+
32
f
2
−
24
f
3
−
10
f
4
+
32
f
5
−
24
f
6
+
8
f
7
−
f
8
)
+
1
4
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{3})={\frac {1}{4}}(3f_{0}-16f_{1}+32f_{2}-24f_{3}-10f_{4}+32f_{5}-24f_{6}+8f_{7}-f_{8})+{\frac {1}{4}}h^{9}f^{(9)}(\xi )}
h
6
f
(
6
)
(
x
4
)
=
1
4
(
−
f
0
+
12
f
1
−
52
f
2
+
116
f
3
−
150
f
4
+
116
f
5
−
52
f
6
+
12
f
7
−
f
8
)
+
13
240
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{4})={\frac {1}{4}}(-f_{0}+12f_{1}-52f_{2}+116f_{3}-150f_{4}+116f_{5}-52f_{6}+12f_{7}-f_{8})+{\frac {13}{240}}h^{10}f^{(10)}(\xi )}
h
6
f
(
6
)
(
x
5
)
=
1
4
(
−
f
0
+
8
f
1
−
24
f
2
+
32
f
3
−
10
f
4
−
24
f
5
+
32
f
6
−
16
f
7
+
3
f
8
)
−
1
4
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{5})={\frac {1}{4}}(-f_{0}+8f_{1}-24f_{2}+32f_{3}-10f_{4}-24f_{5}+32f_{6}-16f_{7}+3f_{8})-{\frac {1}{4}}h^{9}f^{(9)}(\xi )}
h
6
f
(
6
)
(
x
6
)
=
1
4
(
3
f
0
−
28
f
1
+
116
f
2
−
276
f
3
+
410
f
4
−
388
f
5
+
228
f
6
−
76
f
7
+
11
f
8
)
+
1
2
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{6})={\frac {1}{4}}(3f_{0}-28f_{1}+116f_{2}-276f_{3}+410f_{4}-388f_{5}+228f_{6}-76f_{7}+11f_{8})+{\frac {1}{2}}h^{9}f^{(9)}(\xi )}
h
6
f
(
6
)
(
x
7
)
=
1
4
(
11
f
0
−
96
f
1
+
368
f
2
−
808
f
3
+
1110
f
4
−
976
f
5
+
536
f
6
−
168
f
7
+
23
f
8
)
+
13
4
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{7})={\frac {1}{4}}(11f_{0}-96f_{1}+368f_{2}-808f_{3}+1110f_{4}-976f_{5}+536f_{6}-168f_{7}+23f_{8})+{\frac {13}{4}}h^{9}f^{(9)}(\xi )}
h
6
f
(
6
)
(
x
8
)
=
1
4
(
23
f
0
−
196
f
1
+
732
f
2
−
1564
f
3
+
2090
f
4
−
1788
f
5
+
956
f
6
−
292
f
7
+
39
f
8
)
+
9
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{8})={\frac {1}{4}}(23f_{0}-196f_{1}+732f_{2}-1564f_{3}+2090f_{4}-1788f_{5}+956f_{6}-292f_{7}+39f_{8})+9h^{9}f^{(9)}(\xi )}
9-punts 7e afgeleide:
h
7
f
(
7
)
(
x
0
)
=
1
2
(
−
9
f
0
+
70
f
1
−
238
f
2
+
462
f
3
−
560
f
4
+
434
f
5
−
210
f
6
+
58
f
7
−
7
f
8
)
+
91
12
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{7}f^{(7)}(x_{0})={\frac {1}{2}}(-9f_{0}+70f_{1}-238f_{2}+462f_{3}-560f_{4}+434f_{5}-210f_{6}+58f_{7}-7f_{8})+{\frac {91}{12}}h^{9}f^{(9)}(\xi )}
h
7
f
(
7
)
(
x
1
)
=
1
2
(
−
7
f
0
+
54
f
1
−
182
f
2
+
350
f
3
−
420
f
4
+
322
f
5
−
154
f
6
+
42
f
7
−
5
f
8
)
+
49
12
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{7}f^{(7)}(x_{1})={\frac {1}{2}}(-7f_{0}+54f_{1}-182f_{2}+350f_{3}-420f_{4}+322f_{5}-154f_{6}+42f_{7}-5f_{8})+{\frac {49}{12}}h^{9}f^{(9)}(\xi )}
h
7
f
(
7
)
(
x
2
)
=
1
2
(
−
5
f
0
+
38
f
1
−
126
f
2
+
238
f
3
−
280
f
4
+
210
f
5
−
98
f
6
+
26
f
7
−
3
f
8
)
+
19
12
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{7}f^{(7)}(x_{2})={\frac {1}{2}}(-5f_{0}+38f_{1}-126f_{2}+238f_{3}-280f_{4}+210f_{5}-98f_{6}+26f_{7}-3f_{8})+{\frac {19}{12}}h^{9}f^{(9)}(\xi )}
h
7
f
(
7
)
(
x
3
)
=
1
2
(
−
3
f
0
+
22
f
1
−
70
f
2
+
126
f
3
−
140
f
4
+
98
f
5
−
42
f
6
+
10
f
7
−
f
8
)
+
1
12
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{7}f^{(7)}(x_{3})={\frac {1}{2}}(-3f_{0}+22f_{1}-70f_{2}+126f_{3}-140f_{4}+98f_{5}-42f_{6}+10f_{7}-f_{8})+{\frac {1}{12}}h^{9}f^{(9)}(\xi )}
h
7
f
(
7
)
(
x
4
)
=
1
2
(
−
f
0
+
6
f
1
−
14
f
2
+
14
f
3
−
14
f
5
+
14
f
6
−
6
f
7
+
f
8
)
−
5
12
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{7}f^{(7)}(x_{4})={\frac {1}{2}}(-f_{0}+6f_{1}-14f_{2}+14f_{3}-14f_{5}+14f_{6}-6f_{7}+f_{8})-{\frac {5}{12}}h^{9}f^{(9)}(\xi )}
h
7
f
(
7
)
(
x
5
)
=
1
2
(
f
0
−
10
f
1
+
42
f
2
−
98
f
3
+
140
f
4
−
126
f
5
+
70
f
6
−
22
f
7
+
3
f
8
)
+
1
12
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{7}f^{(7)}(x_{5})={\frac {1}{2}}(f_{0}-10f_{1}+42f_{2}-98f_{3}+140f_{4}-126f_{5}+70f_{6}-22f_{7}+3f_{8})+{\frac {1}{12}}h^{9}f^{(9)}(\xi )}
h
7
f
(
7
)
(
x
6
)
=
1
2
(
3
f
0
−
26
f
1
+
98
f
2
−
210
f
3
+
280
f
4
−
238
f
5
+
126
f
6
−
38
f
7
+
5
f
8
)
+
19
12
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{7}f^{(7)}(x_{6})={\frac {1}{2}}(3f_{0}-26f_{1}+98f_{2}-210f_{3}+280f_{4}-238f_{5}+126f_{6}-38f_{7}+5f_{8})+{\frac {19}{12}}h^{9}f^{(9)}(\xi )}
h
7
f
(
7
)
(
x
7
)
=
1
2
(
5
f
0
−
42
f
1
+
154
f
2
−
322
f
3
+
420
f
4
−
350
f
5
+
182
f
6
−
54
f
7
+
7
f
8
)
+
49
12
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{7}f^{(7)}(x_{7})={\frac {1}{2}}(5f_{0}-42f_{1}+154f_{2}-322f_{3}+420f_{4}-350f_{5}+182f_{6}-54f_{7}+7f_{8})+{\frac {49}{12}}h^{9}f^{(9)}(\xi )}
h
7
f
(
7
)
(
x
8
)
=
1
2
(
7
f
0
−
58
f
1
+
210
f
2
−
434
f
3
+
560
f
4
−
462
f
5
+
238
f
6
−
70
f
7
+
9
f
8
)
+
91
12
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{7}f^{(7)}(x_{8})={\frac {1}{2}}(7f_{0}-58f_{1}+210f_{2}-434f_{3}+560f_{4}-462f_{5}+238f_{6}-70f_{7}+9f_{8})+{\frac {91}{12}}h^{9}f^{(9)}(\xi )}
9-punts 8e afgeleide:
h
8
f
(
8
)
(
x
0
)
=
f
0
−
8
f
1
+
28
f
2
−
56
f
3
+
70
f
4
−
56
f
5
+
28
f
6
−
8
f
7
+
f
8
−
4
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{8}f^{(8)}(x_{0})=f_{0}-8f_{1}+28f_{2}-56f_{3}+70f_{4}-56f_{5}+28f_{6}-8f_{7}+f_{8}-4h^{9}f^{(9)}(\xi )}
h
8
f
(
8
)
(
x
1
)
=
f
0
−
8
f
1
+
28
f
2
−
56
f
3
+
70
f
4
−
56
f
5
+
28
f
6
−
8
f
7
+
f
8
−
3
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{8}f^{(8)}(x_{1})=f_{0}-8f_{1}+28f_{2}-56f_{3}+70f_{4}-56f_{5}+28f_{6}-8f_{7}+f_{8}-3h^{9}f^{(9)}(\xi )}
h
8
f
(
8
)
(
x
2
)
=
f
0
−
8
f
1
+
28
f
2
−
56
f
3
+
70
f
4
−
56
f
5
+
28
f
6
−
8
f
7
+
f
8
−
2
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{8}f^{(8)}(x_{2})=f_{0}-8f_{1}+28f_{2}-56f_{3}+70f_{4}-56f_{5}+28f_{6}-8f_{7}+f_{8}-2h^{9}f^{(9)}(\xi )}
h
8
f
(
8
)
(
x
3
)
=
f
0
−
8
f
1
+
28
f
2
−
56
f
3
+
70
f
4
−
56
f
5
+
28
f
6
−
8
f
7
+
f
8
−
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{8}f^{(8)}(x_{3})=f_{0}-8f_{1}+28f_{2}-56f_{3}+70f_{4}-56f_{5}+28f_{6}-8f_{7}+f_{8}-h^{9}f^{(9)}(\xi )}
h
8
f
(
8
)
(
x
4
)
=
f
0
−
8
f
1
+
28
f
2
−
56
f
3
+
70
f
4
−
56
f
5
+
28
f
6
−
8
f
7
+
f
8
−
1
3
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{8}f^{(8)}(x_{4})=f_{0}-8f_{1}+28f_{2}-56f_{3}+70f_{4}-56f_{5}+28f_{6}-8f_{7}+f_{8}-{\frac {1}{3}}h^{10}f^{(10)}(\xi )}
h
8
f
(
8
)
(
x
5
)
=
f
0
−
8
f
1
+
28
f
2
−
56
f
3
+
70
f
4
−
56
f
5
+
28
f
6
−
8
f
7
+
f
8
+
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{8}f^{(8)}(x_{5})=f_{0}-8f_{1}+28f_{2}-56f_{3}+70f_{4}-56f_{5}+28f_{6}-8f_{7}+f_{8}+h^{9}f^{(9)}(\xi )}
h
8
f
(
8
)
(
x
6
)
=
f
0
−
8
f
1
+
28
f
2
−
56
f
3
+
70
f
4
−
56
f
5
+
28
f
6
−
8
f
7
+
f
8
+
2
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{8}f^{(8)}(x_{6})=f_{0}-8f_{1}+28f_{2}-56f_{3}+70f_{4}-56f_{5}+28f_{6}-8f_{7}+f_{8}+2h^{9}f^{(9)}(\xi )}
h
8
f
(
8
)
(
x
7
)
=
f
0
−
8
f
1
+
28
f
2
−
56
f
3
+
70
f
4
−
56
f
5
+
28
f
6
−
8
f
7
+
f
8
+
3
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{8}f^{(8)}(x_{7})=f_{0}-8f_{1}+28f_{2}-56f_{3}+70f_{4}-56f_{5}+28f_{6}-8f_{7}+f_{8}+3h^{9}f^{(9)}(\xi )}
h
8
f
(
8
)
(
x
8
)
=
f
0
−
8
f
1
+
28
f
2
−
56
f
3
+
70
f
4
−
56
f
5
+
28
f
6
−
8
f
7
+
f
8
+
4
h
9
f
(
9
)
(
ξ
)
{\displaystyle h^{8}f^{(8)}(x_{8})=f_{0}-8f_{1}+28f_{2}-56f_{3}+70f_{4}-56f_{5}+28f_{6}-8f_{7}+f_{8}+4h^{9}f^{(9)}(\xi )}
10-punts 1e afgeleide:
h
f
(
1
)
(
x
0
)
=
1
2520
(
−
7129
f
0
+
22680
f
1
−
45360
f
2
+
70560
f
3
−
79380
f
4
+
63504
f
5
−
35280
f
6
+
12960
f
7
−
2835
f
8
+
280
f
9
)
−
1
10
h
10
f
(
10
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{0})={\frac {1}{2520}}(-7129f_{0}+22680f_{1}-45360f_{2}+70560f_{3}-79380f_{4}+63504f_{5}-35280f_{6}+12960f_{7}-2835f_{8}+280f_{9})-{\frac {1}{10}}h^{10}f^{(10)}(\xi )}
h
f
(
1
)
(
x
1
)
=
1
2520
(
−
280
f
0
−
4329
f
1
+
10080
f
2
−
11760
f
3
+
11760
f
4
−
8820
f
5
+
4704
f
6
−
1680
f
7
+
360
f
8
−
35
f
9
)
+
1
90
h
10
f
(
10
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{1})={\frac {1}{2520}}(-280f_{0}-4329f_{1}+10080f_{2}-11760f_{3}+11760f_{4}-8820f_{5}+4704f_{6}-1680f_{7}+360f_{8}-35f_{9})+{\frac {1}{90}}h^{10}f^{(10)}(\xi )}
h
f
(
1
)
(
x
2
)
=
1
2520
(
35
f
0
−
630
f
1
−
2754
f
2
+
5880
f
3
−
4410
f
4
+
2940
f
5
−
1470
f
6
+
504
f
7
−
105
f
8
+
10
f
9
)
−
1
360
h
10
f
(
10
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{2})={\frac {1}{2520}}(35f_{0}-630f_{1}-2754f_{2}+5880f_{3}-4410f_{4}+2940f_{5}-1470f_{6}+504f_{7}-105f_{8}+10f_{9})-{\frac {1}{360}}h^{10}f^{(10)}(\xi )}
h
f
(
1
)
(
x
3
)
=
1
2520
(
−
10
f
0
+
135
f
1
−
1080
f
2
−
1554
f
3
+
3780
f
4
−
1890
f
5
+
840
f
6
−
270
f
7
+
54
f
8
−
5
f
9
)
+
1
840
h
10
f
(
10
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{3})={\frac {1}{2520}}(-10f_{0}+135f_{1}-1080f_{2}-1554f_{3}+3780f_{4}-1890f_{5}+840f_{6}-270f_{7}+54f_{8}-5f_{9})+{\frac {1}{840}}h^{10}f^{(10)}(\xi )}
h
f
(
1
)
(
x
4
)
=
1
2520
(
5
f
0
−
60
f
1
+
360
f
2
−
1680
f
3
−
504
f
4
+
2520
f
5
−
840
f
6
+
240
f
7
−
45
f
8
+
4
f
9
)
−
1
1260
h
10
f
(
10
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{4})={\frac {1}{2520}}(5f_{0}-60f_{1}+360f_{2}-1680f_{3}-504f_{4}+2520f_{5}-840f_{6}+240f_{7}-45f_{8}+4f_{9})-{\frac {1}{1260}}h^{10}f^{(10)}(\xi )}
h
f
(
1
)
(
x
5
)
=
1
2520
(
−
4
f
0
+
45
f
1
−
240
f
2
+
840
f
3
−
2520
f
4
+
504
f
5
+
1680
f
6
−
360
f
7
+
60
f
8
−
5
f
9
)
+
1
1260
h
10
f
(
10
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{5})={\frac {1}{2520}}(-4f_{0}+45f_{1}-240f_{2}+840f_{3}-2520f_{4}+504f_{5}+1680f_{6}-360f_{7}+60f_{8}-5f_{9})+{\frac {1}{1260}}h^{10}f^{(10)}(\xi )}
h
f
(
1
)
(
x
6
)
=
1
2520
(
5
f
0
−
54
f
1
+
270
f
2
−
840
f
3
+
1890
f
4
−
3780
f
5
+
1554
f
6
+
1080
f
7
−
135
f
8
+
10
f
9
)
−
1
840
h
10
f
(
10
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{6})={\frac {1}{2520}}(5f_{0}-54f_{1}+270f_{2}-840f_{3}+1890f_{4}-3780f_{5}+1554f_{6}+1080f_{7}-135f_{8}+10f_{9})-{\frac {1}{840}}h^{10}f^{(10)}(\xi )}
h
f
(
1
)
(
x
7
)
=
1
2520
(
−
10
f
0
+
105
f
1
−
504
f
2
+
1470
f
3
−
2940
f
4
+
4410
f
5
−
5880
f
6
+
2754
f
7
+
630
f
8
−
35
f
9
)
+
1
360
h
10
f
(
10
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{7})={\frac {1}{2520}}(-10f_{0}+105f_{1}-504f_{2}+1470f_{3}-2940f_{4}+4410f_{5}-5880f_{6}+2754f_{7}+630f_{8}-35f_{9})+{\frac {1}{360}}h^{10}f^{(10)}(\xi )}
h
f
(
1
)
(
x
8
)
=
1
2520
(
35
f
0
−
360
f
1
+
1680
f
2
−
4704
f
3
+
8820
f
4
−
11760
f
5
+
11760
f
6
−
10080
f
7
+
4329
f
8
+
280
f
9
)
−
1
90
h
10
f
(
10
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{8})={\frac {1}{2520}}(35f_{0}-360f_{1}+1680f_{2}-4704f_{3}+8820f_{4}-11760f_{5}+11760f_{6}-10080f_{7}+4329f_{8}+280f_{9})-{\frac {1}{90}}h^{10}f^{(10)}(\xi )}
