should do for your example.
Suppose f: (a,b) -> R is differentiable and |f'(x)|<=M for all x in (a,b). Prove that f is uniformly continuous on (a,b). Give an example of a function f: (0,1)->R that is differentiable and uniformly continuous on (0,1) but such that f' is unbounded.
I have no clue.