Formula

Newton's Divided Difference formula

1. Find equation using Newton's Divided Difference Interpolation formula `f(x) = y_0 + (x - x_0) f[x_0, x_1] + (x - x_0)(x - x_1) f[x_0, x_1, x_2] + (x - x_0)(x - x_1)(x - x_2) f[x_0, x_1, x_2, x_3] + ...` 2. Now, differentiate f(x) with respect to x to get f'(x) and f''(x) 3. Now, substitute value of `x` in f'(x) and f''(x)

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Examples
1. Using Newton's Divided Difference formula to find solution

x	f(x)
2	4
4	56
9	711
10	980

x = 5

Solution:
The value of table for `x` and `y`

x	2	4	9	10
y	4	56	711	980

Numerical divided differences method to find solution

Newton's divided difference table is

x	y	`1^(st)` order	`2^(nd)` order	`3^(rd)` order
2	4
		26
4	56		15
		131		1
9	711		23
		269
10	980

Newton's divided difference interpolation formula is
`f(x) = y_0 + (x - x_0) f[x_0, x_1] + (x - x_0)(x - x_1) f[x_0, x_1, x_2] + (x - x_0)(x - x_1)(x - x_2) f[x_0, x_1, x_2, x_3]`

`f(x) = 4 + (x -2) xx 26 + (x -2)(x -4) xx 15 + (x -2)(x -4)(x -9) xx 1`

`f(x) = 4 + (x-2) xx 26 + (x^2-6x+8) xx 15 + (x^3-15x^2+62x-72) xx 1`

`f(x) = 4 + (26x-52) + (15x^2-90x+120) + (x^3-15x^2+62x-72) `

`f(x) = x^3-2x `

Now, differentiate with x
`f'(x)=3x^2-2`

`f''(x)=6x`

Now, substitute `x=5`

`f'(5)=3 xx 5^2-2=73`

`f''(5)=6 xx 5=30`

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