The definition of derivatives is useful in exercises about functional equations. Ever solve $f(x+y)=f(x)f(y)$ ? A more elaborate one is $[f(x):f(y):f(z):f(t)]=[x:y:z:t]$, functions preserving the cross-ratio (= anharmonic ratio).
However, we should not neglect the interest of the black-box side of mathematics. We should remember that it is this aspect which has made mathematics so much unavoidable in Science. Somehow, it participates to the unreasonable effectiveness of Mathematics in the Natural Sciences'' (E. Wigner's famous statement). After all, the definition of derivatives has the same status as the constructions of ${\mathbb Z},{\mathbb Q},{\mathbb R},{\mathbb C}$. One can spend a year without thinking about them, while using these fundamental objects every hour, by applying rules. Do you remember the construction of the polynomial algebra $k[X]$ ? How would you define $\pi$ ? In a more advanced situation, chemists have efficient rules to deal with characters of representations of finite groups, and they do not need to read a justification, or to remember it, even though the first Chapter of J.-P. Serre's book was intended to be read by his chemist wife. Mathematics is the tool box of Science. It is even a tool box for itself, in the sense that new topics use the older ones. To go further, we must accept older truths. Of course, it is way better to accept them for good reasons, that is, because we have completely understood the definitions. But if the half of a classroom, who does not intend to do mathematical research, neglects the definition and prefer focussing on the rules, there is no problem at all, provided they apply the rules correctly. There are many ways to learn rules, one of them being solving a lot of exercises.