h
f
(
1
)
(
x
9
)
=
1
2520
(
−
280
f
0
+
2835
f
1
−
12960
f
2
+
35280
f
3
−
63504
f
4
+
79380
f
5
−
70560
f
6
+
45360
f
7
−
22680
f
8
+
7129
f
9
)
+
1
10
h
10
f
(
10
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{9})={\frac {1}{2520}}(-280f_{0}+2835f_{1}-12960f_{2}+35280f_{3}-63504f_{4}+79380f_{5}-70560f_{6}+45360f_{7}-22680f_{8}+7129f_{9})+{\frac {1}{10}}h^{10}f^{(10)}(\xi )}
10-punts 2e afgeleide:
h
2
f
(
2
)
(
x
0
)
=
1
5040
(
32575
f
0
−
165924
f
1
+
422568
f
2
−
704368
f
3
+
818874
f
4
−
667800
f
5
+
375704
f
6
−
139248
f
7
+
30663
f
8
−
3044
f
9
)
+
7129
12600
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{0})={\frac {1}{5040}}(32575f_{0}-165924f_{1}+422568f_{2}-704368f_{3}+818874f_{4}-667800f_{5}+375704f_{6}-139248f_{7}+30663f_{8}-3044f_{9})+{\frac {7129}{12600}}h^{10}f^{(10)}(\xi )}
h
2
f
(
2
)
(
x
1
)
=
1
5040
(
3044
f
0
+
2135
f
1
−
28944
f
2
+
57288
f
3
−
65128
f
4
+
51786
f
5
−
28560
f
6
+
10424
f
7
−
2268
f
8
+
223
f
9
)
−
481
12600
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{1})={\frac {1}{5040}}(3044f_{0}+2135f_{1}-28944f_{2}+57288f_{3}-65128f_{4}+51786f_{5}-28560f_{6}+10424f_{7}-2268f_{8}+223f_{9})-{\frac {481}{12600}}h^{10}f^{(10)}(\xi )}
h
2
f
(
2
)
(
x
2
)
=
1
5040
(
−
223
f
0
+
5274
f
1
−
7900
f
2
−
2184
f
3
+
10458
f
4
−
8932
f
5
+
4956
f
6
−
1800
f
7
+
389
f
8
−
38
f
9
)
+
17
2800
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{2})={\frac {1}{5040}}(-223f_{0}+5274f_{1}-7900f_{2}-2184f_{3}+10458f_{4}-8932f_{5}+4956f_{6}-1800f_{7}+389f_{8}-38f_{9})+{\frac {17}{2800}}h^{10}f^{(10)}(\xi )}
h
2
f
(
2
)
(
x
3
)
=
1
5040
(
38
f
0
−
603
f
1
+
6984
f
2
−
12460
f
3
+
5796
f
4
+
882
f
5
−
952
f
6
+
396
f
7
−
90
f
8
+
9
f
9
)
−
37
25200
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{3})={\frac {1}{5040}}(38f_{0}-603f_{1}+6984f_{2}-12460f_{3}+5796f_{4}+882f_{5}-952f_{6}+396f_{7}-90f_{8}+9f_{9})-{\frac {37}{25200}}h^{10}f^{(10)}(\xi )}
h
2
f
(
2
)
(
x
4
)
=
1
5040
(
−
9
f
0
+
128
f
1
−
1008
f
2
+
8064
f
3
−
14350
f
4
+
8064
f
5
−
1008
f
6
+
128
f
7
−
9
f
8
)
+
1
3150
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{4})={\frac {1}{5040}}(-9f_{0}+128f_{1}-1008f_{2}+8064f_{3}-14350f_{4}+8064f_{5}-1008f_{6}+128f_{7}-9f_{8})+{\frac {1}{3150}}h^{10}f^{(10)}(\xi )}
h
2
f
(
2
)
(
x
5
)
=
1
5040
(
−
9
f
1
+
128
f
2
−
1008
f
3
+
8064
f
4
−
14350
f
5
+
8064
f
6
−
1008
f
7
+
128
f
8
−
9
f
9
)
+
1
3150
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{5})={\frac {1}{5040}}(-9f_{1}+128f_{2}-1008f_{3}+8064f_{4}-14350f_{5}+8064f_{6}-1008f_{7}+128f_{8}-9f_{9})+{\frac {1}{3150}}h^{10}f^{(10)}(\xi )}
h
2
f
(
2
)
(
x
6
)
=
1
5040
(
9
f
0
−
90
f
1
+
396
f
2
−
952
f
3
+
882
f
4
+
5796
f
5
−
12460
f
6
+
6984
f
7
−
603
f
8
+
38
f
9
)
−
37
25200
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{6})={\frac {1}{5040}}(9f_{0}-90f_{1}+396f_{2}-952f_{3}+882f_{4}+5796f_{5}-12460f_{6}+6984f_{7}-603f_{8}+38f_{9})-{\frac {37}{25200}}h^{10}f^{(10)}(\xi )}
h
2
f
(
2
)
(
x
7
)
=
1
5040
(
−
38
f
0
+
389
f
1
−
1800
f
2
+
4956
f
3
−
8932
f
4
+
10458
f
5
−
2184
f
6
−
7900
f
7
+
5274
f
8
−
223
f
9
)
+
17
2800
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{7})={\frac {1}{5040}}(-38f_{0}+389f_{1}-1800f_{2}+4956f_{3}-8932f_{4}+10458f_{5}-2184f_{6}-7900f_{7}+5274f_{8}-223f_{9})+{\frac {17}{2800}}h^{10}f^{(10)}(\xi )}
h
2
f
(
2
)
(
x
8
)
=
1
5040
(
223
f
0
−
2268
f
1
+
10424
f
2
−
28560
f
3
+
51786
f
4
−
65128
f
5
+
57288
f
6
−
28944
f
7
+
2135
f
8
+
3044
f
9
)
−
481
12600
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{8})={\frac {1}{5040}}(223f_{0}-2268f_{1}+10424f_{2}-28560f_{3}+51786f_{4}-65128f_{5}+57288f_{6}-28944f_{7}+2135f_{8}+3044f_{9})-{\frac {481}{12600}}h^{10}f^{(10)}(\xi )}
h
2
f
(
2
)
(
x
9
)
=
1
5040
(
−
3044
f
0
+
30663
f
1
−
139248
f
2
+
375704
f
3
−
667800
f
4
+
818874
f
5
−
704368
f
6
+
422568
f
7
−
165924
f
8
+
32575
f
9
)
+
7129
12600
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{9})={\frac {1}{5040}}(-3044f_{0}+30663f_{1}-139248f_{2}+375704f_{3}-667800f_{4}+818874f_{5}-704368f_{6}+422568f_{7}-165924f_{8}+32575f_{9})+{\frac {7129}{12600}}h^{10}f^{(10)}(\xi )}
10-punts 3e afgeleide:
h
3
f
(
3
)
(
x
0
)
=
1
15120
(
−
180920
f
0
+
1145259
f
1
−
3375594
f
2
+
6095796
f
3
−
7392546
f
4
+
6185970
f
5
−
3540894
f
6
+
1328724
f
7
−
295326
f
8
+
29531
f
9
)
−
1303
672
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{0})={\frac {1}{15120}}(-180920f_{0}+1145259f_{1}-3375594f_{2}+6095796f_{3}-7392546f_{4}+6185970f_{5}-3540894f_{6}+1328724f_{7}-295326f_{8}+29531f_{9})-{\frac {1303}{672}}h^{10}f^{(10)}(\xi )}
h
3
f
(
3
)
(
x
1
)
=
1
15120
(
−
29531
f
0
+
114390
f
1
−
183636
f
2
+
168126
f
3
−
105714
f
4
+
49266
f
5
−
15540
f
6
+
2826
f
7
−
171
f
8
−
16
f
9
)
+
61
4320
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{1})={\frac {1}{15120}}(-29531f_{0}+114390f_{1}-183636f_{2}+168126f_{3}-105714f_{4}+49266f_{5}-15540f_{6}+2826f_{7}-171f_{8}-16f_{9})+{\frac {61}{4320}}h^{10}f^{(10)}(\xi )}
h
3
f
(
3
)
(
x
2
)
=
1
15120
(
16
f
0
−
29691
f
1
+
115110
f
2
−
185556
f
3
+
171486
f
4
−
109746
f
5
+
52626
f
6
−
17460
f
7
+
3546
f
8
−
331
f
9
)
+
79
6048
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{2})={\frac {1}{15120}}(16f_{0}-29691f_{1}+115110f_{2}-185556f_{3}+171486f_{4}-109746f_{5}+52626f_{6}-17460f_{7}+3546f_{8}-331f_{9})+{\frac {79}{6048}}h^{10}f^{(10)}(\xi )}
h
3
f
(
3
)
(
x
3
)
=
1
15120
(
331
f
0
−
3294
f
1
−
14796
f
2
+
75390
f
3
−
116046
f
4
+
88074
f
5
−
40236
f
6
+
12906
f
7
−
2565
f
8
+
236
f
9
)
−
89
10080
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{3})={\frac {1}{15120}}(331f_{0}-3294f_{1}-14796f_{2}+75390f_{3}-116046f_{4}+88074f_{5}-40236f_{6}+12906f_{7}-2565f_{8}+236f_{9})-{\frac {89}{10080}}h^{10}f^{(10)}(\xi )}
h
3
f
(
3
)
(
x
4
)
=
1
15120
(
−
236
f
0
+
2691
f
1
−
13914
f
2
+
13524
f
3
+
25830
f
4
−
56574
f
5
+
38514
f
6
−
11916
f
7
+
2286
f
8
−
205
f
9
)
+
41
6048
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{4})={\frac {1}{15120}}(-236f_{0}+2691f_{1}-13914f_{2}+13524f_{3}+25830f_{4}-56574f_{5}+38514f_{6}-11916f_{7}+2286f_{8}-205f_{9})+{\frac {41}{6048}}h^{10}f^{(10)}(\xi )}
h
3
f
(
3
)
(
x
5
)
=
1
15120
(
205
f
0
−
2286
f
1
+
11916
f
2
−
38514
f
3
+
56574
f
4
−
25830
f
5
−
13524
f
6
+
13914
f
7
−
2691
f
8
+
236
f
9
)
−
41
6048
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{5})={\frac {1}{15120}}(205f_{0}-2286f_{1}+11916f_{2}-38514f_{3}+56574f_{4}-25830f_{5}-13524f_{6}+13914f_{7}-2691f_{8}+236f_{9})-{\frac {41}{6048}}h^{10}f^{(10)}(\xi )}
h
3
f
(
3
)
(
x
6
)
=
1
15120
(
−
236
f
0
+
2565
f
1
−
12906
f
2
+
40236
f
3
−
88074
f
4
+
116046
f
5
−
75390
f
6
+
14796
f
7
+
3294
f
8
−
331
f
9
)
+
89
10080
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{6})={\frac {1}{15120}}(-236f_{0}+2565f_{1}-12906f_{2}+40236f_{3}-88074f_{4}+116046f_{5}-75390f_{6}+14796f_{7}+3294f_{8}-331f_{9})+{\frac {89}{10080}}h^{10}f^{(10)}(\xi )}
h
3
f
(
3
)
(
x
7
)
=
1
15120
(
331
f
0
−
3546
f
1
+
17460
f
2
−
52626
f
3
+
109746
f
4
−
171486
f
5
+
185556
f
6
−
115110
f
7
+
29691
f
8
−
16
f
9
)
−
79
6048
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{7})={\frac {1}{15120}}(331f_{0}-3546f_{1}+17460f_{2}-52626f_{3}+109746f_{4}-171486f_{5}+185556f_{6}-115110f_{7}+29691f_{8}-16f_{9})-{\frac {79}{6048}}h^{10}f^{(10)}(\xi )}
h
3
f
(
3
)
(
x
8
)
=
1
15120
(
16
f
0
+
171
f
1
−
2826
f
2
+
15540
f
3
−
49266
f
4
+
105714
f
5
−
168126
f
6
+
183636
f
7
−
114390
f
8
+
29531
f
9
)
−
61
4320
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{8})={\frac {1}{15120}}(16f_{0}+171f_{1}-2826f_{2}+15540f_{3}-49266f_{4}+105714f_{5}-168126f_{6}+183636f_{7}-114390f_{8}+29531f_{9})-{\frac {61}{4320}}h^{10}f^{(10)}(\xi )}
h
3
f
(
3
)
(
x
9
)
=
1
15120
(
−
29531
f
0
+
295326
f
1
−
1328724
f
2
+
3540894
f
3
−
6185970
f
4
+
7392546
f
5
−
6095796
f
6
+
3375594
f
7
−
1145259
f
8
+
180920
f
9
)
+
1303
672
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{9})={\frac {1}{15120}}(-29531f_{0}+295326f_{1}-1328724f_{2}+3540894f_{3}-6185970f_{4}+7392546f_{5}-6095796f_{6}+3375594f_{7}-1145259f_{8}+180920f_{9})+{\frac {1303}{672}}h^{10}f^{(10)}(\xi )}
10-punts 4e afgeleide:
h
4
f
(
4
)
(
x
0
)
=
1
240
(
4275
f
0
−
30668
f
1
+
99604
f
2
−
192624
f
3
+
244498
f
4
−
210920
f
5
+
123348
f
6
−
47024
f
7
+
10579
f
8
−
1068
f
9
)
+
4523
945
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{0})={\frac {1}{240}}(4275f_{0}-30668f_{1}+99604f_{2}-192624f_{3}+244498f_{4}-210920f_{5}+123348f_{6}-47024f_{7}+10579f_{8}-1068f_{9})+{\frac {4523}{945}}h^{10}f^{(10)}(\xi )}
h
4
f
(
4
)
(
x
1
)
=
1
240
(
1068
f
0
−
6405
f
1
+
17392
f
2
−
28556
f
3
+
31656
f
4
−
24638
f
5
+
13360
f
6
−
4812
f
7
+
1036
f
8
−
101
f
9
)
+
1271
3780
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{1})={\frac {1}{240}}(1068f_{0}-6405f_{1}+17392f_{2}-28556f_{3}+31656f_{4}-24638f_{5}+13360f_{6}-4812f_{7}+1036f_{8}-101f_{9})+{\frac {1271}{3780}}h^{10}f^{(10)}(\xi )}
h
4
f
(
4
)
(
x
2
)
=
1
240
(
101
f
0
+
58
f
1
−
1860
f
2
+
5272
f
3
−
7346
f
4
+
6204
f
5
−
3428
f
6
+
1240
f
7
−
267
f
8
+
26
f
9
)
−
1279
15120
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{2})={\frac {1}{240}}(101f_{0}+58f_{1}-1860f_{2}+5272f_{3}-7346f_{4}+6204f_{5}-3428f_{6}+1240f_{7}-267f_{8}+26f_{9})-{\frac {1279}{15120}}h^{10}f^{(10)}(\xi )}
h
4
f
(
4
)
(
x
3
)
=
1
240
(
−
26
f
0
+
361
f
1
−
1112
f
2
+
1260
f
3
−
188
f
4
−
794
f
5
+
744
f
6
−
308
f
7
+
70
f
8
−
7
f
9
)
+
359
15120
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{3})={\frac {1}{240}}(-26f_{0}+361f_{1}-1112f_{2}+1260f_{3}-188f_{4}-794f_{5}+744f_{6}-308f_{7}+70f_{8}-7f_{9})+{\frac {359}{15120}}h^{10}f^{(10)}(\xi )}
h
4
f
(
4
)
(
x
4
)
=
1
240
(
7
f
0
−
96
f
1
+
676
f
2
−
1952
f
3
+
2730
f
4
−
1952
f
5
+
676
f
6
−
96
f
7
+
7
f
8
)
−
41
7560
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{4})={\frac {1}{240}}(7f_{0}-96f_{1}+676f_{2}-1952f_{3}+2730f_{4}-1952f_{5}+676f_{6}-96f_{7}+7f_{8})-{\frac {41}{7560}}h^{10}f^{(10)}(\xi )}
h
4
f
(
4
)
(
x
5
)
=
1
240
(
7
f
1
−
96
f
2
+
676
f
3
−
1952
f
4
+
2730
f
5
−
1952
f
6
+
676
f
7
−
96
f
8
+
7
f
9
)
−
41
7560
