# 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 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

## 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