Formula

1. For `x=x_0` `[(dy)/(dx)]_(x=x_0) = 1/h * [1/2 * (Delta y_0 + Delta y_(-1)) - 1/12 * (Delta^3 y_(-1) + Delta^3 y_(-2)) + 1/60 * (Delta^5 y_(-2) + Delta^5 y_(-3)) + ...]` `[(d^2y)/(dx^2)]_(x=x_0) = 1/h^2 * [Delta^2 y_(-1) - 1/12 Delta^4 y_(-2) + 1/90 * Delta^6 y_(-3) + ...]`

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Examples
1. Using Stirling's formula to find solution

x	f(x)
7.47	0.193
7.48	0.195
7.49	0.198
7.50	0.201
7.51	0.203
7.52	0.206
7.53	0.208

x = 7.5

Solution:
Stirling's formula (central difference formula).
The value of table for `x` and `y`

x	7.47	7.48	7.49	7.5	7.51	7.52	7.53
y	0.193	0.195	0.198	0.201	0.203	0.206	0.208

Difference table is

x	y	`Deltay`	`Delta^2y`	`Delta^3y`	`Delta^4y`	`Delta^5y`	`Delta^6y`
7.47	0.193
		0.002
7.48	0.195		0.001
		0.003		-0.001
7.49	0.198		0		0
		0.003		-0.001		0.003
7.5	0.201		-0.001		0.003		-0.01
		0.002		0.002		-0.007
7.51	0.203		0.001		-0.004
		0.003		-0.002
7.52	0.206		-0.001
		0.002
7.53	0.208

The value of `x` at you want to find `f(x) : x_0 = 7.5`

`h = x_1 - x_0 = 7.48 - 7.47 = 0.01`


Stirling's Formula is
`[(dy)/(dx)]_(x=x_0) = 1/h * [1/2 * (Delta y_0 + Delta y_(-1)) - 1/12 * (Delta^3 y_(-1) + Delta^3 y_(-2)) + 1/60 * (Delta^5 y_(-2) + Delta^5 y_(-3)) + ...]`

`:.[(dy)/(dx)]_(x=7.5) = 1/0.01 * [1/2 * (0.002 +0.003) - 1/12 * (0.002 -0.001)+ 1/60 * (-0.007 +0.003)]`

`:.[(dy)/(dx)]_(x=7.5) = 1/0.01 * [0.0025-0.0000833333-0.0000666667]`

`:.[(dy)/(dx)]_(x=7.5) = 0.235`


`[(d^2y)/(dx^2)]_(x=x_0) = 1/h^2 * [Delta^2 y_(-1) - 1/12 Delta^4 y_(-2) + 1/90 * Delta^6 y_(-3) + ...]`

`:.[(d^2y)/(dx^2)]_(x=7.5) = 1/0.0001 * [-0.001 - 1/12 * 0.003+ 1/90 * -0.01]`

`:.[(d^2y)/(dx^2)]_(x=7.5) = 1/0.0001 * [-0.001-0.00025-0.0001111111]`

`:.[(d^2y)/(dx^2)]_(x=7.5) = -13.61111`

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