# Chain Rule (with Examples and Proof)

by Dave
(DAVE)—

Okay, so you know how to differentiation a function using a definition and some derivative rules. On the other hand, simple basic functions such as the fifth root of twice an input does not fall under these techniques. The chain rule gives another method to find the derivative of a function whose input is another function. Under certain conditions, such as differentiability, the result is fantastic, but you should practice using it.

With a lot of work, we can sometimes find derivatives without using the chain rule either by expanding a polynomial, by using another differentiation rule, or maybe by using a trigonometric identity. The derivative would be the same in either approach; however, the chain rule allows us to find derivatives that would otherwise be very difficult to handle.

## The Chain Rule and Its Proof

This section gives plenty of examples of the use of the chain rule as well as an easily understandable proof of the chain rule. In order to understand the chin rule the reader must be aware of composition of functions.

## Examples Using The Chain Rule

In the following examples we continue to illustrate the chain rule.

## Calculus 1 (Explore, Discover, Learn) Series

If you would like me to make a video with the solutions to some of the exercises let me know in the comments.

This article (and accompanying video) are a part of a series of articles (and videos) called the Calculus 1 (Explore, Discover, Learn) Series. Also, I put together for you a getting started with calculus 1 page and a video playlist for calculus one.