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{5})={\frac {1}{240}}(7f_{1}-96f_{2}+676f_{3}-1952f_{4}+2730f_{5}-1952f_{6}+676f_{7}-96f_{8}+7f_{9})-{\frac {41}{7560}}h^{10}f^{(10)}(\xi )}
h
4
f
(
4
)
(
x
6
)
=
1
240
(
−
7
f
0
+
70
f
1
−
308
f
2
+
744
f
3
−
794
f
4
−
188
f
5
+
1260
f
6
−
1112
f
7
+
361
f
8
−
26
f
9
)
+
359
15120
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{6})={\frac {1}{240}}(-7f_{0}+70f_{1}-308f_{2}+744f_{3}-794f_{4}-188f_{5}+1260f_{6}-1112f_{7}+361f_{8}-26f_{9})+{\frac {359}{15120}}h^{10}f^{(10)}(\xi )}
h
4
f
(
4
)
(
x
7
)
=
1
240
(
26
f
0
−
267
f
1
+
1240
f
2
−
3428
f
3
+
6204
f
4
−
7346
f
5
+
5272
f
6
−
1860
f
7
+
58
f
8
+
101
f
9
)
−
1279
15120
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{7})={\frac {1}{240}}(26f_{0}-267f_{1}+1240f_{2}-3428f_{3}+6204f_{4}-7346f_{5}+5272f_{6}-1860f_{7}+58f_{8}+101f_{9})-{\frac {1279}{15120}}h^{10}f^{(10)}(\xi )}
h
4
f
(
4
)
(
x
8
)
=
1
240
(
−
101
f
0
+
1036
f
1
−
4812
f
2
+
13360
f
3
−
24638
f
4
+
31656
f
5
−
28556
f
6
+
17392
f
7
−
6405
f
8
+
1068
f
9
)
+
1271
3780
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{8})={\frac {1}{240}}(-101f_{0}+1036f_{1}-4812f_{2}+13360f_{3}-24638f_{4}+31656f_{5}-28556f_{6}+17392f_{7}-6405f_{8}+1068f_{9})+{\frac {1271}{3780}}h^{10}f^{(10)}(\xi )}
h
4
f
(
4
)
(
x
9
)
=
1
240
(
−
1068
f
0
+
10579
f
1
−
47024
f
2
+
123348
f
3
−
210920
f
4
+
244498
f
5
−
192624
f
6
+
99604
f
7
−
30668
f
8
+
4275
f
9
)
+
4523
945
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{9})={\frac {1}{240}}(-1068f_{0}+10579f_{1}-47024f_{2}+123348f_{3}-210920f_{4}+244498f_{5}-192624f_{6}+99604f_{7}-30668f_{8}+4275f_{9})+{\frac {4523}{945}}h^{10}f^{(10)}(\xi )}
10-punts 5e afgeleide:
h
5
f
(
5
)
(
x
0
)
=
1
144
(
−
3013
f
0
+
23421
f
1
−
81444
f
2
+
166476
f
3
−
220614
f
4
+
196638
f
5
−
117876
f
6
+
45804
f
7
−
10461
f
8
+
1069
f
9
)
−
285
32
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{0})={\frac {1}{144}}(-3013f_{0}+23421f_{1}-81444f_{2}+166476f_{3}-220614f_{4}+196638f_{5}-117876f_{6}+45804f_{7}-10461f_{8}+1069f_{9})-{\frac {285}{32}}h^{10}f^{(10)}(\xi )}
h
5
f
(
5
)
(
x
1
)
=
1
144
(
−
1069
f
0
+
7677
f
1
−
24684
f
2
+
46836
f
3
−
58014
f
4
+
48774
f
5
−
27852
f
6
+
10404
f
7
−
2301
f
8
+
229
f
9
)
−
427
288
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{1})={\frac {1}{144}}(-1069f_{0}+7677f_{1}-24684f_{2}+46836f_{3}-58014f_{4}+48774f_{5}-27852f_{6}+10404f_{7}-2301f_{8}+229f_{9})-{\frac {427}{288}}h^{10}f^{(10)}(\xi )}
h
5
f
(
5
)
(
x
2
)
=
1
144
(
−
229
f
0
+
1221
f
1
−
2628
f
2
+
2796
f
3
−
1254
f
4
−
306
f
5
+
684
f
6
−
372
f
7
+
99
f
8
−
11
f
9
)
+
31
288
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{2})={\frac {1}{144}}(-229f_{0}+1221f_{1}-2628f_{2}+2796f_{3}-1254f_{4}-306f_{5}+684f_{6}-372f_{7}+99f_{8}-11f_{9})+{\frac {31}{288}}h^{10}f^{(10)}(\xi )}
h
5
f
(
5
)
(
x
3
)
=
1
144
(
11
f
0
−
339
f
1
+
1716
f
2
−
3948
f
3
+
5106
f
4
−
4026
f
5
+
2004
f
6
−
636
f
7
+
123
f
8
−
11
f
9
)
+
1
32
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{3})={\frac {1}{144}}(11f_{0}-339f_{1}+1716f_{2}-3948f_{3}+5106f_{4}-4026f_{5}+2004f_{6}-636f_{7}+123f_{8}-11f_{9})+{\frac {1}{32}}h^{10}f^{(10)}(\xi )}
h
5
f
(
5
)
(
x
4
)
=
1
144
(
11
f
0
−
99
f
1
+
156
f
2
+
396
f
3
−
1638
f
4
+
2334
f
5
−
1716
f
6
+
684
f
7
−
141
f
8
+
13
f
9
)
−
13
288
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{4})={\frac {1}{144}}(11f_{0}-99f_{1}+156f_{2}+396f_{3}-1638f_{4}+2334f_{5}-1716f_{6}+684f_{7}-141f_{8}+13f_{9})-{\frac {13}{288}}h^{10}f^{(10)}(\xi )}
h
5
f
(
5
)
(
x
5
)
=
1
144
(
−
13
f
0
+
141
f
1
−
684
f
2
+
1716
f
3
−
2334
f
4
+
1638
f
5
−
396
f
6
−
156
f
7
+
99
f
8
−
11
f
9
)
+
13
288
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{5})={\frac {1}{144}}(-13f_{0}+141f_{1}-684f_{2}+1716f_{3}-2334f_{4}+1638f_{5}-396f_{6}-156f_{7}+99f_{8}-11f_{9})+{\frac {13}{288}}h^{10}f^{(10)}(\xi )}
h
5
f
(
5
)
(
x
6
)
=
1
144
(
11
f
0
−
123
f
1
+
636
f
2
−
2004
f
3
+
4026
f
4
−
5106
f
5
+
3948
f
6
−
1716
f
7
+
339
f
8
−
11
f
9
)
−
1
32
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{6})={\frac {1}{144}}(11f_{0}-123f_{1}+636f_{2}-2004f_{3}+4026f_{4}-5106f_{5}+3948f_{6}-1716f_{7}+339f_{8}-11f_{9})-{\frac {1}{32}}h^{10}f^{(10)}(\xi )}
h
5
f
(
5
)
(
x
7
)
=
1
144
(
11
f
0
−
99
f
1
+
372
f
2
−
684
f
3
+
306
f
4
+
1254
f
5
−
2796
f
6
+
2628
f
7
−
1221
f
8
+
229
f
9
)
−
31
288
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{7})={\frac {1}{144}}(11f_{0}-99f_{1}+372f_{2}-684f_{3}+306f_{4}+1254f_{5}-2796f_{6}+2628f_{7}-1221f_{8}+229f_{9})-{\frac {31}{288}}h^{10}f^{(10)}(\xi )}
h
5
f
(
5
)
(
x
8
)
=
1
144
(
−
229
f
0
+
2301
f
1
−
10404
f
2
+
27852
f
3
−
48774
f
4
+
58014
f
5
−
46836
f
6
+
24684
f
7
−
7677
f
8
+
1069
f
9
)
+
427
288
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{8})={\frac {1}{144}}(-229f_{0}+2301f_{1}-10404f_{2}+27852f_{3}-48774f_{4}+58014f_{5}-46836f_{6}+24684f_{7}-7677f_{8}+1069f_{9})+{\frac {427}{288}}h^{10}f^{(10)}(\xi )}
h
5
f
(
5
)
(
x
9
)
=
1
144
(
−
1069
f
0
+
10461
f
1
−
45804
f
2
+
117876
f
3
−
196638
f
4
+
220614
f
5
−
166476
f
6
+
81444
f
7
−
23421
f
8
+
3013
f
9
)
+
285
32
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{9})={\frac {1}{144}}(-1069f_{0}+10461f_{1}-45804f_{2}+117876f_{3}-196638f_{4}+220614f_{5}-166476f_{6}+81444f_{7}-23421f_{8}+3013f_{9})+{\frac {285}{32}}h^{10}f^{(10)}(\xi )}
10-punts 6e afgeleide:
h
6
f
(
6
)
(
x
0
)
=
1
4
(
75
f
0
−
616
f
1
+
2252
f
2
−
4812
f
3
+
6626
f
4
−
6100
f
5
+
3756
f
6
−
1492
f
7
+
347
f
8
−
36
f
9
)
+
3013
240
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{0})={\frac {1}{4}}(75f_{0}-616f_{1}+2252f_{2}-4812f_{3}+6626f_{4}-6100f_{5}+3756f_{6}-1492f_{7}+347f_{8}-36f_{9})+{\frac {3013}{240}}h^{10}f^{(10)}(\xi )}
h
6
f
(
6
)
(
x
1
)
=
1
4
(
36
f
0
−
285
f
1
+
1004
f
2
−
2068
f
3
+
2748
f
4
−
2446
f
5
+
1460
f
6
−
564
f
7
+
128
f
8
−
13
f
9
)
+
853
240
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{1})={\frac {1}{4}}(36f_{0}-285f_{1}+1004f_{2}-2068f_{3}+2748f_{4}-2446f_{5}+1460f_{6}-564f_{7}+128f_{8}-13f_{9})+{\frac {853}{240}}h^{10}f^{(10)}(\xi )}
h
6
f
(
6
)
(
x
2
)
=
1
4
(
13
f
0
−
94
f
1
+
300
f
2
−
556
f
3
+
662
f
4
−
528
f
5
+
284
f
6
−
100
f
7
+
21
f
8
−
2
f
9
)
+
73
240
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{2})={\frac {1}{4}}(13f_{0}-94f_{1}+300f_{2}-556f_{3}+662f_{4}-528f_{5}+284f_{6}-100f_{7}+21f_{8}-2f_{9})+{\frac {73}{240}}h^{10}f^{(10)}(\xi )}
h
6
f
(
6
)
(
x
3
)
=
1
4
(
2
f
0
−
7
f
1
−
4
f
2
+
60
f
3
−
136
f
4
+
158
f
5
−
108
f
6
+
44
f
7
−
10
f
8
+
f
9
)
−
47
240
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{3})={\frac {1}{4}}(2f_{0}-7f_{1}-4f_{2}+60f_{3}-136f_{4}+158f_{5}-108f_{6}+44f_{7}-10f_{8}+f_{9})-{\frac {47}{240}}h^{10}f^{(10)}(\xi )}
h
6
f
(
6
)
(
x
4
)
=
1
4
(
−
f
0
+
12
f
1
−
52
f
2
+
116
f
3
−
150
f
4
+
116
f
5
−
52
f
6
+
12
f
7
−
f
8
)
+
13
240
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{4})={\frac {1}{4}}(-f_{0}+12f_{1}-52f_{2}+116f_{3}-150f_{4}+116f_{5}-52f_{6}+12f_{7}-f_{8})+{\frac {13}{240}}h^{10}f^{(10)}(\xi )}
h
6
f
(
6
)
(
x
5
)
=
1
4
(
−
f
1
+
12
f
2
−
52
f
3
+
116
f
4
−
150
f
5
+
116
f
6
−
52
f
7
+
12
f
8
−
f
9
)
+
13
240
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{5})={\frac {1}{4}}(-f_{1}+12f_{2}-52f_{3}+116f_{4}-150f_{5}+116f_{6}-52f_{7}+12f_{8}-f_{9})+{\frac {13}{240}}h^{10}f^{(10)}(\xi )}
h
6
f
(
6
)
(
x
6
)
=
1
4
(
f
0
−
10
f
1
+
44
f
2
−
108
f
3
+
158
f
4
−
136
f
5
+
60
f
6
−
4
f
7
−
7
f
8
+
2
f
9
)
−
47
240
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{6})={\frac {1}{4}}(f_{0}-10f_{1}+44f_{2}-108f_{3}+158f_{4}-136f_{5}+60f_{6}-4f_{7}-7f_{8}+2f_{9})-{\frac {47}{240}}h^{10}f^{(10)}(\xi )}
h
6
f
(
6
)
(
x
7
)
=
1
4
(
−
2
f
0
+
21
f
1
−
100
f
2
+
284
f
3
−
528
f
4
+
662
f
5
−
556
f
6
+
300
f
7
−
94
f
8
+
13
f
9
)
+
73
240
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{7})={\frac {1}{4}}(-2f_{0}+21f_{1}-100f_{2}+284f_{3}-528f_{4}+662f_{5}-556f_{6}+300f_{7}-94f_{8}+13f_{9})+{\frac {73}{240}}h^{10}f^{(10)}(\xi )}
h
6
f
(
6
)
(
x
8
)
=
1
4
(
−
13
f
0
+
128
f
1
−
564
f
2
+
1460
f
3
−
2446
f
4
+
2748
f
5
−
2068
f
6
+
1004
f
7
−
285
f
8
+
36
f
9
)
+
853
240
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{8})={\frac {1}{4}}(-13f_{0}+128f_{1}-564f_{2}+1460f_{3}-2446f_{4}+2748f_{5}-2068f_{6}+1004f_{7}-285f_{8}+36f_{9})+{\frac {853}{240}}h^{10}f^{(10)}(\xi )}
h
6
f
(
6
)
(
x
9
)
=
1
4
(
−
36
f
0
+
347
f
1
−
1492
f
2
+
3756
f
3
−
6100
f
4
+
6626
f
5
−
4812
f
6
+
2252
f
7
−
616
f
8
+
75
f
9
)
+
3013
240
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{9})={\frac {1}{4}}(-36f_{0}+347f_{1}-1492f_{2}+3756f_{3}-6100f_{4}+6626f_{5}-4812f_{6}+2252f_{7}-616f_{8}+75f_{9})+{\frac {3013}{240}}h^{10}f^{(10)}(\xi )}
10-punts 7e afgeleide:
h
7
f
(
7
)
(
x
0
)
=
1
12
(
−
145
f
0
+
1239
f
1
−
4704
f
2
+
10416
f
3
−
14826
f
4
+
14070
f
5
−
8904
f
6
+
3624
f
7
−
861
f
8
+
91
f
9
)
−
105
8
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{7}f^{(7)}(x_{0})={\frac {1}{12}}(-145f_{0}+1239f_{1}-4704f_{2}+10416f_{3}-14826f_{4}+14070f_{5}-8904f_{6}+3624f_{7}-861f_{8}+91f_{9})-{\frac {105}{8}}h^{10}f^{(10)}(\xi )}
h
7
f
(
7
)
(
x
1
)
=
1
12
(
−
91
f
0
+
765
f
1
−
2856
f
2
+
6216
f
3
−
8694
f
4
+
8106
f
5
−
5040
f
6
+
2016
f
7
−
471
f
8
+
49
f
9
)
−
133
24
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{7}f^{(7)}(x_{1})={\frac {1}{12}}(-91f_{0}+765f_{1}-2856f_{2}+6216f_{3}-8694f_{4}+8106f_{5}-5040f_{6}+2016f_{7}-471f_{8}+49f_{9})-{\frac {133}{24}}h^{10}f^{(10)}(\xi )}
h
7
f
(
7
)
(
x
2
)
=
1
12
(
−
49
f
0
+
399
f
1
−
1440
f
2
+
3024
f
3
−
4074
f
4
+
3654
f
5
−
2184
f
6
+
840
f
7
−
189
f
8
+
19
f
9
)
−
35
24
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{7}f^{(7)}(x_{2})={\frac {1}{12}}(-49f_{0}+399f_{1}-1440f_{2}+3024f_{3}-4074f_{4}+3654f_{5}-2184f_{6}+840f_{7}-189f_{8}+19f_{9})-{\frac {35}{24}}h^{10}f^{(10)}(\xi )}
h
7
f
(
7
)
(
x
3
)
=
1
12
(
−
19
f
0
+
141
f
1
−
456
f
2
+
840
f
3
−
966
f
4
+
714
f
5
−
336
f
6
+
96
f
7
−
15
f
8
+
f
9
)
+
1
8
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{7}f^{(7)}(x_{3})={\frac {1}{12}}(-19f_{0}+141f_{1}-456f_{2}+840f_{3}-966f_{4}+714f_{5}-336f_{6}+96f_{7}-15f_{8}+f_{9})+{\frac {1}{8}}h^{10}f^{(10)}(\xi )}
h
7
f
(
7
)
(
x
4
)
=
1
12
(
−
f
0
−
9
f
1
+
96
f
2
−
336
f
3
+
630
f
4
−
714
f
5
+
504
