ap calc study guides

๐Ÿ‘‘ย  Unit 1: Limits & Continuity

๐Ÿถย  Unit 8: Applications of Integration

โ™พย  Unit 10: Infinite Sequences and Series (BC Only)

3.1 The Chain Rule

#chainrule

โฑ๏ธย ย 2 min read

written by

Jillian Holbrook

jillian holbrook


What is the Chain Rule? ๐Ÿ”—

๐ŸŽฅWatch: 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).

When to Use the Chain Rule ๐Ÿ“†

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 โ€œinnerโ€ and โ€œouterโ€ functions.

A common example is f(g(x)).ย  Questions will be written with this form where you will see f(x) =ย  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!

How to Apply the Chain Rule ๐Ÿค”

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

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(411).png?alt=media&token=22109ba8-38fe-4ebc-bf43-e510eb44a5b4

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

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(412).png?alt=media&token=96ad7362-8a5d-45e0-8ecc-e7108a4aed4d

In other words, the Chain Rule multiplies the derivative of the โ€œinner functionโ€ by the derivative of the โ€œouter function.โ€

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

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(413).png?alt=media&token=92a89ea9-692e-4e27-9c20-366938706f2c

First, we must identify what the โ€œinnerโ€ and โ€œouterโ€ functions are.ย 

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(414).png?alt=media&token=e35caace-48b7-4a60-92d7-d53972f801f1

Can be identified as the โ€œinner function,โ€ or g(x), because it is inside the โ€œouter function,โ€ f(x):

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(415).png?alt=media&token=7fdec20f-a467-45f2-8bf0-e55819448768

With our knowledge of the Power Rule, we know that the โ€œouter functionโ€™sโ€ derivative is:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(416).png?alt=media&token=558e71ca-0713-4fb0-b9b7-ceb72b27d01e

However, the Chain Rule dictates that we also need to take the derivative of the โ€œinner functionโ€ as well:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(417).png?alt=media&token=fe584e7d-49b7-4be9-92eb-97f721689b69

Using the Power Rule, we can take gโ€™(x):

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(418).png?alt=media&token=3e11da3a-3c24-4ea8-a2df-cf170b75fcdf

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

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(419).png?alt=media&token=0f193534-1862-4b74-b817-c25de0bf8efc

Chain Rule Practice ๐Ÿ“

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-eUbputCbsmVa.webp?alt=media&token=fc97c774-227a-4cd0-88ec-76f7602d7dba

Image Courtesy of Giphy (Try to guess this reference!)

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

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(409).png?alt=media&token=8ee50d56-c4f4-4614-8cdc-dbc0041a72bd

Answers ๐Ÿ“

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(410).png?alt=media&token=85d32b4d-3e30-498f-9320-4e23988d06db

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