A problem I like to give students to solve shortly after introducing the derivative is to evaluate $f'(2)$ for $f(x) = x^x$. Of course, this function can be rewritten as $f(x) = e^{x\ln x}$ but in my experience students don't think of this. In fact, students who have seen Calculus before almost universally reach the solution $f'(2) = 4$ which they get from the mistaken idea that $f'(x) = x\cdot x^{x-1} = x^x$. The only students that usually get this problem correct are those that haven't yet learned any of the computational methods and only know the definition.

I teach the limit definition and emphasize the physical and geometric interpretations, and then move from that to the concept of the tangent line and linear approximation. I think these concepts encapsulate most of what is significant (intuitively) about the definition. I dislike exam questions that require students to compute derivatives using the limit definition when they know a "better" way to do it. It isn't too hard to write a problem where no formula for the function is given and ask students questions about the sign or approximate magnitude of the derivative or whether or not the function should even have a derivative. For students who to do not intend to pursue mathematics, this seems appropriate to me. Even those who become mathematicians will almost surely see these ideas again in complete detail in an elementary analysis course.