Question: Give an example of a function f that is differentiable on [0,1] but its derivative is not bounded on [0,1].

Ok, I know that the derivative f' cannot be continuous, because then it would be bounded on [0,1]. I also know that it cannot be increasing with jump discontinuities, because the derivative has to have the intermediate value property. I also do not think (but am not 100 percent sure) that it can have any infinite discontinuities, because I think that would make f unbounded, which it cannot be on [0,1] if its differentiable on [0,1].

Any ideas?