Forward and backward differentiation?

Question

Given a set of data
example: x = [a b c d e f] & y = [g h i j k l]
a question reads: find the most accurate approximation of dy/dx evaluated at x = a or b or c or d or e or f
how do you know to use the backwards or forwards differentiation formula. ( I think it has to do with where your asked to take the derivative) Please help!

Answer 1

Clearly, at the end points, you don't have much choice. But in the middle you have several choices.

One of the better choices is to run a parabola through three adjacent points and evaluate the derivative of the parabola at the middle point. This is equivalent to taking the weighted average of the forward and backward formlua results

If the points are evenly spaced, this is just the normal average.

If not, then you want to weigh the ratio to the closer point more heavily.

That is, if D01 is the result of the backward formula at X1, and D12 is the result of the forward formula at X1 (equal to the backward formula at X2), then try:

D = D12(X1 - X0) / (X2 - X0) + D01(X2 - X1) / (X2 - X0)

As noted, this is equivalent to running a parabola through the three points and taking the derivative of the parabola at X1.

This approach can be extended to use the cubic instead; or to use all the points to create a cubic spline; etc.