Derivative Proof of Power Rule
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 xn would be as follows
Using the Binomial Theorem, we get
Subtract the xn
Factor out an h
All of the terms with an h will go to 0, and then we are left with
Implicit Differentiation Proof of Power Rule
If we don't want to get messy with the Binomial Theorem, we can simply use implicit differentiation, which is basically treating y as f(x) and using Chain rule.
Take the natural log of both sides
Take the derivative with respect to x
Notice that we took the derivative of lny and used chain rule as well to take the derivative of the inside function y.
Multiply both sides by y
Substitute xc back in for y