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The Chain Rule

Hannah Hill

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 🚀

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!

🔄 Composite Functions

Key Concept

Composite functions are functions inside other functions. Think of it like a set of Russian nesting dolls!

Given two functions, f(x)f(x) and g(x)g(x), the composite function (f∘g)(x)(f \circ g)(x) is formed by applying ff to the output of gg. Mathematically:

(f∘g)(x)=f(g(x))(f \circ g)(x) = f(g(x))

  • g(x)g(x) is the inner function.
  • f(x)f(x) is the outer function.

Composite Function Example

Let's break it down with an example:

f(x)=x2\textcolor{red}{f(x) = x^2}

g(x)=3x+1\textcolor{blue}{g(x) = 3x + 1}

For f(g(x))f(g(x)), we have:

  • The inner function, g(x)g(x), takes xx, multiplies it by 3, and adds 1. - The outer function, f(x)f(x), takes the result from g(x)g(x) and squares it.

Therefore:

f(g(x))=(3x+1)2f(g(x)) = \textcolor{red}{(\textcolor{blue}{3x + 1})^2}

Understanding this is key to using the Chain Rule!


🔗 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:

  1. Leibniz Notation:

    dydx=dyduâ‹…dudx\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}

    • dydx\frac{dy}{dx} is the overall derivative.
    • uu is the ...

Question 1 of 9

🚀 Given f(x)=x3f(x) = x^3 and g(x)=2x+1g(x) = 2x + 1, what is the inner function in the composite function f(g(x))f(g(x))?

x3x^3

2x+1

(2x+1)3(2x+1)^3

3x^2