Formula

1. Four-point CDF (Central difference formula) `f^'(x)=1/(12h)[f(x-2h)-8f(x-h)+8f(x+h)-f(x+2h)]`

2. Four-point FDF (Forward difference formula) for second derivatives `f^('')(x)=(2f(x)-5f(x+h)+4f(x+2h)-f(x+3h))/(h^2)`

3. Four-point BDF (Backward difference formula) for second derivatives `f^('')(x)=(-f(x-3h)+4f(x-2h)-5f(x-h)+2f(x))/(h^2)`

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Examples
1. Using Four point Forward difference, Backward difference, Central difference formula numerical differentiation to find solution

x	1	1.05	1.10	1.15	1.20	1.25	1.30
f(x)	1	1.02470	1.04881	1.07238	1.09545	1.11803	1.14018

`f^'(1.10) and f^('')(1.10)`

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

x	1	1.05	1.1	1.15	1.2	1.25	1.3
y	1	1.0247	1.0488	1.0724	1.0954	1.118	1.1402

Four-point CDF (Central difference formula)
`f^'(x)=1/(12h)[f(x-2h)-8f(x-h)+8f(x+h)-f(x+2h)]`

`f^'(1.10)=1/(12*0.05)[f(1.10-2*0.05)-8f(1.10-0.05)+8f(1.10+0.05)-f(1.10+2*0.05)]`

`f^'(1.10)=1/0.6[f(1)-8f(1.05)+8f(1.15)-f(1.2)]`

`f^'(1.10)=1/0.6[1-8(1.0247)+8(1.0724)-1.0954]`

`f^'(1.10)=0.4767`

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Four-point FDF (Forward difference formula) for second derivatives
`f^('')(x)=(2f(x)-5f(x+h)+4f(x+2h)-f(x+3h))/(h^2)`

`f^('')(1.10)=(2f(1.10)-5f(1.10+0.05)+4f(1.10+2*0.05)-f(1.10+3*0.05))/((0.05)^2)`

`f^('')(1.10)=(2f(1.10)-5f(1.15)+4f(1.2)-f(1.25))/(0.0025)`

`f^('')(1.10)=(2(1.0488)-5(1.0724)+4(1.0954)-(1.118))/(0.0025)`

`f^('')(1.10)=-0.204`

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