Sounds like Mathematica is the way to go then? Because it turned out students who could do the problems. The 'old paradigm' leaves most behind, which serves no one.
Students could not do the computer based problems. Save for a few. More students get through with the old paradigm.
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.
To find out whether f is differentiable at some point, I'd naively draw f around 2 and look for obvious discontinuities. Why encrypt the problem in such a way? Alternatively, finding f' and explaining if it is undefined at the point in question would work even more reliable - for corner cases the visualization of a graph might hide the fact, i.e. if the resolution is too poor.
Regarding the first sentence your wrote. That's how you solve the problem. Graph the difference quotient for x=2 and h in [-0.1, 0.1] or some such suitably small region. It's quite easy to do in the computer. One just needs to know that this is what you need to do. I did not encrypt the problem. It's quite straightforward provided one knows the concepts.
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.
I see. Then I don't understand what point you were trying to make. There are lots of different ways of testing understanding of a concept. I presented one way of doing so with students having access to a computer.
The point of the problem I gave was not to find the derivative or even to know if a function is differentiable. The point is to test their understanding of the definition of derivative. It's a good problem in that is tests whether a student understands that in the two variable expression we call the difference quotient one of the variables is fixed for purposes of the definition. The students need to know that it is h that varies and not x. This is an unusual occurrence and requires some getting used to.