f
6
−
216
f
7
+
51
f
8
−
5
f
9
)
+
5
24
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{7}f^{(7)}(x_{4})={\frac {1}{12}}(-f_{0}-9f_{1}+96f_{2}-336f_{3}+630f_{4}-714f_{5}+504f_{6}-216f_{7}+51f_{8}-5f_{9})+{\frac {5}{24}}h^{10}f^{(10)}(\xi )}
h
7
f
(
7
)
(
x
5
)
=
1
12
(
5
f
0
−
51
f
1
+
216
f
2
−
504
f
3
+
714
f
4
−
630
f
5
+
336
f
6
−
96
f
7
+
9
f
8
+
f
9
)
−
5
24
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{7}f^{(7)}(x_{5})={\frac {1}{12}}(5f_{0}-51f_{1}+216f_{2}-504f_{3}+714f_{4}-630f_{5}+336f_{6}-96f_{7}+9f_{8}+f_{9})-{\frac {5}{24}}h^{10}f^{(10)}(\xi )}
h
7
f
(
7
)
(
x
6
)
=
1
12
(
−
f
0
+
15
f
1
−
96
f
2
+
336
f
3
−
714
f
4
+
966
f
5
−
840
f
6
+
456
f
7
−
141
f
8
+
19
f
9
)
−
1
8
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{7}f^{(7)}(x_{6})={\frac {1}{12}}(-f_{0}+15f_{1}-96f_{2}+336f_{3}-714f_{4}+966f_{5}-840f_{6}+456f_{7}-141f_{8}+19f_{9})-{\frac {1}{8}}h^{10}f^{(10)}(\xi )}
h
7
f
(
7
)
(
x
7
)
=
1
12
(
−
19
f
0
+
189
f
1
−
840
f
2
+
2184
f
3
−
3654
f
4
+
4074
f
5
−
3024
f
6
+
1440
f
7
−
399
f
8
+
49
f
9
)
+
35
24
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{7}f^{(7)}(x_{7})={\frac {1}{12}}(-19f_{0}+189f_{1}-840f_{2}+2184f_{3}-3654f_{4}+4074f_{5}-3024f_{6}+1440f_{7}-399f_{8}+49f_{9})+{\frac {35}{24}}h^{10}f^{(10)}(\xi )}
h
7
f
(
7
)
(
x
8
)
=
1
12
(
−
49
f
0
+
471
f
1
−
2016
f
2
+
5040
f
3
−
8106
f
4
+
8694
f
5
−
6216
f
6
+
2856
f
7
−
765
f
8
+
91
f
9
)
+
133
24
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{7}f^{(7)}(x_{8})={\frac {1}{12}}(-49f_{0}+471f_{1}-2016f_{2}+5040f_{3}-8106f_{4}+8694f_{5}-6216f_{6}+2856f_{7}-765f_{8}+91f_{9})+{\frac {133}{24}}h^{10}f^{(10)}(\xi )}
h
7
f
(
7
)
(
x
9
)
=
1
12
(
−
91
f
0
+
861
f
1
−
3624
f
2
+
8904
f
3
−
14070
f
4
+
14826
f
5
−
10416
f
6
+
4704
f
7
−
1239
f
8
+
145
f
9
)
+
105
8
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{7}f^{(7)}(x_{9})={\frac {1}{12}}(-91f_{0}+861f_{1}-3624f_{2}+8904f_{3}-14070f_{4}+14826f_{5}-10416f_{6}+4704f_{7}-1239f_{8}+145f_{9})+{\frac {105}{8}}h^{10}f^{(10)}(\xi )}
10-punts 8e afgeleide:
h
8
f
(
8
)
(
x
0
)
=
5
f
0
−
44
f
1
+
172
f
2
−
392
f
3
+
574
f
4
−
560
f
5
+
364
f
6
−
152
f
7
+
37
f
8
−
4
f
9
+
29
3
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{8}f^{(8)}(x_{0})=5f_{0}-44f_{1}+172f_{2}-392f_{3}+574f_{4}-560f_{5}+364f_{6}-152f_{7}+37f_{8}-4f_{9}+{\frac {29}{3}}h^{10}f^{(10)}(\xi )}
h
8
f
(
8
)
(
x
1
)
=
4
f
0
−
35
f
1
+
136
f
2
−
308
f
3
+
448
f
4
−
434
f
5
+
280
f
6
−
116
f
7
+
28
f
8
−
3
f
9
+
17
3
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{8}f^{(8)}(x_{1})=4f_{0}-35f_{1}+136f_{2}-308f_{3}+448f_{4}-434f_{5}+280f_{6}-116f_{7}+28f_{8}-3f_{9}+{\frac {17}{3}}h^{10}f^{(10)}(\xi )}
h
8
f
(
8
)
(
x
2
)
=
3
f
0
−
26
f
1
+
100
f
2
−
224
f
3
+
322
f
4
−
308
f
5
+
196
f
6
−
80
f
7
+
19
f
8
−
2
f
9
+
8
3
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{8}f^{(8)}(x_{2})=3f_{0}-26f_{1}+100f_{2}-224f_{3}+322f_{4}-308f_{5}+196f_{6}-80f_{7}+19f_{8}-2f_{9}+{\frac {8}{3}}h^{10}f^{(10)}(\xi )}
h
8
f
(
8
)
(
x
3
)
=
2
f
0
−
17
f
1
+
64
f
2
−
140
f
3
+
196
f
4
−
182
f
5
+
112
f
6
−
44
f
7
+
10
f
8
−
f
9
+
2
3
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{8}f^{(8)}(x_{3})=2f_{0}-17f_{1}+64f_{2}-140f_{3}+196f_{4}-182f_{5}+112f_{6}-44f_{7}+10f_{8}-f_{9}+{\frac {2}{3}}h^{10}f^{(10)}(\xi )}
h
8
f
(
8
)
(
x
4
)
=
f
0
−
8
f
1
+
28
f
2
−
56
f
3
+
70
f
4
−
56
f
5
+
28
f
6
−
8
f
7
+
f
8
−
1
3
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{8}f^{(8)}(x_{4})=f_{0}-8f_{1}+28f_{2}-56f_{3}+70f_{4}-56f_{5}+28f_{6}-8f_{7}+f_{8}-{\frac {1}{3}}h^{10}f^{(10)}(\xi )}
h
8
f
(
8
)
(
x
5
)
=
f
1
−
8
f
2
+
28
f
3
−
56
f
4
+
70
f
5
−
56
f
6
+
28
f
7
−
8
f
8
+
f
9
−
1
3
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{8}f^{(8)}(x_{5})=f_{1}-8f_{2}+28f_{3}-56f_{4}+70f_{5}-56f_{6}+28f_{7}-8f_{8}+f_{9}-{\frac {1}{3}}h^{10}f^{(10)}(\xi )}
h
8
f
(
8
)
(
x
6
)
=
−
f
0
+
10
f
1
−
44
f
2
+
112
f
3
−
182
f
4
+
196
f
5
−
140
f
6
+
64
f
7
−
17
f
8
+
2
f
9
+
2
3
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{8}f^{(8)}(x_{6})=-f_{0}+10f_{1}-44f_{2}+112f_{3}-182f_{4}+196f_{5}-140f_{6}+64f_{7}-17f_{8}+2f_{9}+{\frac {2}{3}}h^{10}f^{(10)}(\xi )}
h
8
f
(
8
)
(
x
7
)
=
−
2
f
0
+
19
f
1
−
80
f
2
+
196
f
3
−
308
f
4
+
322
f
5
−
224
f
6
+
100
f
7
−
26
f
8
+
3
f
9
+
8
3
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{8}f^{(8)}(x_{7})=-2f_{0}+19f_{1}-80f_{2}+196f_{3}-308f_{4}+322f_{5}-224f_{6}+100f_{7}-26f_{8}+3f_{9}+{\frac {8}{3}}h^{10}f^{(10)}(\xi )}
h
8
f
(
8
)
(
x
8
)
=
−
3
f
0
+
28
f
1
−
116
f
2
+
280
f
3
−
434
f
4
+
448
f
5
−
308
f
6
+
136
f
7
−
35
f
8
+
4
f
9
+
17
3
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{8}f^{(8)}(x_{8})=-3f_{0}+28f_{1}-116f_{2}+280f_{3}-434f_{4}+448f_{5}-308f_{6}+136f_{7}-35f_{8}+4f_{9}+{\frac {17}{3}}h^{10}f^{(10)}(\xi )}
h
8
f
(
8
)
(
x
9
)
=
−
4
f
0
+
37
f
1
−
152
f
2
+
364
f
3
−
560
f
4
+
574
f
5
−
392
f
6
+
172
f
7
−
44
f
8
+
5
f
9
+
29
3
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{8}f^{(8)}(x_{9})=-4f_{0}+37f_{1}-152f_{2}+364f_{3}-560f_{4}+574f_{5}-392f_{6}+172f_{7}-44f_{8}+5f_{9}+{\frac {29}{3}}h^{10}f^{(10)}(\xi )}
10-punts 9e afgeleide:
h
9
f
(
9
)
(
x
0
)
=
−
f
0
+
9
f
1
−
36
f
2
+
84
f
3
−
126
f
4
+
126
f
5
−
84
f
6
+
36
f
7
−
9
f
8
+
f
9
−
9
2
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{9}f^{(9)}(x_{0})=-f_{0}+9f_{1}-36f_{2}+84f_{3}-126f_{4}+126f_{5}-84f_{6}+36f_{7}-9f_{8}+f_{9}-{\frac {9}{2}}h^{10}f^{(10)}(\xi )}
h
9
f
(
9
)
(
x
1
)
=
−
f
0
+
9
f
1
−
36
f
2
+
84
f
3
−
126
f
4
+
126
f
5
−
84
f
6
+
36
f
7
−
9
f
8
+
f
9
−
7
2
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{9}f^{(9)}(x_{1})=-f_{0}+9f_{1}-36f_{2}+84f_{3}-126f_{4}+126f_{5}-84f_{6}+36f_{7}-9f_{8}+f_{9}-{\frac {7}{2}}h^{10}f^{(10)}(\xi )}
h
9
f
(
9
)
(
x
2
)
=
−
f
0
+
9
f
1
−
36
f
2
+
84
f
3
−
126
f
4
+
126
f
5
−
84
f
6
+
36
f
7
−
9
f
8
+
f
9
−
5
2
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{9}f^{(9)}(x_{2})=-f_{0}+9f_{1}-36f_{2}+84f_{3}-126f_{4}+126f_{5}-84f_{6}+36f_{7}-9f_{8}+f_{9}-{\frac {5}{2}}h^{10}f^{(10)}(\xi )}
h
9
f
(
9
)
(
x
3
)
=
−
f
0
+
9
f
1
−
36
f
2
+
84
f
3
−
126
f
4
+
126
f
5
−
84
f
6
+
36
f
7
−
9
f
8
+
f
9
−
3
2
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{9}f^{(9)}(x_{3})=-f_{0}+9f_{1}-36f_{2}+84f_{3}-126f_{4}+126f_{5}-84f_{6}+36f_{7}-9f_{8}+f_{9}-{\frac {3}{2}}h^{10}f^{(10)}(\xi )}
h
9
f
(
9
)
(
x
4
)
=
−
f
0
+
9
f
1
−
36
f
2
+
84
f
3
−
126
f
4
+
126
f
5
−
84
f
6
+
36
f
7
−
9
f
8
+
f
9
−
1
2
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{9}f^{(9)}(x_{4})=-f_{0}+9f_{1}-36f_{2}+84f_{3}-126f_{4}+126f_{5}-84f_{6}+36f_{7}-9f_{8}+f_{9}-{\frac {1}{2}}h^{10}f^{(10)}(\xi )}
h
9
f
(
9
)
(
x
5
)
=
−
f
0
+
9
f
1
−
36
f
2
+
84
f
3
−
126
f
4
+
126
f
5
−
84
f
6
+
36
f
7
−
9
f
8
+
f
9
+
1
2
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{9}f^{(9)}(x_{5})=-f_{0}+9f_{1}-36f_{2}+84f_{3}-126f_{4}+126f_{5}-84f_{6}+36f_{7}-9f_{8}+f_{9}+{\frac {1}{2}}h^{10}f^{(10)}(\xi )}
h
9
f
(
9
)
(
x
6
)
=
−
f
0
+
9
f
1
−
36
f
2
+
84
f
3
−
126
f
4
+
126
f
5
−
84
f
6
+
36
f
7
−
9
f
8
+
f
9
+
3
2
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{9}f^{(9)}(x_{6})=-f_{0}+9f_{1}-36f_{2}+84f_{3}-126f_{4}+126f_{5}-84f_{6}+36f_{7}-9f_{8}+f_{9}+{\frac {3}{2}}h^{10}f^{(10)}(\xi )}
h
9
f
(
9
)
(
x
7
)
=
−
f
0
+
9
f
1
−
36
f
2
+
84
f
3
−
126
f
4
+
126
f
5
−
84
f
6
+
36
f
7
−
9
f
8
+
f
9
+
5
2
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{9}f^{(9)}(x_{7})=-f_{0}+9f_{1}-36f_{2}+84f_{3}-126f_{4}+126f_{5}-84f_{6}+36f_{7}-9f_{8}+f_{9}+{\frac {5}{2}}h^{10}f^{(10)}(\xi )}
h
9
f
(
9
)
(
x
8
)
=
−
f
0
+
9
f
1
−
36
f
2
+
84
f
3
−
126
f
4
+
126
f
5
−
84
f
6
+
36
f
7
−
9
f
8
+
f
9
+
7
2
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{9}f^{(9)}(x_{8})=-f_{0}+9f_{1}-36f_{2}+84f_{3}-126f_{4}+126f_{5}-84f_{6}+36f_{7}-9f_{8}+f_{9}+{\frac {7}{2}}h^{10}f^{(10)}(\xi )}
h
9
f
(
9
)
(
x
9
)
=
−
f
0
+
9
f
1
−
36
f
2
+
84
f
3
−
126
f
4
+
126
f
5
−
84
f
6
+
36
f
7
−
9
f
8
+
f
9
+
9
2
h
10
f
(
10
)
(
ξ
)
{\displaystyle h^{9}f^{(9)}(x_{9})=-f_{0}+9f_{1}-36f_{2}+84f_{3}-126f_{4}+126f_{5}-84f_{6}+36f_{7}-9f_{8}+f_{9}+{\frac {9}{2}}h^{10}f^{(10)}(\xi )}
11-punts 1e afgeleide:
h
f
(
1
)
(
x
0
)
=
1
2520
(
−
7381
f
0
+
25200
f
1
−
56700
f
2
+
100800
f
3
−
132300
f
4
+
127008
f
5
−
88200
f
6
+
43200
f
7
−
14175
f
8
+
2800
f
9
−
252
f
10
)
+
1
11
h
11
f
(
11
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{0})={\frac {1}{2520}}(-7381f_{0}+25200f_{1}-56700f_{2}+100800f_{3}-132300f_{4}+127008f_{5}-88200f_{6}+43200f_{7}-14175f_{8}+2800f_{9}-252f_{10})+{\frac {1}{11}}h^{11}f^{(11)}(\xi )}
h
f
(
1
)
(
x
1
)
=
1
2520
(
−
252
f
0
−
4609
f
1
+
11340
f
2
−
15120
f
3
+
17640
f
4
−
15876
f
5
+
10584
f
6
−
5040
f
7
+
1620
f
8
−
315
f
9
+
28
f
10
)
−
1
110
h
11
f
(
11
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{1})={\frac {1}{2520}}(-252f_{0}-4609f_{1}+11340f_{2}-15120f_{3}+17640f_{4}-15876f_{5}+10584f_{6}-5040f_{7}+1620f_{8}-315f_{9}+28f_{10})-{\frac {1}{110}}h^{11}f^{(11)}(\xi )}
h
f
(
1
)
(
x
2
)
=
1
2520
(
28
f
0
−
560
f
1
−
3069
f
2
+
6720
f
3
−
5880
f
4
+
4704
f
5
−
2940
f
6
+
1344
f
7
−
420
f
8
+
80
f
9
−
7
f
10
)
+
1
495
h
11
f
(
11
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{2})={\frac {1}{2520}}(28f_{0}-560f_{1}-3069f_{2}+6720f_{3}-5880f_{4}+4704f_{5}-2940f_{6}+1344f_{7}-420f_{8}+80f_{9}-7f_{10})+{\frac {1}{495}}h^{11}f^{(11)}(\xi )}
h
f
(
1
)
(
x
3
)
=
1
2520
(
−
7
f
0
+
105
f
1
−
945
f
2
−
1914
f
3
+
4410
f
4
−
2646
f
5
+
1470
f
6
−
630
f
7
+
189
f
8
−
35
f
9
+
3
f
10
)
−
1
1320
h
11
f
(
11
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{3})={\frac {1}{2520}}(-7f_{0}+105f_{1}-945f_{2}-1914f_{3}+4410f_{4}-2646f_{5}+1470f_{6}-630f_{7}+189f_{8}-35f_{9}+3f_{10})-{\frac {1}{1320}}h^{11}f^{(11)}(\xi )}
h
f
(
1
)
(
x
4
)
=
1
2520
(
3
f
0
−
40
f
1
+
270
f
2
−
1440
f
3
−
924
f
4
+
3024
f
5
−
1260
f
6
+
480
f
7
−
135
f
8
+
24
f
9
−
2
f
10
)
+
1
2310
h
11
f
(
11
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{4})={\frac {1}{2520}}(3f_{0}-40f_{1}+270f_{2}-1440f_{3}-924f_{4}+3024f_{5}-1260f_{6}+480f_{7}-135f_{8}+24f_{9}-2f_{10})+{\frac {1}{2310}}h^{11}f^{(11)}(\xi )}
h
f
(
1
)
(
x
5
)
=
1
2520
(
−
2
f
0
+
25
f
1
−
150
f
2
+
600
f
3
−
2100
f
4
+
2100
f
6
−
600
f
7
+
150
f
