Though there are many different ways to prove the rules for finding a derivative, the most common way to set up a proof of these rules is to go back to the limit definition. This way, we can see how the limit definition works for various functions.
We must remember that mathematics is a succession. It builds on itself, so many proofs rely on results of other proofs - more specifically, complex proofs of derivatives rely on knowing basic derivatives. We can also use derivative rules to prove derivatives, but even those are build off of basic principles in Calculus. For the sake of brevity, we won't go through every proof, but it is important to know how many of these derivatives were obtained.
It is important to understand that we are not simply "proving a derivative," but seeing how various rules work for computing the derivative.