In practice there is no need for ideal differentiators because usually signals contain noise at high frequencies which should be suppresed. From the plot we can see that derivative and integration formulas pdf differences don’t resemble such behavior, all they care about is to get as closer as possible to the response of ideal differentiator, without supression of noisy high frequencies.
As a consequence they perform well only on exact values, which contain no noise. Different technique is needed for robust derivative estimation of noisy signals. Second order central difference is simple to derive. I just wanted to say how much i enjoyed finding this resource as i am taking my first course in numerical differential equations. I am having some confusion based on the definitions for the central difference operator that i am given and the one you are using.
I once wrote a Mathematica script to compute central differences and get your results except for that case. Also I have used least-squares instead of interpolation. 5 grid has only 25 degrees of freedom. Thanks for your quick reply. I can cite the reference in a paper I’m writing.