Most general form for calculating derivatives using limits.

[xwanderingpoetx]

Right now I am working on finding the most general form for calculating a derivative using limits.

Right now I have:
\(\displaystyle \lim_{h\rightarrow0} \frac{f(x+nh)-f(x-zh)}{h(n+z)} = f'(x)\)


I need to incorporate the finite differencing expression below into the above formula.
\(\displaystyle \frac{-f(x+2h)+8f(x+h)-8f(x-h)+f(x-2h)}{12h}\)


The finite differencing expression is also equal to the derivative.

I know this has something to do with weighted averages, but I can't figure it out.

Any help would be greatly appreciated. Thank you.

 

[tkhunny]

The finite differencing expression is also equal to the derivative.

You should first work on this statement. Only the derivative is equal to the derivative. ANY finite difference, at best, will be equivalent over part of the Domain of the function. If you have a specific application, over a ver small Domain, perhaps you can invent something like that and consdier it generally applicable.

In any case, why can't you just apply the general derivatve formulation to f'(x) and produce a result close to your second derivative finite difference?