Old paradigm problem. Here is f, using the definition of derivative find f'.
New paradigm problem. Here is f. Here is a Mathematica function defining f(x+h) - f(x) divided by h. Graph this function for appropriate values of x and h to show whether or not f is differentiable at 2.
The old paradigm problem you just proceed as in all the examples. In the new problem they get intimidated because it involves using a computer and not a graphing calculator. They don't understand that you keep x fixed at 2 and vary h around 0. When they start the problem they give nonsensical input to the computer, get nonsensical output and promptly blame the stupid program.
If students are allowed to use Mathematica during a test then asking them to find f' using the definition of f is not helpful in determining if they understand the concepts. They, for all reasonable problems, just need to execute a single command:
Limit[ (f[x+h]-f[x])/h, h->0]
This doesn't really test their understanding. If one is going to allow students to use Mathematica on a test then the problems need to be adjusted.
