I've been told that I can't, in general, differentiate functions involving absolute values (e.g. f(x) = |x|). Why is that?

When you differentiate a function which has only one parameter, like f(x), you are finding a new function, f'(x), which gives the gradient of your original function at every point. You'll probably remember this from when you first defined differentiation: you started by working out the average gradient for your function over a finite interval, then shrank that interval down 'til it was infinitesimally small to give you the gradient of f(x) at every point.

The problem with functions involving absolute values is that lots of them don't have a well defined gradient at every point. Look at the graph of f(x)=|x|. [I'll draw it on the whiteboard.] When x< 0 the gradient is -1, and when x>0 the gradient is 1. But what happens at x=0? It's a spike: the gradient is snapping between -1 and 1, but there is no single gradient here. [Draw some possible tangents on the graph to demonstrate.] So there can be no function which gives the gradient of |x| at every point; you can't differentiate |x|.

[If they showed that they understood this then I'd move on to discuss the smoothness of functions and how functions, like |x|, are smooth over certain ranges.]