What is meant when $f:[a,b] \to \mathbb R$ is said to be differentiable? Sometimes I see an exercise like this: 

Let $f:[a,b] \to \mathbb R$ be differentiable. (A few more givens here.) Show that $f'$ has such-and-such property.

What is usually meant by that? Should $f'$ be defined on $[a,b]$ with one-sided derivatives in $a,b$? Or should one rather assume $(a,b)$ to be the domain of $f'$?

Usually this would mean that $f$ has one-sided derivatives at the endpoints.
That's the natural result of defining $f'(x_0)$ as the limit $\lim_{x\to x_0}\frac{f(x)-f(x_0)}{x-x_0}$ when $x\to x_0$ is taken to mean that $x$ approaches $x_0$ within the domain of the function.
