Hi Guys currently doing some Numerical Differentiation and I was looking at the different formulas such as 1) forward difference method 2) backward difference method 3) central difference method
$$\frac{f(x+h)-f(x)}{h}$$
$$\frac{f(x)-f(x-h)}{h}$$
$$\frac{f(x+h)-f(x-h)}{2}$$
Now based on using taylor's series, I know we can derive different or higher orders for the derivatives for example if i required $$f''(x)$$
using Taylor's series
$$f(x+h) = f(x)+f'(x)h+\frac{f''(x)h^2}{2!}+\frac{f'''(x)h^3}{3!}$$
$$f(x-h) = f(x)-f'(x)h+\frac{f''(x)h^2}{2!}-\frac{f'''(x)h^3}{3!}$$
$$\frac{f(x+h)-2f(x)+f(x-h)}{h^2} = f''(x)$$
My question is as I am bot confused to define the third derivative when using the Taylor series when making the substitution from the Taylor series and this is my approach then adding both equations $$f(x+h) + f(x-h)$$
$$f(x+h)-2f(x)+h^2f''(x)+f(x-h) = \frac{h^3f^3(x)}{3!} - \frac{h^3f^3(x)}{3!}$$
then I decided to try $$f(x+h) - f(x-h) = hf(x)+hf(x)+\frac{2h^3f'''(x)}{3!}$$
$$\frac{3![f(x+h)+2h(x)-f(x-h)]}{2h^3} = f^3(x)$$
and this is incorrect so I am hoping someone can help me understand defining higher order derivatives when using the Taylor series for these methods.
Essentially I am trying to understand how to attain
$$f^3(x),f^4(x),.. etc $$
etc using the Taylor series