8
−
25
f
9
+
2
f
10
)
−
1
2772
h
11
f
(
11
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{5})={\frac {1}{2520}}(-2f_{0}+25f_{1}-150f_{2}+600f_{3}-2100f_{4}+2100f_{6}-600f_{7}+150f_{8}-25f_{9}+2f_{10})-{\frac {1}{2772}}h^{11}f^{(11)}(\xi )}
h
f
(
1
)
(
x
6
)
=
1
2520
(
2
f
0
−
24
f
1
+
135
f
2
−
480
f
3
+
1260
f
4
−
3024
f
5
+
924
f
6
+
1440
f
7
−
270
f
8
+
40
f
9
−
3
f
10
)
+
1
2310
h
11
f
(
11
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{6})={\frac {1}{2520}}(2f_{0}-24f_{1}+135f_{2}-480f_{3}+1260f_{4}-3024f_{5}+924f_{6}+1440f_{7}-270f_{8}+40f_{9}-3f_{10})+{\frac {1}{2310}}h^{11}f^{(11)}(\xi )}
h
f
(
1
)
(
x
7
)
=
1
2520
(
−
3
f
0
+
35
f
1
−
189
f
2
+
630
f
3
−
1470
f
4
+
2646
f
5
−
4410
f
6
+
1914
f
7
+
945
f
8
−
105
f
9
+
7
f
10
)
−
1
1320
h
11
f
(
11
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{7})={\frac {1}{2520}}(-3f_{0}+35f_{1}-189f_{2}+630f_{3}-1470f_{4}+2646f_{5}-4410f_{6}+1914f_{7}+945f_{8}-105f_{9}+7f_{10})-{\frac {1}{1320}}h^{11}f^{(11)}(\xi )}
h
f
(
1
)
(
x
8
)
=
1
2520
(
7
f
0
−
80
f
1
+
420
f
2
−
1344
f
3
+
2940
f
4
−
4704
f
5
+
5880
f
6
−
6720
f
7
+
3069
f
8
+
560
f
9
−
28
f
10
)
+
1
495
h
11
f
(
11
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{8})={\frac {1}{2520}}(7f_{0}-80f_{1}+420f_{2}-1344f_{3}+2940f_{4}-4704f_{5}+5880f_{6}-6720f_{7}+3069f_{8}+560f_{9}-28f_{10})+{\frac {1}{495}}h^{11}f^{(11)}(\xi )}
h
f
(
1
)
(
x
9
)
=
1
2520
(
−
28
f
0
+
315
f
1
−
1620
f
2
+
5040
f
3
−
10584
f
4
+
15876
f
5
−
17640
f
6
+
15120
f
7
−
11340
f
8
+
4609
f
9
+
252
f
10
)
−
1
110
h
11
f
(
11
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{9})={\frac {1}{2520}}(-28f_{0}+315f_{1}-1620f_{2}+5040f_{3}-10584f_{4}+15876f_{5}-17640f_{6}+15120f_{7}-11340f_{8}+4609f_{9}+252f_{10})-{\frac {1}{110}}h^{11}f^{(11)}(\xi )}
h
f
(
1
)
(
x
10
)
=
1
2520
(
252
f
0
−
2800
f
1
+
14175
f
2
−
43200
f
3
+
88200
f
4
−
127008
f
5
+
132300
f
6
−
100800
f
7
+
56700
f
8
−
25200
f
9
+
7381
f
10
)
+
1
11
h
11
f
(
11
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{10})={\frac {1}{2520}}(252f_{0}-2800f_{1}+14175f_{2}-43200f_{3}+88200f_{4}-127008f_{5}+132300f_{6}-100800f_{7}+56700f_{8}-25200f_{9}+7381f_{10})+{\frac {1}{11}}h^{11}f^{(11)}(\xi )}
11-punts 2e afgeleide:
h
2
f
(
2
)
(
x
0
)
=
1
25200
(
177133
f
0
−
972200
f
1
+
2754450
f
2
−
5232800
f
3
+
7088550
f
4
−
6932016
f
5
+
4872700
f
6
−
2407200
f
7
+
794925
f
8
−
157800
f
9
+
14258
f
10
)
−
671
1260
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{0})={\frac {1}{25200}}(177133f_{0}-972200f_{1}+2754450f_{2}-5232800f_{3}+7088550f_{4}-6932016f_{5}+4872700f_{6}-2407200f_{7}+794925f_{8}-157800f_{9}+14258f_{10})-{\frac {671}{1260}}h^{11}f^{(11)}(\xi )}
h
2
f
(
2
)
(
x
1
)
=
1
25200
(
14258
f
0
+
20295
f
1
−
188010
f
2
+
401880
f
3
−
527660
f
4
+
501354
f
5
−
344820
f
6
+
167560
f
7
−
54630
f
8
+
10735
f
9
−
962
f
10
)
+
419
12600
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{1})={\frac {1}{25200}}(14258f_{0}+20295f_{1}-188010f_{2}+401880f_{3}-527660f_{4}+501354f_{5}-344820f_{6}+167560f_{7}-54630f_{8}+10735f_{9}-962f_{10})+{\frac {419}{12600}}h^{11}f^{(11)}(\xi )}
h
2
f
(
2
)
(
x
2
)
=
1
25200
(
−
962
f
0
+
24840
f
1
−
32615
f
2
−
29280
f
3
+
84420
f
4
−
83216
f
5
+
56910
f
6
−
27360
f
7
+
8830
f
8
−
1720
f
9
+
153
f
10
)
−
31
6300
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{2})={\frac {1}{25200}}(-962f_{0}+24840f_{1}-32615f_{2}-29280f_{3}+84420f_{4}-83216f_{5}+56910f_{6}-27360f_{7}+8830f_{8}-1720f_{9}+153f_{10})-{\frac {31}{6300}}h^{11}f^{(11)}(\xi )}
h
2
f
(
2
)
(
x
3
)
=
1
25200
(
153
f
0
−
2645
f
1
+
33255
f
2
−
57860
f
3
+
21210
f
4
+
13734
f
5
−
12530
f
6
+
6420
f
7
−
2115
f
8
+
415
f
9
−
37
f
10
)
+
29
25200
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{3})={\frac {1}{25200}}(153f_{0}-2645f_{1}+33255f_{2}-57860f_{3}+21210f_{4}+13734f_{5}-12530f_{6}+6420f_{7}-2115f_{8}+415f_{9}-37f_{10})+{\frac {29}{25200}}h^{11}f^{(11)}(\xi )}
h
2
f
(
2
)
(
x
4
)
=
1
25200
(
−
37
f
0
+
560
f
1
−
4680
f
2
+
39360
f
3
−
70070
f
4
+
38304
f
5
−
3360
f
6
−
320
f
7
+
315
f
8
−
80
f
9
+
8
f
10
)
−
1
3150
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{4})={\frac {1}{25200}}(-37f_{0}+560f_{1}-4680f_{2}+39360f_{3}-70070f_{4}+38304f_{5}-3360f_{6}-320f_{7}+315f_{8}-80f_{9}+8f_{10})-{\frac {1}{3150}}h^{11}f^{(11)}(\xi )}
h
2
f
(
2
)
(
x
5
)
=
1
25200
(
8
f
0
−
125
f
1
+
1000
f
2
−
6000
f
3
+
42000
f
4
−
73766
f
5
+
42000
f
6
−
6000
f
7
+
1000
f
8
−
125
f
9
+
8
f
10
)
−
1
16632
h
12
f
(
12
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{5})={\frac {1}{25200}}(8f_{0}-125f_{1}+1000f_{2}-6000f_{3}+42000f_{4}-73766f_{5}+42000f_{6}-6000f_{7}+1000f_{8}-125f_{9}+8f_{10})-{\frac {1}{16632}}h^{12}f^{(12)}(\xi )}
h
2
f
(
2
)
(
x
6
)
=
1
25200
(
8
f
0
−
80
f
1
+
315
f
2
−
320
f
3
−
3360
f
4
+
38304
f
5
−
70070
f
6
+
39360
f
7
−
4680
f
8
+
560
f
9
−
37
f
10
)
+
1
3150
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{6})={\frac {1}{25200}}(8f_{0}-80f_{1}+315f_{2}-320f_{3}-3360f_{4}+38304f_{5}-70070f_{6}+39360f_{7}-4680f_{8}+560f_{9}-37f_{10})+{\frac {1}{3150}}h^{11}f^{(11)}(\xi )}
h
2
f
(
2
)
(
x
7
)
=
1
25200
(
−
37
f
0
+
415
f
1
−
2115
f
2
+
6420
f
3
−
12530
f
4
+
13734
f
5
+
21210
f
6
−
57860
f
7
+
33255
f
8
−
2645
f
9
+
153
f
10
)
−
29
25200
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{7})={\frac {1}{25200}}(-37f_{0}+415f_{1}-2115f_{2}+6420f_{3}-12530f_{4}+13734f_{5}+21210f_{6}-57860f_{7}+33255f_{8}-2645f_{9}+153f_{10})-{\frac {29}{25200}}h^{11}f^{(11)}(\xi )}
h
2
f
(
2
)
(
x
8
)
=
1
25200
(
153
f
0
−
1720
f
1
+
8830
f
2
−
27360
f
3
+
56910
f
4
−
83216
f
5
+
84420
f
6
−
29280
f
7
−
32615
f
8
+
24840
f
9
−
962
f
10
)
+
31
6300
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{8})={\frac {1}{25200}}(153f_{0}-1720f_{1}+8830f_{2}-27360f_{3}+56910f_{4}-83216f_{5}+84420f_{6}-29280f_{7}-32615f_{8}+24840f_{9}-962f_{10})+{\frac {31}{6300}}h^{11}f^{(11)}(\xi )}
h
2
f
(
2
)
(
x
9
)
=
1
25200
(
−
962
f
0
+
10735
f
1
−
54630
f
2
+
167560
f
3
−
344820
f
4
+
501354
f
5
−
527660
f
6
+
401880
f
7
−
188010
f
8
+
20295
f
9
+
14258
f
10
)
−
419
12600
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{9})={\frac {1}{25200}}(-962f_{0}+10735f_{1}-54630f_{2}+167560f_{3}-344820f_{4}+501354f_{5}-527660f_{6}+401880f_{7}-188010f_{8}+20295f_{9}+14258f_{10})-{\frac {419}{12600}}h^{11}f^{(11)}(\xi )}
h
2
f
(
2
)
(
x
10
)
=
1
25200
(
14258
f
0
−
157800
f
1
+
794925
f
2
−
2407200
f
3
+
4872700
f
4
−
6932016
f
5
+
7088550
f
6
−
5232800
f
7
+
2754450
f
8
−
972200
f
9
+
177133
f
10
)
+
671
1260
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{10})={\frac {1}{25200}}(14258f_{0}-157800f_{1}+794925f_{2}-2407200f_{3}+4872700f_{4}-6932016f_{5}+7088550f_{6}-5232800f_{7}+2754450f_{8}-972200f_{9}+177133f_{10})+{\frac {671}{1260}}h^{11}f^{(11)}(\xi )}
11-punts 3e afgeleide:
h
3
f
(
3
)
(
x
0
)
=
1
30240
(
−
420475
f
0
+
2876868
f
1
−
9389763
f
2
+
19227792
f
3
−
27098442
f
4
+
27147960
f
5
−
19395138
f
6
+
9693648
f
7
−
3229227
f
8
+
645412
f
9
−
58635
f
10
)
+
16103
8400
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{0})={\frac {1}{30240}}(-420475f_{0}+2876868f_{1}-9389763f_{2}+19227792f_{3}-27098442f_{4}+27147960f_{5}-19395138f_{6}+9693648f_{7}-3229227f_{8}+645412f_{9}-58635f_{10})+{\frac {16103}{8400}}h^{11}f^{(11)}(\xi )}
h
3
f
(
3
)
(
x
1
)
=
1
30240
(
−
58635
f
0
+
224510
f
1
−
348057
f
2
+
285012
f
3
−
121758
f
4
−
9072
f
5
+
58590
f
6
−
45588
f
7
+
18873
f
8
−
4302
f
9
+
427
f
10
)
−
123
5600
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{1})={\frac {1}{30240}}(-58635f_{0}+224510f_{1}-348057f_{2}+285012f_{3}-121758f_{4}-9072f_{5}+58590f_{6}-45588f_{7}+18873f_{8}-4302f_{9}+427f_{10})-{\frac {123}{5600}}h^{11}f^{(11)}(\xi )}
h
3
f
(
3
)
(
x
2
)
=
1
30240
(
427
f
0
−
63332
f
1
+
247995
f
2
−
418512
f
3
+
425922
f
4
−
319032
f
5
+
188202
f
6
−
82320
f
7
+
24867
f
8
−
4612
f
9
+
395
f
10
)
−
593
75600
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{2})={\frac {1}{30240}}(427f_{0}-63332f_{1}+247995f_{2}-418512f_{3}+425922f_{4}-319032f_{5}+188202f_{6}-82320f_{7}+24867f_{8}-4612f_{9}+395f_{10})-{\frac {593}{75600}}h^{11}f^{(11)}(\xi )}
h
3
f
(
3
)
(
x
3
)
=
1
30240
(
395
f
0
−
3918
f
1
−
41607
f
2
+
182820
f
3
−
288162
f
4
+
243432
f
5
−
136542
f
6
+
57852
f
7
−
17145
f
8
+
3142
f
9
−
267
f
10
)
+
263
50400
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{3})={\frac {1}{30240}}(395f_{0}-3918f_{1}-41607f_{2}+182820f_{3}-288162f_{4}+243432f_{5}-136542f_{6}+57852f_{7}-17145f_{8}+3142f_{9}-267f_{10})+{\frac {263}{50400}}h^{11}f^{(11)}(\xi )}
h
3
f
(
3
)
(
x
4
)
=
1
30240
(
−
267
f
0
+
3332
f
1
−
18603
f
2
+
2448
f
3
+
94710
f
4
−
164808
f
5
+
120078
f
6
−
48432
f
7
+
13797
f
8
−
2460
f
9
+
205
f
10
)
−
13
3600
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{4})={\frac {1}{30240}}(-267f_{0}+3332f_{1}-18603f_{2}+2448f_{3}+94710f_{4}-164808f_{5}+120078f_{6}-48432f_{7}+13797f_{8}-2460f_{9}+205f_{10})-{\frac {13}{3600}}h^{11}f^{(11)}(\xi )}
h
3
f
(
3
)
(
x
5
)
=
1
30240
(
205
f
0
−
2522
f
1
+
14607
f
2
−
52428
f
3
+
70098
f
4
−
70098
f
6
+
52428
f
7
−
14607
f
8
+
2522
f
9
−
205
f
10
)
+
479
151200
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{5})={\frac {1}{30240}}(205f_{0}-2522f_{1}+14607f_{2}-52428f_{3}+70098f_{4}-70098f_{6}+52428f_{7}-14607f_{8}+2522f_{9}-205f_{10})+{\frac {479}{151200}}h^{11}f^{(11)}(\xi )}
h
3
f
(
3
)
(
x
6
)
=
1
30240
(
−
205
f
0
+
2460
f
1
−
13797
f
2
+
48432
f
3
−
120078
f
4
+
164808
f
5
−
94710
f
6
−
2448
f
7
+
18603
f
8
−
3332
f
9
+
267
f
10
)
−
13
3600
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{6})={\frac {1}{30240}}(-205f_{0}+2460f_{1}-13797f_{2}+48432f_{3}-120078f_{4}+164808f_{5}-94710f_{6}-2448f_{7}+18603f_{8}-3332f_{9}+267f_{10})-{\frac {13}{3600}}h^{11}f^{(11)}(\xi )}
h
3
f
(
3
)
(
x
7
)
=
1
30240
(
267
f
0
−
3142
f
1
+
17145
f
2
−
57852
f
3
+
136542
f
4
−
243432
f
5
+
288162
f
6
−
182820
f
7
+
41607
f
8
+
3918
f
9
−
395
f
10
)
+
263
50400
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{7})={\frac {1}{30240}}(267f_{0}-3142f_{1}+17145f_{2}-57852f_{3}+136542f_{4}-243432f_{5}+288162f_{6}-182820f_{7}+41607f_{8}+3918f_{9}-395f_{10})+{\frac {263}{50400}}h^{11}f^{(11)}(\xi )}
h
3
f
(
3
)
(
x
8
)
=
1
30240
