This has already been said before, but let me iterate this. The value of the derivative of a function at a point is evaluated by computing a limit, if it exists. The limit of a sequence may not exist if at least one of the following happens:
the absolute value of some of the terms grows without bounds;
you can extract two subsequences which both converge, but they converge to different values.
It is important to keep in mind that both behaviors may happen for the same limit, and that for 2 you might be able to extract countably many subsequences each converging to a different value (only "countably many" as I am talking now of a limit of a sequence). EDIT: in fact, you can have uncountably many different limits (see Jason's comment below).
The situation for limits of functions is not simpler than the above, since you can "embed" any sequence into the limit defining the derivative of a continuous function at a point. One of the good consequences of the fact that you are dealing with a limit of a continuous function is that if situation 2 occurs and the two subsequences you found have limits a<
b, then also every intermediate value $c \in [a,b]$ is the limit of a subsequence. An elaboration of this is that the derivative of a once differentiable function satisfies the intermediate value property (Darboux's Theorem).
Thus, if you analyze the behavior of the values of the limit that defines a derivative at a point you may find that, as you zoom in, the values you obtain
a. also zoom in on a single real number - the derivative exists;
b. are always contained in some finite interval, no matter how close to the actual limiting point you get - e.g. the function $x^2 \sin(1/x)$ near 0;
c. are unbounded (possibly with alternating signs) - e.g. the function $\sqrt{x} \sin (1/x)$.
Note also that further pathologies include the fact that the behaviors as you approach the limit from below or from above may well be different. Even crazier, the "overall" interval you find in b, or an infinite interval in the case c, may contain inside it also countably many of other intervals nested more or less arbitrarily.
To conclude, derivatives of continuous functions are so immensely complicated that there is very little you can say about them!