Thread: Most general form for calculating derivatives using limits.

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 this finite differencing expression 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.

Any help would be greatly appreciated. Thank you.

I know it has something to do with weighted averages, but I can't wrap my head around it.

I figured it out, if any of you are interested.

If you set $\displaystyle
lim_{h\rightarrow0} \frac{f(x+nh)-f(x-zh)}{h(n+z)} = P$, then P is the derivative.

Using a weighted average and the finite differencing formula, the most general form for calculating a first derivative is:

$\displaystyle \sum_{i = 1}^{n}\frac{a_{i}P_{i}}{a_{i}}$