# What is the derivative of #x^n#?

##### 2 Answers

For the function *not* equal 0, for reasons which will become clear. n should also be an integer or a rational number (i.e. a fraction).

The rule is:

In other words, we "borrow" the power of x and make it the coefficient of the derivative, and then subtract 1 from the power.

As I mentioned, the special case is where n=0. This means that

We can use our rule and *technically* get the right answer:

However, later on down the track, we will run into complications when we try to use the inverse of this rule.

Below are the proofs for every numbers, but only the proof for all integers use the basic skillset of the definition of derivatives. The proof for all rationals use the chain rule and for irrationals use implicit differentiation.

#### Explanation:

That being said, I'll show them all here, so you can understand the process. Beware that it

From

If

Where

Dividing that

We can take out the first term from the sum

Taking the limit, everything else still in the sum goes to zero. Calculating

For

Take out the first term

Take the limit, Where

For rationals we need to use the chain rule. I.e.:

So, knowing that

If

So, using the chain rule we have

And last but not least, using implicit differentiation we can prove for all real numbers, including the irrationals.