## Chapters

# Power Rule

Now that you've learned some methods of differentiating complex functions, let's take a look at functions of the following form:

How could we find $f'\left(x\right)$ ?

It turns out that there's two ways to do this. We could use either induction or the binomial theorem (or some third method that you can email me about :)). I'm going to go through the induction strategy first, because it's a bit simpler. If you don't like induction or haven't learned it yet (it's super cool!) you can skip ahead to the other method. Otherwise, let's go!

#### Induction

In order to use induction, we need an inductive hypothesis and a few base cases to support it. We can easily compute the following base cases by hand:

And so, we can hypothesize that

is true for all $n\lt k$ (This is something we do a lot in induction).Then, we can try to find the derivative of $x^k.$ (Remember that the inductive hypothesis does not apply here, because $k\nless k.$ To do this, we can use the product rule:

And we are done! Because we've proved that the pattern begins, and we've also proved that it's possible to get from a base case to a non base case while holding the pattern, we've proved that:

#### Binomial Theorem

The proof of this derivative is more commonly done using the Binomial theorem, which says that:

This, as you will see soon, turns out to be quite a useful definition for us. To calculate the derivative this time, we'll be using a first principle limits approach. So, let's set up our limit:

Notice that I'm using $h$ instead of $\delta x.$ It's just a different name I gave it, and you can call it whatever you want. I generally like calling it $h,$ just because it's simpler to write.

Let's (partially) expand our binomial, using the binomial theorem:

Substituting $h=0,$

And we are done!