(
−
395
f
0
+
4612
f
1
−
24867
f
2
+
82320
f
3
−
188202
f
4
+
319032
f
5
−
425922
f
6
+
418512
f
7
−
247995
f
8
+
63332
f
9
−
427
f
10
)
−
593
75600
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{8})={\frac {1}{30240}}(-395f_{0}+4612f_{1}-24867f_{2}+82320f_{3}-188202f_{4}+319032f_{5}-425922f_{6}+418512f_{7}-247995f_{8}+63332f_{9}-427f_{10})-{\frac {593}{75600}}h^{11}f^{(11)}(\xi )}
h
3
f
(
3
)
(
x
9
)
=
1
30240
(
−
427
f
0
+
4302
f
1
−
18873
f
2
+
45588
f
3
−
58590
f
4
+
9072
f
5
+
121758
f
6
−
285012
f
7
+
348057
f
8
−
224510
f
9
+
58635
f
10
)
−
123
5600
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{9})={\frac {1}{30240}}(-427f_{0}+4302f_{1}-18873f_{2}+45588f_{3}-58590f_{4}+9072f_{5}+121758f_{6}-285012f_{7}+348057f_{8}-224510f_{9}+58635f_{10})-{\frac {123}{5600}}h^{11}f^{(11)}(\xi )}
h
3
f
(
3
)
(
x
10
)
=
1
30240
(
58635
f
0
−
645412
f
1
+
3229227
f
2
−
9693648
f
3
+
19395138
f
4
−
27147960
f
5
+
27098442
f
6
−
19227792
f
7
+
9389763
f
8
−
2876868
f
9
+
420475
f
10
)
+
16103
8400
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{10})={\frac {1}{30240}}(58635f_{0}-645412f_{1}+3229227f_{2}-9693648f_{3}+19395138f_{4}-27147960f_{5}+27098442f_{6}-19227792f_{7}+9389763f_{8}-2876868f_{9}+420475f_{10})+{\frac {16103}{8400}}h^{11}f^{(11)}(\xi )}
11-punts 4e afgeleide:
h
4
f
(
4
)
(
x
0
)
=
1
15120
(
341693
f
0
−
2655764
f
1
+
9531612
f
2
−
20819472
f
3
+
30600654
f
4
−
31524696
f
5
+
22968204
f
6
−
11646672
f
7
+
3923037
f
8
−
790964
f
9
+
72368
f
10
)
−
7645
1512
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{0})={\frac {1}{15120}}(341693f_{0}-2655764f_{1}+9531612f_{2}-20819472f_{3}+30600654f_{4}-31524696f_{5}+22968204f_{6}-11646672f_{7}+3923037f_{8}-790964f_{9}+72368f_{10})-{\frac {7645}{1512}}h^{11}f^{(11)}(\xi )}
h
4
f
(
4
)
(
x
1
)
=
1
15120
(
72368
f
0
−
454355
f
1
+
1324476
f
2
−
2409108
f
3
+
3061968
f
4
−
2833362
f
5
+
1909320
f
6
−
913236
f
7
+
294048
f
8
−
57203
f
9
+
5084
f
10
)
−
2041
7560
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{1})={\frac {1}{15120}}(72368f_{0}-454355f_{1}+1324476f_{2}-2409108f_{3}+3061968f_{4}-2833362f_{5}+1909320f_{6}-913236f_{7}+294048f_{8}-57203f_{9}+5084f_{10})-{\frac {2041}{7560}}h^{11}f^{(11)}(\xi )}
h
4
f
(
4
)
(
x
2
)
=
1
15120
(
5084
f
0
+
16444
f
1
−
174735
f
2
+
485616
f
3
−
731388
f
4
+
713160
f
5
−
484554
f
6
+
231600
f
7
−
74376
f
8
+
14428
f
9
−
1279
f
10
)
+
167
2520
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{2})={\frac {1}{15120}}(5084f_{0}+16444f_{1}-174735f_{2}+485616f_{3}-731388f_{4}+713160f_{5}-484554f_{6}+231600f_{7}-74376f_{8}+14428f_{9}-1279f_{10})+{\frac {167}{2520}}h^{11}f^{(11)}(\xi )}
h
4
f
(
4
)
(
x
3
)
=
1
15120
(
−
1279
f
0
+
19153
f
1
−
53901
f
2
+
36300
f
3
+
63546
f
4
−
140490
f
5
+
122262
f
6
−
62484
f
7
+
20565
f
8
−
4031
f
9
+
359
f
10
)
−
277
15120
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{3})={\frac {1}{15120}}(-1279f_{0}+19153f_{1}-53901f_{2}+36300f_{3}+63546f_{4}-140490f_{5}+122262f_{6}-62484f_{7}+20565f_{8}-4031f_{9}+359f_{10})-{\frac {277}{15120}}h^{11}f^{(11)}(\xi )}
h
4
f
(
4
)
(
x
4
)
=
1
15120
(
359
f
0
−
5228
f
1
+
38898
f
2
−
113136
f
3
+
154770
f
4
−
102312
f
5
+
25368
f
6
+
3792
f
7
−
3249
f
8
+
820
f
9
−
82
f
10
)
+
41
7560
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{4})={\frac {1}{15120}}(359f_{0}-5228f_{1}+38898f_{2}-113136f_{3}+154770f_{4}-102312f_{5}+25368f_{6}+3792f_{7}-3249f_{8}+820f_{9}-82f_{10})+{\frac {41}{7560}}h^{11}f^{(11)}(\xi )}
h
4
f
(
4
)
(
x
5
)
=
1
15120
(
−
82
f
0
+
1261
f
1
−
9738
f
2
+
52428
f
3
−
140196
f
4
+
192654
f
5
−
140196
f
6
+
52428
f
7
−
9738
f
8
+
1261
f
9
−
82
f
10
)
+
479
453600
h
12
f
(
12
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{5})={\frac {1}{15120}}(-82f_{0}+1261f_{1}-9738f_{2}+52428f_{3}-140196f_{4}+192654f_{5}-140196f_{6}+52428f_{7}-9738f_{8}+1261f_{9}-82f_{10})+{\frac {479}{453600}}h^{12}f^{(12)}(\xi )}
h
4
f
(
4
)
(
x
6
)
=
1
15120
(
−
82
f
0
+
820
f
1
−
3249
f
2
+
3792
f
3
+
25368
f
4
−
102312
f
5
+
154770
f
6
−
113136
f
7
+
38898
f
8
−
5228
f
9
+
359
f
10
)
−
41
7560
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{6})={\frac {1}{15120}}(-82f_{0}+820f_{1}-3249f_{2}+3792f_{3}+25368f_{4}-102312f_{5}+154770f_{6}-113136f_{7}+38898f_{8}-5228f_{9}+359f_{10})-{\frac {41}{7560}}h^{11}f^{(11)}(\xi )}
h
4
f
(
4
)
(
x
7
)
=
1
15120
(
359
f
0
−
4031
f
1
+
20565
f
2
−
62484
f
3
+
122262
f
4
−
140490
f
5
+
63546
f
6
+
36300
f
7
−
53901
f
8
+
19153
f
9
−
1279
f
10
)
+
277
15120
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{7})={\frac {1}{15120}}(359f_{0}-4031f_{1}+20565f_{2}-62484f_{3}+122262f_{4}-140490f_{5}+63546f_{6}+36300f_{7}-53901f_{8}+19153f_{9}-1279f_{10})+{\frac {277}{15120}}h^{11}f^{(11)}(\xi )}
h
4
f
(
4
)
(
x
8
)
=
1
15120
(
−
1279
f
0
+
14428
f
1
−
74376
f
2
+
231600
f
3
−
484554
f
4
+
713160
f
5
−
731388
f
6
+
485616
f
7
−
174735
f
8
+
16444
f
9
+
5084
f
10
)
−
167
2520
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{8})={\frac {1}{15120}}(-1279f_{0}+14428f_{1}-74376f_{2}+231600f_{3}-484554f_{4}+713160f_{5}-731388f_{6}+485616f_{7}-174735f_{8}+16444f_{9}+5084f_{10})-{\frac {167}{2520}}h^{11}f^{(11)}(\xi )}
h
4
f
(
4
)
(
x
9
)
=
1
15120
(
5084
f
0
−
57203
f
1
+
294048
f
2
−
913236
f
3
+
1909320
f
4
−
2833362
f
5
+
3061968
f
6
−
2409108
f
7
+
1324476
f
8
−
454355
f
9
+
72368
f
10
)
+
2041
7560
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{9})={\frac {1}{15120}}(5084f_{0}-57203f_{1}+294048f_{2}-913236f_{3}+1909320f_{4}-2833362f_{5}+3061968f_{6}-2409108f_{7}+1324476f_{8}-454355f_{9}+72368f_{10})+{\frac {2041}{7560}}h^{11}f^{(11)}(\xi )}
h
4
f
(
4
)
(
x
10
)
=
1
15120
(
72368
f
0
−
790964
f
1
+
3923037
f
2
−
11646672
f
3
+
22968204
f
4
−
31524696
f
5
+
30600654
f
6
−
20819472
f
7
+
9531612
f
8
−
2655764
f
9
+
341693
f
10
)
+
7645
1512
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{10})={\frac {1}{15120}}(72368f_{0}-790964f_{1}+3923037f_{2}-11646672f_{3}+22968204f_{4}-31524696f_{5}+30600654f_{6}-20819472f_{7}+9531612f_{8}-2655764f_{9}+341693f_{10})+{\frac {7645}{1512}}h^{11}f^{(11)}(\xi )}
11-punts 5e afgeleide:
h
5
f
(
5
)
(
x
0
)
=
1
288
(
−
8591
f
0
+
72492
f
1
−
278313
f
2
+
640752
f
3
−
979878
f
4
+
1039656
f
5
−
774402
f
6
+
399408
f
7
−
136347
f
8
+
27788
f
9
−
2565
f
10
)
+
31063
3024
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{0})={\frac {1}{288}}(-8591f_{0}+72492f_{1}-278313f_{2}+640752f_{3}-979878f_{4}+1039656f_{5}-774402f_{6}+399408f_{7}-136347f_{8}+27788f_{9}-2565f_{10})+{\frac {31063}{3024}}h^{11}f^{(11)}(\xi )}
h
5
f
(
5
)
(
x
1
)
=
1
288
(
−
2565
f
0
+
19624
f
1
−
68583
f
2
+
144912
f
3
−
205698
f
4
+
205152
f
5
−
145374
f
6
+
72048
f
7
−
23817
f
8
+
4728
f
9
−
427
f
10
)
+
8261
6048
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{1})={\frac {1}{288}}(-2565f_{0}+19624f_{1}-68583f_{2}+144912f_{3}-205698f_{4}+205152f_{5}-145374f_{6}+72048f_{7}-23817f_{8}+4728f_{9}-427f_{10})+{\frac {8261}{6048}}h^{11}f^{(11)}(\xi )}
h
5
f
(
5
)
(
x
2
)
=
1
288
(
−
427
f
0
+
2132
f
1
−
3861
f
2
+
1872
f
3
+
4002
f
4
−
8424
f
5
+
7878
f
6
−
4464
f
7
+
1593
f
8
−
332
f
9
+
31
f
10
)
−
353
3024
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{2})={\frac {1}{288}}(-427f_{0}+2132f_{1}-3861f_{2}+1872f_{3}+4002f_{4}-8424f_{5}+7878f_{6}-4464f_{7}+1593f_{8}-332f_{9}+31f_{10})-{\frac {353}{3024}}h^{11}f^{(11)}(\xi )}
h
5
f
(
5
)
(
x
3
)
=
1
288
(
31
f
0
−
768
f
1
+
3837
f
2
−
8976
f
3
+
12102
f
4
−
10320
f
5
+
5898
f
6
−
2352
f
7
+
651
f
8
−
112
f
9
+
9
f
10
)
−
55
6048
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{3})={\frac {1}{288}}(31f_{0}-768f_{1}+3837f_{2}-8976f_{3}+12102f_{4}-10320f_{5}+5898f_{6}-2352f_{7}+651f_{8}-112f_{9}+9f_{10})-{\frac {55}{6048}}h^{11}f^{(11)}(\xi )}
h
5
f
(
5
)
(
x
4
)
=
1
288
(
9
f
0
−
68
f
1
−
273
f
2
+
2352
f
3
−
6006
f
4
+
7944
f
5
−
6162
f
6
+
2928
f
7
−
867
f
8
+
156
f
9
−
13
f
10
)
+
67
3024
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{4})={\frac {1}{288}}(9f_{0}-68f_{1}-273f_{2}+2352f_{3}-6006f_{4}+7944f_{5}-6162f_{6}+2928f_{7}-867f_{8}+156f_{9}-13f_{10})+{\frac {67}{3024}}h^{11}f^{(11)}(\xi )}
h
5
f
(
5
)
(
x
5
)
=
1
288
(
−
13
f
0
+
152
f
1
−
783
f
2
+
1872
f
3
−
1938
f
4
+
1938
f
6
−
1872
f
7
+
783
f
8
−
152
f
9
+
13
f
10
)
−
139
6048
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{5})={\frac {1}{288}}(-13f_{0}+152f_{1}-783f_{2}+1872f_{3}-1938f_{4}+1938f_{6}-1872f_{7}+783f_{8}-152f_{9}+13f_{10})-{\frac {139}{6048}}h^{11}f^{(11)}(\xi )}
h
5
f
(
5
)
(
x
6
)
=
1
288
(
13
f
0
−
156
f
1
+
867
f
2
−
2928
f
3
+
6162
f
4
−
7944
f
5
+
6006
f
6
−
2352
f
7
+
273
f
8
+
68
f
9
−
9
f
10
)
+
67
3024
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{6})={\frac {1}{288}}(13f_{0}-156f_{1}+867f_{2}-2928f_{3}+6162f_{4}-7944f_{5}+6006f_{6}-2352f_{7}+273f_{8}+68f_{9}-9f_{10})+{\frac {67}{3024}}h^{11}f^{(11)}(\xi )}
h
5
f
(
5
)
(
x
7
)
=
1
288
(
−
9
f
0
+
112
f
1
−
651
f
2
+
2352
f
3
−
5898
f
4
+
10320
f
5
−
12102
f
6
+
8976
f
7
−
3837
f
8
+
768
f
9
−
31
f
10
)
−
55
6048
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{7})={\frac {1}{288}}(-9f_{0}+112f_{1}-651f_{2}+2352f_{3}-5898f_{4}+10320f_{5}-12102f_{6}+8976f_{7}-3837f_{8}+768f_{9}-31f_{10})-{\frac {55}{6048}}h^{11}f^{(11)}(\xi )}
h
5
f
(
5
)
(
x
8
)
=
1
288
(
−
31
f
0
+
332
f
1
−
1593
f
2
+
4464
f
3
−
7878
f
4
+
8424
f
5
−
4002
f
6
−
1872
f
7
+
3861
f
8
−
2132
f
9
+
427
f
10
)
−
353
3024
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{8})={\frac {1}{288}}(-31f_{0}+332f_{1}-1593f_{2}+4464f_{3}-7878f_{4}+8424f_{5}-4002f_{6}-1872f_{7}+3861f_{8}-2132f_{9}+427f_{10})-{\frac {353}{3024}}h^{11}f^{(11)}(\xi )}
h
5
f
(
5
)
(
x
9
)
=
1
288
(
427
f
0
−
4728
f
1
+
23817
f
2
−
72048
f
3
+
145374
f
4
−
205152
f
5
+
205698
f
6
−
144912
f
7
+
68583
f
8
−
19624
f
9
+
2565
f
10
)
+
8261
6048
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{9})={\frac {1}{288}}(427f_{0}-4728f_{1}+23817f_{2}-72048f_{3}+145374f_{4}-205152f_{5}+205698f_{6}-144912f_{7}+68583f_{8}-19624f_{9}+2565f_{10})+{\frac {8261}{6048}}h^{11}f^{(11)}(\xi )}
h
5
f
(
5
)
(
x
10
)
=
1
288
(
2565
f
0
−
27788
f
1
+
136347
f
2
−
399408
f
3
+
774402
f
4
−
1039656
f
5
+
979878
f
6
−
640752
f
7
+
278313
f
8
−
72492
f
9
+
8591
f
10
)
+
31063
