Calculus Lesson 3.1: Chain Rule

​Topic 2.3: The Chain Rule

By Jesus Barrientos

We will use the Chain Rule to find the derivative of multiple composite functions.

Content Objective #2

We will differentiate a composite function using the Chain Rule

​Content Objective #1

Objectives

In Topic 2, we discovered the multiple methods that can be used to calculate derivatives.

Of course we know that functions will come in different shapes and sizes so we need to expand our derivative techniques.

​

​Introduction

Take the derivative of the outside function, times the derivative of the inside function.

Process

Derivative rule used to find derivatives of composite functions. (i.e. functions inside of other functions)

Definition

Chain Rule

Composite Functions

Discuss: When composing functions, what do we usually replace? What is a clue that something has been replaced?

Identifying functions

Where f(x) is the outside function and g(x) is the inside function.

Example of a Composite Function

Given h(x)
- What is the inside function?
- What is the outside function?

Example of a Composite Function

Math Response

Given $$h\left(x\right)=\left(9x+10\right)^{11}$$ what is the inside function?

Type answer here

Correct Answer: 9x+10

Deg^°

Rad

Math Response

Given $$h\left(x\right)=\left(9x+10\right)^{11}$$ what is the outside function?

Type answer here

Correct Answer: x^{11}

Deg^°

Rad

Rule

Chain Rule

Steps using Example #1

Step 1: Identify the composite function. Find f and g. Label

Steps using Example #1

Step 2: Find f' and g'

Steps using Example #1

Step 3: Set up f' with g function inside.

Steps using Example #1

Step 4: Find g'. Multiply and simplify.

Example #2

Example #3

Math Response

Given $$h\left(x\right)=\left(8x^2+11\right)^7$$ find h'(x). Simplify as needed.

Type answer here

Correct Answer: 112x\left(8x^2+11\right)^6

Deg^°

Rad

Math Response

Given $$h\left(x\right)=\sqrt[]{5x^2-11}$$ find h'(x). Simplify as needed.

Type answer here

Correct Answer: \frac{5x}{\sqrt{5x^2-11}}

Deg^°

Rad

Applying the Chain Rule

Applying the Chain Rule

Applying the Chain Rule (Table)

Applying the Chain Rule (Table)

Applying the Chain Rule (Table)

Math Response

Given $$f\left(x\right)=\sqrt[]{g\left(x\right)}$$ find f'(5)

Type answer here

Correct Answer: 1

Deg^°

Rad

Math Response

Given $$f\left(x\right)=g\left(h\left(x\right)\right)$$ find f'(5).

Type answer here

Correct Answer: -24

Deg^°

Rad

Applying the Chain Rule (Trig)

Applying the Chain Rule (Trig)

Applying the Chain Rule (Trig)

Math Response

Given $$y=\cos\left(3x^2\right)$$ find $$\frac{dy}{dx}$$ . (Only type the derivative) Simplify as needed.

Type answer here

Correct Answer: -6x\sin\left(3x^2\right)

Deg^°

Rad

Applying the Chain Rule (Exp)

Applying the Chain Rule (Exp)

Applying the Chain Rule (Exp)

Math Response

Given $$y=3^{\sin x}$$ . Simplify as needed.

Type answer here

Correct Answer: 3^{\sin x}\cos x\ln3

Deg^°

Rad

Math Response

Given $$f\left(x\right)=\ln\left(\cos x\right)$$ find f'(x)

Type answer here

Correct Answer: -\tan x

Deg^°

Rad

Applying the Chain Rule (Exp)

Applying the Chain Rule (Exp)

Applying the Chain Rule (Exp)

Applying the Chain Rule (AP)

Applying the Chain Rule (AP)