Finding the derivative at a point turns into a viewing window problem.
I want to find a delta-epsilon window so that the curve looks like a line.
(I should not be able to tell the difference between the curve and the secant line from
(c,f(c)) to (c+delta, f(c+delta)).)
I then approximate the derivative by finding rise/run for the line.

For most functions, it is easiest to use the calculator definition of derivative. Set x-scale to 0.001, and set y-scale to something that keeps the curve in the viewing window.
Use this method to find the derivative of the given function at three points. then try with another function.
If you look at a badly behaved function, like at x=4/101, you need a smaller epsilon.
This applet is meant as an illustration of the definition of a derivative at a point.
It has the advantage of showing that numeric differentiation is quite robust.
The default curve is a parabola, where students will be able to find the derivative symbolically.