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.

Let

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 x^c back in for y