3024
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{5}f^{(5)}(x_{10})={\frac {1}{288}}(2565f_{0}-27788f_{1}+136347f_{2}-399408f_{3}+774402f_{4}-1039656f_{5}+979878f_{6}-640752f_{7}+278313f_{8}-72492f_{9}+8591f_{10})+{\frac {31063}{3024}}h^{11}f^{(11)}(\xi )}
11-punts 6e afgeleide:
h
6
f
(
6
)
(
x
0
)
=
1
240
(
7513
f
0
−
67090
f
1
+
270705
f
2
−
650280
f
3
+
1030290
f
4
−
1125276
f
5
+
858090
f
6
−
451080
f
7
+
156405
f
8
−
32290
f
9
+
3013
f
10
)
−
781
48
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{0})={\frac {1}{240}}(7513f_{0}-67090f_{1}+270705f_{2}-650280f_{3}+1030290f_{4}-1125276f_{5}+858090f_{6}-451080f_{7}+156405f_{8}-32290f_{9}+3013f_{10})-{\frac {781}{48}}h^{11}f^{(11)}(\xi )}
h
6
f
(
6
)
(
x
1
)
=
1
240
(
3013
f
0
−
25630
f
1
+
98625
f
2
−
226440
f
3
+
344010
f
4
−
361716
f
5
+
266730
f
6
−
136200
f
7
+
46065
f
8
−
9310
f
9
+
853
f
10
)
−
223
60
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{1})={\frac {1}{240}}(3013f_{0}-25630f_{1}+98625f_{2}-226440f_{3}+344010f_{4}-361716f_{5}+266730f_{6}-136200f_{7}+46065f_{8}-9310f_{9}+853f_{10})-{\frac {223}{60}}h^{11}f^{(11)}(\xi )}
h
6
f
(
6
)
(
x
2
)
=
1
240
(
853
f
0
−
6370
f
1
+
21285
f
2
−
42120
f
3
+
55050
f
4
−
50076
f
5
+
32370
f
6
−
14760
f
7
+
4545
f
8
−
850
f
9
+
73
f
10
)
−
13
80
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{2})={\frac {1}{240}}(853f_{0}-6370f_{1}+21285f_{2}-42120f_{3}+55050f_{4}-50076f_{5}+32370f_{6}-14760f_{7}+4545f_{8}-850f_{9}+73f_{10})-{\frac {13}{80}}h^{11}f^{(11)}(\xi )}
h
6
f
(
6
)
(
x
3
)
=
1
240
(
73
f
0
+
50
f
1
−
2355
f
2
+
9240
f
3
−
18030
f
4
+
21324
f
5
−
16350
f
6
+
8280
f
7
−
2715
f
8
+
530
f
9
−
47
f
10
)
+
17
120
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{3})={\frac {1}{240}}(73f_{0}+50f_{1}-2355f_{2}+9240f_{3}-18030f_{4}+21324f_{5}-16350f_{6}+8280f_{7}-2715f_{8}+530f_{9}-47f_{10})+{\frac {17}{120}}h^{11}f^{(11)}(\xi )}
h
6
f
(
6
)
(
x
4
)
=
1
240
(
−
47
f
0
+
590
f
1
−
2535
f
2
+
5400
f
3
−
6270
f
4
+
3684
f
5
−
390
f
6
−
840
f
7
+
525
f
8
−
130
f
9
+
13
f
10
)
−
13
240
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{4})={\frac {1}{240}}(-47f_{0}+590f_{1}-2535f_{2}+5400f_{3}-6270f_{4}+3684f_{5}-390f_{6}-840f_{7}+525f_{8}-130f_{9}+13f_{10})-{\frac {13}{240}}h^{11}f^{(11)}(\xi )}
h
6
f
(
6
)
(
x
5
)
=
1
240
(
13
f
0
−
190
f
1
+
1305
f
2
−
4680
f
3
+
9690
f
4
−
12276
f
5
+
9690
f
6
−
4680
f
7
+
1305
f
8
−
190
f
9
+
13
f
10
)
−
139
12096
h
12
f
(
12
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{5})={\frac {1}{240}}(13f_{0}-190f_{1}+1305f_{2}-4680f_{3}+9690f_{4}-12276f_{5}+9690f_{6}-4680f_{7}+1305f_{8}-190f_{9}+13f_{10})-{\frac {139}{12096}}h^{12}f^{(12)}(\xi )}
h
6
f
(
6
)
(
x
6
)
=
1
240
(
13
f
0
−
130
f
1
+
525
f
2
−
840
f
3
−
390
f
4
+
3684
f
5
−
6270
f
6
+
5400
f
7
−
2535
f
8
+
590
f
9
−
47
f
10
)
+
13
240
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{6})={\frac {1}{240}}(13f_{0}-130f_{1}+525f_{2}-840f_{3}-390f_{4}+3684f_{5}-6270f_{6}+5400f_{7}-2535f_{8}+590f_{9}-47f_{10})+{\frac {13}{240}}h^{11}f^{(11)}(\xi )}
h
6
f
(
6
)
(
x
7
)
=
1
240
(
−
47
f
0
+
530
f
1
−
2715
f
2
+
8280
f
3
−
16350
f
4
+
21324
f
5
−
18030
f
6
+
9240
f
7
−
2355
f
8
+
50
f
9
+
73
f
10
)
−
17
120
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{7})={\frac {1}{240}}(-47f_{0}+530f_{1}-2715f_{2}+8280f_{3}-16350f_{4}+21324f_{5}-18030f_{6}+9240f_{7}-2355f_{8}+50f_{9}+73f_{10})-{\frac {17}{120}}h^{11}f^{(11)}(\xi )}
h
6
f
(
6
)
(
x
8
)
=
1
240
(
73
f
0
−
850
f
1
+
4545
f
2
−
14760
f
3
+
32370
f
4
−
50076
f
5
+
55050
f
6
−
42120
f
7
+
21285
f
8
−
6370
f
9
+
853
f
10
)
+
13
80
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{8})={\frac {1}{240}}(73f_{0}-850f_{1}+4545f_{2}-14760f_{3}+32370f_{4}-50076f_{5}+55050f_{6}-42120f_{7}+21285f_{8}-6370f_{9}+853f_{10})+{\frac {13}{80}}h^{11}f^{(11)}(\xi )}
h
6
f
(
6
)
(
x
9
)
=
1
240
(
853
f
0
−
9310
f
1
+
46065
f
2
−
136200
f
3
+
266730
f
4
−
361716
f
5
+
344010
f
6
−
226440
f
7
+
98625
f
8
−
25630
f
9
+
3013
f
10
)
+
223
60
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{9})={\frac {1}{240}}(853f_{0}-9310f_{1}+46065f_{2}-136200f_{3}+266730f_{4}-361716f_{5}+344010f_{6}-226440f_{7}+98625f_{8}-25630f_{9}+3013f_{10})+{\frac {223}{60}}h^{11}f^{(11)}(\xi )}
h
6
f
(
6
)
(
x
10
)
=
1
240
(
3013
f
0
−
32290
f
1
+
156405
f
2
−
451080
f
3
+
858090
f
4
−
1125276
f
5
+
1030290
f
6
−
650280
f
7
+
270705
f
8
−
67090
f
9
+
7513
f
10
)
+
781
48
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{6}f^{(6)}(x_{10})={\frac {1}{240}}(3013f_{0}-32290f_{1}+156405f_{2}-451080f_{3}+858090f_{4}-1125276f_{5}+1030290f_{6}-650280f_{7}+270705f_{8}-67090f_{9}+7513f_{10})+{\frac {781}{48}}h^{11}f^{(11)}(\xi )}
11-punts 7e afgeleide:
h
7
f
(
7
)
(
x
0
)
=
1
24
(
−
605
f
0
+
5628
f
1
−
23583
f
2
+
58632
f
3
−
95802
f
4
+
107520
f
5
−
83958
f
6
+
45048
f
7
−
15897
f
8
+
3332
f
9
−
315
f
10
)
+
4781
240
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{7}f^{(7)}(x_{0})={\frac {1}{24}}(-605f_{0}+5628f_{1}-23583f_{2}+58632f_{3}-95802f_{4}+107520f_{5}-83958f_{6}+45048f_{7}-15897f_{8}+3332f_{9}-315f_{10})+{\frac {4781}{240}}h^{11}f^{(11)}(\xi )}
h
7
f
(
7
)
(
x
1
)
=
1
24
(
−
315
f
0
+
2860
f
1
−
11697
f
2
+
28392
f
3
−
45318
f
4
+
49728
f
5
−
38010
f
6
+
19992
f
7
−
6927
f
8
+
1428
f
9
−
133
f
10
)
+
1631
240
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{7}f^{(7)}(x_{1})={\frac {1}{24}}(-315f_{0}+2860f_{1}-11697f_{2}+28392f_{3}-45318f_{4}+49728f_{5}-38010f_{6}+19992f_{7}-6927f_{8}+1428f_{9}-133f_{10})+{\frac {1631}{240}}h^{11}f^{(11)}(\xi )}
h
7
f
(
7
)
(
x
2
)
=
1
24
(
−
133
f
0
+
1148
f
1
−
4455
f
2
+
10248
f
3
−
15498
f
4
+
16128
f
5
−
11718
f
6
+
5880
f
7
−
1953
f
8
+
388
f
9
−
35
f
10
)
+
301
240
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{7}f^{(7)}(x_{2})={\frac {1}{24}}(-133f_{0}+1148f_{1}-4455f_{2}+10248f_{3}-15498f_{4}+16128f_{5}-11718f_{6}+5880f_{7}-1953f_{8}+388f_{9}-35f_{10})+{\frac {301}{240}}h^{11}f^{(11)}(\xi )}
h
7
f
(
7
)
(
x
3
)
=
1
24
(
−
35
f
0
+
252
f
1
−
777
f
2
+
1320
f
3
−
1302
f
4
+
672
f
5
−
42
f
6
−
168
f
7
+
105
f
8
−
28
f
9
+
3
f
10
)
−
49
240
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{7}f^{(7)}(x_{3})={\frac {1}{24}}(-35f_{0}+252f_{1}-777f_{2}+1320f_{3}-1302f_{4}+672f_{5}-42f_{6}-168f_{7}+105f_{8}-28f_{9}+3f_{10})-{\frac {49}{240}}h^{11}f^{(11)}(\xi )}
h
7
f
(
7
)
(
x
4
)
=
1
24
(
3
f
0
−
68
f
1
+
417
f
2
−
1272
f
3
+
2310
f
4
−
2688
f
5
+
2058
f
6
−
1032
f
7
+
327
f
8
−
60
f
9
+
5
f
10
)
−
19
240
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{7}f^{(7)}(x_{4})={\frac {1}{24}}(3f_{0}-68f_{1}+417f_{2}-1272f_{3}+2310f_{4}-2688f_{5}+2058f_{6}-1032f_{7}+327f_{8}-60f_{9}+5f_{10})-{\frac {19}{240}}h^{11}f^{(11)}(\xi )}
h
7
f
(
7
)
(
x
5
)
=
1
24
(
5
f
0
−
52
f
1
+
207
f
2
−
408
f
3
+
378
f
4
−
378
f
6
+
408
f
7
−
207
f
8
+
52
f
9
−
5
f
10
)
+
31
240
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{7}f^{(7)}(x_{5})={\frac {1}{24}}(5f_{0}-52f_{1}+207f_{2}-408f_{3}+378f_{4}-378f_{6}+408f_{7}-207f_{8}+52f_{9}-5f_{10})+{\frac {31}{240}}h^{11}f^{(11)}(\xi )}
h
7
f
(
7
)
(
x
6
)
=
1
24
(
−
5
f
0
+
60
f
1
−
327
f
2
+
1032
f
3
−
2058
f
4
+
2688
f
5
−
2310
f
6
+
1272
f
7
−
417
f
8
+
68
f
9
−
3
f
10
)
−
19
240
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{7}f^{(7)}(x_{6})={\frac {1}{24}}(-5f_{0}+60f_{1}-327f_{2}+1032f_{3}-2058f_{4}+2688f_{5}-2310f_{6}+1272f_{7}-417f_{8}+68f_{9}-3f_{10})-{\frac {19}{240}}h^{11}f^{(11)}(\xi )}
h
7
f
(
7
)
(
x
7
)
=
1
24
(
−
3
f
0
+
28
f
1
−
105
f
2
+
168
f
3
+
42
f
4
−
672
f
5
+
1302
f
6
−
1320
f
7
+
777
f
8
−
252
f
9
+
35
f
10
)
−
49
240
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{7}f^{(7)}(x_{7})={\frac {1}{24}}(-3f_{0}+28f_{1}-105f_{2}+168f_{3}+42f_{4}-672f_{5}+1302f_{6}-1320f_{7}+777f_{8}-252f_{9}+35f_{10})-{\frac {49}{240}}h^{11}f^{(11)}(\xi )}
h
7
f
(
7
)
(
x
8
)
=
1
24
(
35
f
0
−
388
f
1
+
1953
f
2
−
5880
f
3
+
11718
f
4
−
16128
f
5
+
15498
f
6
−
10248
f
7
+
4455
f
8
−
1148
f
9
+
133
f
10
)
+
301
240
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{7}f^{(7)}(x_{8})={\frac {1}{24}}(35f_{0}-388f_{1}+1953f_{2}-5880f_{3}+11718f_{4}-16128f_{5}+15498f_{6}-10248f_{7}+4455f_{8}-1148f_{9}+133f_{10})+{\frac {301}{240}}h^{11}f^{(11)}(\xi )}
h
7
f
(
7
)
(
x
9
)
=
1
24
(
133
f
0
−
1428
f
1
+
6927
f
2
−
19992
f
3
+
38010
f
4
−
49728
f
5
+
45318
f
6
−
28392
f
7
+
11697
f
8
−
2860
f
9
+
315
f
10
)
+
1631
240
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{7}f^{(7)}(x_{9})={\frac {1}{24}}(133f_{0}-1428f_{1}+6927f_{2}-19992f_{3}+38010f_{4}-49728f_{5}+45318f_{6}-28392f_{7}+11697f_{8}-2860f_{9}+315f_{10})+{\frac {1631}{240}}h^{11}f^{(11)}(\xi )}
h
7
f
(
7
)
(
x
10
)
=
1
24
(
315
f
0
−
3332
f
1
+
15897
f
2
−
45048
f
3
+
83958
f
4
−
107520
f
5
+
95802
f
6
−
58632
f
7
+
23583
f
8
−
5628
f
9
+
605
f
10
)
+
4781
240
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{7}f^{(7)}(x_{10})={\frac {1}{24}}(315f_{0}-3332f_{1}+15897f_{2}-45048f_{3}+83958f_{4}-107520f_{5}+95802f_{6}-58632f_{7}+23583f_{8}-5628f_{9}+605f_{10})+{\frac {4781}{240}}h^{11}f^{(11)}(\xi )}
11-punts 8e afgeleide:
h
8
f
(
8
)
(
x
0
)
=
1
3
(
44
f
0
−
422
f
1
+
1821
f
2
−
4656
f
3
+
7812
f
4
−
8988
f
5
+
7182
f
6
−
3936
f
7
+
1416
f
8
−
302
f
9
+
29
f
10
)
−
55
3
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{8}f^{(8)}(x_{0})={\frac {1}{3}}(44f_{0}-422f_{1}+1821f_{2}-4656f_{3}+7812f_{4}-8988f_{5}+7182f_{6}-3936f_{7}+1416f_{8}-302f_{9}+29f_{10})-{\frac {55}{3}}h^{11}f^{(11)}(\xi )}
h
8
f
(
8
)
(
x
1
)
=
1
3
(
29
f
0
−
275
f
1
+
1173
f
2
−
2964
f
3
+
4914
f
4
−
5586
f
5
+
4410
f
6
−
2388
f
7
+
849
f
8
−
179
f
9
+
17
f
10
)
−
26
3
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{8}f^{(8)}(x_{1})={\frac {1}{3}}(29f_{0}-275f_{1}+1173f_{2}-2964f_{3}+4914f_{4}-5586f_{5}+4410f_{6}-2388f_{7}+849f_{8}-179f_{9}+17f_{10})-{\frac {26}{3}}h^{11}f^{(11)}(\xi )}
h
8
f
(
8
)
(
x
2
)
=
1
3
(
17
f
0
−
158
f
1
+
660
f
2
−
1632
f
3
+
2646
f
4
−
2940
f
5
+
2268
f
6
−
1200
f
7
+
417
f
8
−
86
f
9
+
8
f
10
)
−
3
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{8}f^{(8)}(x_{2})={\frac {1}{3}}(17f_{0}-158f_{1}+660f_{2}-1632f_{3}+2646f_{4}-2940f_{5}+2268f_{6}-1200f_{7}+417f_{8}-86f_{9}+8f_{10})-3h^{11}f^{(11)}(\xi )}
h
8
f
(
8
)
