Most general form for calculating derivatives using limits.

**xwanderingpoetx** · November 5th 2009, 03:25 PM

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

Right now I have:

$\lim_{h\rightarrow0} \frac{f(x+nh)-f(x-zh)}{h(n+z)} = f'(x)$

I need to incorporate this finite differencing expression into the above formula.

$\frac{-f(x+2h)+8f(x+h)-8f(x-h)+f(x-2h)}{12h}$

The finite differencing expression is also equal to the derivative.

Any help would be greatly appreciated. Thank you.

**xwanderingpoetx** · November 6th 2009, 05:25 AM

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

**xwanderingpoetx** · November 6th 2009, 11:45 AM

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

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

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

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

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