# Most general form for calculating derivatives using limits.

• Nov 5th 2009, 04:25 PM
xwanderingpoetx
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:

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

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.
• Nov 6th 2009, 06:25 AM
xwanderingpoetx
I know it has something to do with weighted averages, but I can't wrap my head around it.
• Nov 6th 2009, 12:45 PM
xwanderingpoetx
I figured it out, if any of you are interested.

If you set $
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:

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