Formula

1. For `x=x_0` `[(dy)/(dx)]_(x=x_0) = 1/h * [Delta y_0 - 1/4 * (Delta^2 y_(0) + Delta^2 y_(-1)) + 1/12 Delta^3 y_(-1) + 1/24 * (Delta^4 y_(-1) + Delta^4 y_(-2)) - 1/120 Delta^5 y_(-2) - 1/120 * (Delta^6 y_(-2) + Delta^6 y_(-3)) + ...]` `[(d^2y)/(dx^2)]_(x=x_0) = 1/h^2 * [1/2 * (Delta^2 y_(0) + Delta^2 y_(-1)) - 1/2 Delta^3 y_(-1) - 1/24 * (Delta^4 y_(-1) + Delta^4 y_(-2)) + ...]`

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Examples
1. Using Bessel'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:
Bessel'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`


Bessel's Formula is
`[(dy)/(dx)]_(x=x_0) = 1/h * [Delta y_0 - 1/4 * (Delta^2 y_(0) + Delta^2 y_(-1)) + 1/12 Delta^3 y_(-1) + 1/24 * (Delta^4 y_(-1) + Delta^4 y_(-2)) - 1/120 Delta^5 y_(-2) - 1/120 * (Delta^6 y_(-2) + Delta^6 y_(-3)) + ...]`

`:.[(dy)/(dx)]_(x=7.5) = 1/0.01 * [0.002 - 1/4 * (0.001 -0.001) + 1/12 * (0.002) + 1/24 * (-0.004 +0.003) - 1/120 * (-0.007) - 1/120 * (0 -0.01) + ...]`

`:.[(dy)/(dx)]_(x=7.5) = 1/0.01 * [0.002+0+0.0001666667-0.0000416667+0.0000583333+0.0000833333]`

`:.[(dy)/(dx)]_(x=7.5) = 0.2267`


`[(d^2y)/(dx^2)]_(x=x_0) = 1/h^2 * [1/2 * (Delta^2 y_(0) + Delta^2 y_(-1)) - 1/3 Delta^3 y_(-1) - 1/24 * (Delta^4 y_(-1) + Delta^4 y_(-2)) + 1/24 Delta^5 y_(-2) + 1/180 * (Delta^6 y_(-2) + Delta^6 y_(-3)) + ...]`

`:.[(d^2y)/(dx^2)]_(x=7.5) = 1/0.0001 * [1/2 * (0.001 -0.001) - 1/2 * (0.002) - 1/24 * (-0.004 +0.003) + 1/24 * (-0.007) + 1/180 * (0 -0.01)]`

`:.[(d^2y)/(dx^2)]_(x=7.5) = 1/0.0001 * [0-0.001+0.0000416667-0.0002916667-0.0000555556]`

`:.[(d^2y)/(dx^2)]_(x=7.5) = -13.0556`

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