The Chain Rule

Hannah Hill
6 min read
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Study Guide Overview
This study guide covers the Chain Rule for differentiating composite functions. It explains how to identify inner and outer functions, apply the chain rule using both Leibniz and function notations, and work through practice problems. The guide also emphasizes common mistakes and provides practice questions for the final exam.
Unit 3: Differentiation - Mastering the Chain Rule ๐
Welcome to Unit 3! This unit is all about mastering different differentiation techniques, and we're kicking it off with the Chain Rule, a fundamental concept for handling composite functions. Let's dive in!
๐ Composite Functions
Composite functions are functions inside other functions. Think of it like a set of Russian nesting dolls!
Given two functions, and , the composite function is formed by applying to the output of . Mathematically:
- is the inner function.
- is the outer function.
Composite Function Example
Let's break it down with an example:
For , we have:
- The inner function, , takes , multiplies it by 3, and adds 1. - The outer function, , takes the result from and squares it.
Therefore:
Understanding this is key to using the Chain Rule!
๐ Definition of The Chain Rule
The Chain Rule is your go-to method for differentiating composite functions. It's used a lot, so make sure you know it well!
There are two common notations:
-
Leibniz Notation:
- is the overall derivative.
- is the inner function.
- is the derivative of the outer function with respect to the inner function.
- is the derivative of the inner function with respect to .
-
Function Notation:
- Take the derivative from the outside in โฌ ๏ธโก๏ธ.
- If there's another function inside , youโd repeat the process.
๐ช Steps to Chain Rule
- Identify: Define your inner and outer functions.
- Outer Derivative: Take the derivative of the outer function, leaving the inner function as is.
- Inner Derivative: Take the derivative of the inner function with respect to .
- Multiply: Multiply the two derivatives together.
Think of the Chain Rule as peeling an onion: you differentiate the outer layer, then the next layer, and so on, multiplying each derivative as you go.
Image Courtesy of Geeks for Geeks
๐ข The Chain Rule: Practice Problems
Let's solidify your understanding with some examples.
The Chain Rule: Example 1
Find the derivative of
- Inner/Outer: Inner: , Outer:
- Outer Derivative:
- Inner Derivative:
- Multiply:
The Chain Rule: Example 2
Let and . Find the derivative of .
- Inner/Outer: Outer: , Inner:
- Outer Derivative:
- Inner Derivative:
- Multiply:
The Chain Rule: Example 3
Find the derivative of .
- Inner/Outer: Inner: , Outer:
- Outer Derivative:
- Inner Derivative:
- Multiply:
The Chain Rule: Example 4
Differentiate .
- Inner/Outer: Inner: , Outer:
- Outer Derivative:
- Inner Derivative:
- Multiply:
The Chain Rule: Example 5
Try this one on your own!
Hint: Inner function is , outer function is .
Answer:
Remember to apply the chain rule to every layer of the function. Don't forget the derivative of the innermost function!
๐ Closing
You've made it through the first key topic of Unit 3! The Chain Rule will be your trusty sidekick throughout your AP Calculus journey. Keep practicing, and you'll become a Chain Rule master! ๐
๐ฏ Final Exam Focus
- High-Priority: Chain Rule is everywhere! Expect to see it in multiple-choice and free-response questions.
- Connections: Often combined with other differentiation rules (Product, Quotient).
- Common Pitfalls: Forgetting the inner derivative, not applying the chain rule to every layer.
- Time Management: Practice identifying inner and outer functions quickly.
๐ Practice Questions
Practice Question
Multiple Choice Questions
-
If , then is: (A) (B)
2x\cos(x^2)
(C) (D) -
The derivative of is: (A) (B) (C) (D)
2\sqrt{x}e^{\sqrt{x}}
Free Response Question
Question:
Consider the function , where and .
(a) Find .
(b) Find .
(c) Find .
Scoring Breakdown:
(a) 1 point: Correct evaluation of .
<math-inline>h(1) = \sqrt{4 + f(1)} = \sqrt{4 + 5} = \sqrt{9} = 3</math-inline>
(b) 2 points: 1 point for applying the chain rule correctly, 1 point for the derivative of the square root.
<math-inline>h'(x) = \frac{1}{2}(4 + f(x))^{-\frac{1}{2}} \cdot f'(x) = \frac{f'(x)}{2\sqrt{4 + f(x)}}</math-inline>
(c) 2 points: 1 point for substituting the values correctly, 1 point for the correct answer.
<math-inline>h'(1) = \frac{f'(1)}{2\sqrt{4 + f(1)}} = \frac{-3}{2\sqrt{4 + 5}} = \frac{-3}{2\sqrt{9}} = \frac{-3}{2 \cdot 3} = -\frac{1}{2}</math-inline>

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Question 1 of 9
๐ Given and , what is the inner function in the composite function ?
2x+1
3x^2