3.1 The Chain Rule

🎥Watch: AP Calculus AB/BC - The Chain Rule

The Chain Rule is another mode of application for taking derivatives just like its friends, the Power Rule, the Product Rule, and the Quotient Rule (which you should be familiar with from Unit 2).

Using the Chain Rule is necessary when you encounter a composite function. Composite functions are functions inside of other functions. This is where we see “inner” and “outer” functions.

A common example is f(g(x)).  Questions will be written with this form where you will see f(x) =  and g(x) = , then it will ask you to find the value of f(g(x)).

Composite functions are notorious for popping up repeatedly on both AP Calculus AB and BC exams, so it is important to become familiar and confident with the Chain Rule!

Leibniz was the first to use the Chain Rule to differentiate composite functions. His notation for the Chain Rule can be defined as:

However, we can also view the Chain Rule in the following notation:

In other words, the Chain Rule multiplies the derivative of the “inner function” by the derivative of the “outer function.”

Let’s look at an example to show that this rule works.  We can use the Chain Rule to take the derivative of the following function in polynomial form:

First, we must identify what the “inner” and “outer” functions are. 

Can be identified as the “inner function,” or g(x), because it is inside the “outer function,” f(x):

With our knowledge of the Power Rule, we know that the “outer function’s” derivative is:

However, the Chain Rule dictates that we also need to take the derivative of the “inner function” as well:

Using the Power Rule, we can take g’(x):

Thus, we can combine with can use our original notation to get our final answer:

Now that you understand the basics of the Chain Rule, let’s practice applying it in the problems below.  When you are finished, you can check your answers at the end of the guide afterwards!