This proof requires a lot of work if you are not familiar with implicit differentiation, which is basically differentiating a variable in terms of x. Some may try to prove the power rule by repeatedly using product rule. Though it is not a "proper proof," it can still be good practice using mathematical induction. A common proof that is used is using the Binomial Theorem:

The limit definition for $x^n$ would be as follows

Using the Binomial Theorem, we get

Subtract the $x^n$.

Factor out an h

All of the terms with an h will go to 0, and then we are left with