Differentiation is a mechanical process. You follow the steps and you get the right answer. Performing differentiation is mostly an exercise in following rule

Differentiation is a mechanical process. You follow the steps and you get the right answer. Performing differentiation is mostly an exercise in following rules. Understanding the proofs of the rules does not help in knowing how to apply the rules and for the vast majority of calculus students they need to learn how to differentiate rather than how to prove the rules of differentiation. Also, the geometric idea behind the proofs is nice but does not constitute what we consider rigorous. Such geometric intuition quickly fails in more complicated situations. If one has to learn the proofs they are better off learning the analytic proofs and how to operate with and reason about limits. On the first encounter with the subject almost no one is able to appreciate the intuition behind it. Sometimes people go back to freshen up their skills and are able appreciate and desire a deeper understanding. They wonder about the approach taken to teach the subject and ask why we do things the way we do them. They don’t see that they are no longer at all like the average student who is 18 or so learning it for the first time.