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.

the statement of the chain rule and some examples
the chain rule and the start of its proof
the proof of the chain rule auxiliary function properties
the final steps in the proof of the chain rule

Examples Using The Chain Rule

find the derivative of a function using the chain rule
use the chain rule to find the derivative of a power of a rational function using the quotient rule
find the derivative of a function involving the cotangent function
find the derivative of a function involving a power of the sine function plus a power of the tangent function of a function
find the derivative of a function involving a power of the sine function plus a power of the tangent function of a function
you can not see the functions but use the chain rule to find the derivative at a number
find the derivative of a composition of function even though the functions are not given explicitly by using the chain rule

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

find the derivative of the derivative evaluated at the function and similarly find the derivative of the function that is evaluated at the derivative
modeling the motion of a particle that moves along a line using the position velocity and acceleration
find an equation of a tangent line by using differentiation and the chain rule a sketch of the graph is included
find the horizontal tangent lines a sketch of the graph is included

Exercises on the Chain Rule

use the chain rule exercises
exercises on the chin rule
exercises find the derivative using the chain rule
Exercises on the Chain Rule
use the chain rule to find the derivative and answer the questions

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.

Have fun in your studies!