(
x
3
)
=
1
3
(
8
f
0
−
71
f
1
+
282
f
2
−
660
f
3
+
1008
f
4
−
1050
f
5
+
756
f
6
−
372
f
7
+
120
f
8
−
23
f
9
+
2
f
10
)
−
1
3
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{8}f^{(8)}(x_{3})={\frac {1}{3}}(8f_{0}-71f_{1}+282f_{2}-660f_{3}+1008f_{4}-1050f_{5}+756f_{6}-372f_{7}+120f_{8}-23f_{9}+2f_{10})-{\frac {1}{3}}h^{11}f^{(11)}(\xi )}
h
8
f
(
8
)
(
x
4
)
=
1
3
(
2
f
0
−
14
f
1
+
39
f
2
−
48
f
3
+
84
f
5
−
126
f
6
+
96
f
7
−
42
f
8
+
10
f
9
−
f
10
)
+
1
3
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{8}f^{(8)}(x_{4})={\frac {1}{3}}(2f_{0}-14f_{1}+39f_{2}-48f_{3}+84f_{5}-126f_{6}+96f_{7}-42f_{8}+10f_{9}-f_{10})+{\frac {1}{3}}h^{11}f^{(11)}(\xi )}
h
8
f
(
8
)
(
x
5
)
=
1
3
(
−
f
0
+
13
f
1
−
69
f
2
+
204
f
3
−
378
f
4
+
462
f
5
−
378
f
6
+
204
f
7
−
69
f
8
+
13
f
9
−
f
10
)
+
31
360
h
12
f
(
12
)
(
ξ
)
{\displaystyle h^{8}f^{(8)}(x_{5})={\frac {1}{3}}(-f_{0}+13f_{1}-69f_{2}+204f_{3}-378f_{4}+462f_{5}-378f_{6}+204f_{7}-69f_{8}+13f_{9}-f_{10})+{\frac {31}{360}}h^{12}f^{(12)}(\xi )}
h
8
f
(
8
)
(
x
6
)
=
1
3
(
−
f
0
+
10
f
1
−
42
f
2
+
96
f
3
−
126
f
4
+
84
f
5
−
48
f
7
+
39
f
8
−
14
f
9
+
2
f
10
)
−
1
3
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{8}f^{(8)}(x_{6})={\frac {1}{3}}(-f_{0}+10f_{1}-42f_{2}+96f_{3}-126f_{4}+84f_{5}-48f_{7}+39f_{8}-14f_{9}+2f_{10})-{\frac {1}{3}}h^{11}f^{(11)}(\xi )}
h
8
f
(
8
)
(
x
7
)
=
1
3
(
2
f
0
−
23
f
1
+
120
f
2
−
372
f
3
+
756
f
4
−
1050
f
5
+
1008
f
6
−
660
f
7
+
282
f
8
−
71
f
9
+
8
f
10
)
+
1
3
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{8}f^{(8)}(x_{7})={\frac {1}{3}}(2f_{0}-23f_{1}+120f_{2}-372f_{3}+756f_{4}-1050f_{5}+1008f_{6}-660f_{7}+282f_{8}-71f_{9}+8f_{10})+{\frac {1}{3}}h^{11}f^{(11)}(\xi )}
h
8
f
(
8
)
(
x
8
)
=
1
3
(
8
f
0
−
86
f
1
+
417
f
2
−
1200
f
3
+
2268
f
4
−
2940
f
5
+
2646
f
6
−
1632
f
7
+
660
f
8
−
158
f
9
+
17
f
10
)
+
3
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{8}f^{(8)}(x_{8})={\frac {1}{3}}(8f_{0}-86f_{1}+417f_{2}-1200f_{3}+2268f_{4}-2940f_{5}+2646f_{6}-1632f_{7}+660f_{8}-158f_{9}+17f_{10})+3h^{11}f^{(11)}(\xi )}
h
8
f
(
8
)
(
x
9
)
=
1
3
(
17
f
0
−
179
f
1
+
849
f
2
−
2388
f
3
+
4410
f
4
−
5586
f
5
+
4914
f
6
−
2964
f
7
+
1173
f
8
−
275
f
9
+
29
f
10
)
+
26
3
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{8}f^{(8)}(x_{9})={\frac {1}{3}}(17f_{0}-179f_{1}+849f_{2}-2388f_{3}+4410f_{4}-5586f_{5}+4914f_{6}-2964f_{7}+1173f_{8}-275f_{9}+29f_{10})+{\frac {26}{3}}h^{11}f^{(11)}(\xi )}
h
8
f
(
8
)
(
x
10
)
=
1
3
(
29
f
0
−
302
f
1
+
1416
f
2
−
3936
f
3
+
7182
f
4
−
8988
f
5
+
7812
f
6
−
4656
f
7
+
1821
f
8
−
422
f
9
+
44
f
10
)
+
55
3
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{8}f^{(8)}(x_{10})={\frac {1}{3}}(29f_{0}-302f_{1}+1416f_{2}-3936f_{3}+7182f_{4}-8988f_{5}+7812f_{6}-4656f_{7}+1821f_{8}-422f_{9}+44f_{10})+{\frac {55}{3}}h^{11}f^{(11)}(\xi )}
11-punts 9e afgeleide:
h
9
f
(
9
)
(
x
0
)
=
1
2
(
−
11
f
0
+
108
f
1
−
477
f
2
+
1248
f
3
−
2142
f
4
+
2520
f
5
−
2058
f
6
+
1152
f
7
−
423
f
8
+
92
f
9
−
9
f
10
)
+
12
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{9}f^{(9)}(x_{0})={\frac {1}{2}}(-11f_{0}+108f_{1}-477f_{2}+1248f_{3}-2142f_{4}+2520f_{5}-2058f_{6}+1152f_{7}-423f_{8}+92f_{9}-9f_{10})+12h^{11}f^{(11)}(\xi )}
h
9
f
(
9
)
(
x
1
)
=
1
2
(
−
9
f
0
+
88
f
1
−
387
f
2
+
1008
f
3
−
1722
f
4
+
2016
f
5
−
1638
f
6
+
912
f
7
−
333
f
8
+
72
f
9
−
7
f
10
)
+
15
2
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{9}f^{(9)}(x_{1})={\frac {1}{2}}(-9f_{0}+88f_{1}-387f_{2}+1008f_{3}-1722f_{4}+2016f_{5}-1638f_{6}+912f_{7}-333f_{8}+72f_{9}-7f_{10})+{\frac {15}{2}}h^{11}f^{(11)}(\xi )}
h
9
f
(
9
)
(
x
2
)
=
1
2
(
−
7
f
0
+
68
f
1
−
297
f
2
+
768
f
3
−
1302
f
4
+
1512
f
5
−
1218
f
6
+
672
f
7
−
243
f
8
+
52
f
9
−
5
f
10
)
+
4
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{9}f^{(9)}(x_{2})={\frac {1}{2}}(-7f_{0}+68f_{1}-297f_{2}+768f_{3}-1302f_{4}+1512f_{5}-1218f_{6}+672f_{7}-243f_{8}+52f_{9}-5f_{10})+4h^{11}f^{(11)}(\xi )}
h
9
f
(
9
)
(
x
3
)
=
1
2
(
−
5
f
0
+
48
f
1
−
207
f
2
+
528
f
3
−
882
f
4
+
1008
f
5
−
798
f
6
+
432
f
7
−
153
f
8
+
32
f
9
−
3
f
10
)
+
3
2
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{9}f^{(9)}(x_{3})={\frac {1}{2}}(-5f_{0}+48f_{1}-207f_{2}+528f_{3}-882f_{4}+1008f_{5}-798f_{6}+432f_{7}-153f_{8}+32f_{9}-3f_{10})+{\frac {3}{2}}h^{11}f^{(11)}(\xi )}
h
9
f
(
9
)
(
x
4
)
=
1
2
(
−
3
f
0
+
28
f
1
−
117
f
2
+
288
f
3
−
462
f
4
+
504
f
5
−
378
f
6
+
192
f
7
−
63
f
8
+
12
f
9
−
f
10
)
+
1
4
h
12
f
(
12
)
(
ξ
)
{\displaystyle h^{9}f^{(9)}(x_{4})={\frac {1}{2}}(-3f_{0}+28f_{1}-117f_{2}+288f_{3}-462f_{4}+504f_{5}-378f_{6}+192f_{7}-63f_{8}+12f_{9}-f_{10})+{\frac {1}{4}}h^{12}f^{(12)}(\xi )}
h
9
f
(
9
)
(
x
5
)
=
1
2
(
−
f
0
+
8
f
1
−
27
f
2
+
48
f
3
−
42
f
4
+
42
f
6
−
48
f
7
+
27
f
8
−
8
f
9
+
f
10
)
−
1
2
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{9}f^{(9)}(x_{5})={\frac {1}{2}}(-f_{0}+8f_{1}-27f_{2}+48f_{3}-42f_{4}+42f_{6}-48f_{7}+27f_{8}-8f_{9}+f_{10})-{\frac {1}{2}}h^{11}f^{(11)}(\xi )}
h
9
f
(
9
)
(
x
6
)
=
1
2
(
f
0
−
12
f
1
+
63
f
2
−
192
f
3
+
378
f
4
−
504
f
5
+
462
f
6
−
288
f
7
+
117
f
8
−
28
f
9
+
3
f
10
)
−
1
4
h
12
f
(
12
)
(
ξ
)
{\displaystyle h^{9}f^{(9)}(x_{6})={\frac {1}{2}}(f_{0}-12f_{1}+63f_{2}-192f_{3}+378f_{4}-504f_{5}+462f_{6}-288f_{7}+117f_{8}-28f_{9}+3f_{10})-{\frac {1}{4}}h^{12}f^{(12)}(\xi )}
h
9
f
(
9
)
(
x
7
)
=
1
2
(
3
f
0
−
32
f
1
+
153
f
2
−
432
f
3
+
798
f
4
−
1008
f
5
+
882
f
6
−
528
f
7
+
207
f
8
−
48
f
9
+
5
f
10
)
+
3
2
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{9}f^{(9)}(x_{7})={\frac {1}{2}}(3f_{0}-32f_{1}+153f_{2}-432f_{3}+798f_{4}-1008f_{5}+882f_{6}-528f_{7}+207f_{8}-48f_{9}+5f_{10})+{\frac {3}{2}}h^{11}f^{(11)}(\xi )}
h
9
f
(
9
)
(
x
8
)
=
1
2
(
5
f
0
−
52
f
1
+
243
f
2
−
672
f
3
+
1218
f
4
−
1512
f
5
+
1302
f
6
−
768
f
7
+
297
f
8
−
68
f
9
+
7
f
10
)
+
4
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{9}f^{(9)}(x_{8})={\frac {1}{2}}(5f_{0}-52f_{1}+243f_{2}-672f_{3}+1218f_{4}-1512f_{5}+1302f_{6}-768f_{7}+297f_{8}-68f_{9}+7f_{10})+4h^{11}f^{(11)}(\xi )}
h
9
f
(
9
)
(
x
9
)
=
1
2
(
7
f
0
−
72
f
1
+
333
f
2
−
912
f
3
+
1638
f
4
−
2016
f
5
+
1722
f
6
−
1008
f
7
+
387
f
8
−
88
f
9
+
9
f
10
)
+
15
2
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{9}f^{(9)}(x_{9})={\frac {1}{2}}(7f_{0}-72f_{1}+333f_{2}-912f_{3}+1638f_{4}-2016f_{5}+1722f_{6}-1008f_{7}+387f_{8}-88f_{9}+9f_{10})+{\frac {15}{2}}h^{11}f^{(11)}(\xi )}
h
9
f
(
9
)
(
x
10
)
=
1
2
(
9
f
0
−
92
f
1
+
423
f
2
−
1152
f
3
+
2058
f
4
−
2520
f
5
+
2142
f
6
−
1248
f
7
+
477
f
8
−
108
f
9
+
11
f
10
)
+
12
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{9}f^{(9)}(x_{10})={\frac {1}{2}}(9f_{0}-92f_{1}+423f_{2}-1152f_{3}+2058f_{4}-2520f_{5}+2142f_{6}-1248f_{7}+477f_{8}-108f_{9}+11f_{10})+12h^{11}f^{(11)}(\xi )}
11-punts 10e afgeleide:
h
10
f
(
10
)
(
x
0
)
=
f
0
−
10
f
1
+
45
f
2
−
120
f
3
+
210
f
4
−
252
f
5
+
210
f
6
−
120
f
7
+
45
f
8
−
10
f
9
+
f
10
−
5
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{10}f^{(10)}(x_{0})=f_{0}-10f_{1}+45f_{2}-120f_{3}+210f_{4}-252f_{5}+210f_{6}-120f_{7}+45f_{8}-10f_{9}+f_{10}-5h^{11}f^{(11)}(\xi )}
h
10
f
(
10
)
(
x
1
)
=
f
0
−
10
f
1
+
45
f
2
−
120
f
3
+
210
f
4
−
252
f
5
+
210
f
6
−
120
f
7
+
45
f
8
−
10
f
9
+
f
10
−
4
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{10}f^{(10)}(x_{1})=f_{0}-10f_{1}+45f_{2}-120f_{3}+210f_{4}-252f_{5}+210f_{6}-120f_{7}+45f_{8}-10f_{9}+f_{10}-4h^{11}f^{(11)}(\xi )}
h
10
f
(
10
)
(
x
2
)
=
f
0
−
10
f
1
+
45
f
2
−
120
f
3
+
210
f
4
−
252
f
5
+
210
f
6
−
120
f
7
+
45
f
8
−
10
f
9
+
f
10
−
3
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{10}f^{(10)}(x_{2})=f_{0}-10f_{1}+45f_{2}-120f_{3}+210f_{4}-252f_{5}+210f_{6}-120f_{7}+45f_{8}-10f_{9}+f_{10}-3h^{11}f^{(11)}(\xi )}
h
10
f
(
10
)
(
x
3
)
=
f
0
−
10
f
1
+
45
f
2
−
120
f
3
+
210
f
4
−
252
f
5
+
210
f
6
−
120
f
7
+
45
f
8
−
10
f
9
+
f
10
−
2
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{10}f^{(10)}(x_{3})=f_{0}-10f_{1}+45f_{2}-120f_{3}+210f_{4}-252f_{5}+210f_{6}-120f_{7}+45f_{8}-10f_{9}+f_{10}-2h^{11}f^{(11)}(\xi )}
h
10
f
(
10
)
(
x
4
)
=
f
0
−
10
f
1
+
45
f
2
−
120
f
3
+
210
f
4
−
252
f
5
+
210
f
6
−
120
f
7
+
45
f
8
−
10
f
9
+
f
10
−
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{10}f^{(10)}(x_{4})=f_{0}-10f_{1}+45f_{2}-120f_{3}+210f_{4}-252f_{5}+210f_{6}-120f_{7}+45f_{8}-10f_{9}+f_{10}-h^{11}f^{(11)}(\xi )}
h
10
f
(
10
)
(
x
5
)
=
f
0
−
10
f
1
+
45
f
2
−
120
f
3
+
210
f
4
−
252
f
5
+
210
f
6
−
120
f
7
+
45
f
8
−
10
f
9
+
f
10
−
5
12
h
12
f
(
12
)
(
ξ
)
{\displaystyle h^{10}f^{(10)}(x_{5})=f_{0}-10f_{1}+45f_{2}-120f_{3}+210f_{4}-252f_{5}+210f_{6}-120f_{7}+45f_{8}-10f_{9}+f_{10}-{\frac {5}{12}}h^{12}f^{(12)}(\xi )}
h
10
f
(
10
)
(
x
6
)
=
f
0
−
10
f
1
+
45
f
2
−
120
f
3
+
210
f
4
−
252
f
5
+
210
f
6
−
120
f
7
+
45
f
8
−
10
f
9
+
f
10
+
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{10}f^{(10)}(x_{6})=f_{0}-10f_{1}+45f_{2}-120f_{3}+210f_{4}-252f_{5}+210f_{6}-120f_{7}+45f_{8}-10f_{9}+f_{10}+h^{11}f^{(11)}(\xi )}
h
10
f
(
10
)
(
x
7
)
=
f
0
−
10
f
1
+
45
f
2
−
120
f
3
+
210
f
4
−
252
f
5
+
210
f
6
−
120
f
7
+
45
f
8
−
10
f
9
+
f
10
+
2
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{10}f^{(10)}(x_{7})=f_{0}-10f_{1}+45f_{2}-120f_{3}+210f_{4}-252f_{5}+210f_{6}-120f_{7}+45f_{8}-10f_{9}+f_{10}+2h^{11}f^{(11)}(\xi )}
h
10
f
(
10
)
(
x
8
)
=
f
0
−
10
f
1
+
45
f
2
−
120
f
3
+
210
f
4
−
252
f
5
+
210
f
6
−
120
f
7
+
45
f
8
−
10
f
9
+
f
10
+
3
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{10}f^{(10)}(x_{8})=f_{0}-10f_{1}+45f_{2}-120f_{3}+210f_{4}-252f_{5}+210f_{6}-120f_{7}+45f_{8}-10f_{9}+f_{10}+3h^{11}f^{(11)}(\xi )}
h
10
f
(
10
)
(
x
9
)
=
f
0
−
10
f
1
+
45
f
2
−
120
f
3
+
210
f
4
−
252
f
5
+
210
f
6
−
120
f
7
+
45
f
8
−
10
f
9
+
f
10
+
4
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{10}f^{(10)}(x_{9})=f_{0}-10f_{1}+45f_{2}-120f_{3}+210f_{4}-252f_{5}+210f_{6}-120f_{7}+45f_{8}-10f_{9}+f_{10}+4h^{11}f^{(11)}(\xi )}
h
10
f
(
10
)
(
x
10
)
=
f
0
−
10
f
1
+
45
f
2
−
120
f
3
+
210
f
4
−
252
f
5
+
210
f
6
−
120
f
7
+
45
f
8
−
10
f
9
+
f
10
+
5
h
11
f
(
11
)
(
ξ
)
{\displaystyle h^{10}f^{(10)}(x_{10})=f_{0}-10f_{1}+45f_{2}-120f_{3}+210f_{4}-252f_{5}+210f_{6}-120f_{7}+45f_{8}-10f_{9}+f_{10}+5h^{11}f^{(11)}(\xi )}
