# Chain Rule & Implicit Differentiation

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## Lecture Notes

#### The Chain Rule

PDF
Section 4, Page 3

Chain Rule for derivatives, including definition of composite functions and a worked example.

Instructor: Prof. Jason Starr
Prior Knowledge: Derivative as the Limit of a Difference Quotient (section 2 of lecture 1)

#### Implicit Differentiation

PDF
Section 5, Page 3

Implicit differentiation defined.

Instructor: Prof. Jason Starr
Prior Knowledge: None

#### Example of Implicit Differentiation

PDF
Section 1, Page 1

Implicit differentiation demonstrated through an example.

Instructor: Prof. Jason Starr
Prior Knowledge: Implicit Differentiation (section 5 of lecture 4) and Derivative of xn (section 3 of lecture 3)

## Online Textbook Chapter

#### Derivatives of Combinations of Functions

Document

Rules for the derivatives of sums and products of functions, as well as the chain rule and rules for finding the derivative of an inverse function.

Instructor: Prof. Daniel J. Kleitman
Prior Knowledge: Differentiability (OT6.1)

## Practice Problems

#### Chain Rule

PDF
Problem 2 (page 2)

Two part question that involves applying and explaining the chain rule for derivatives.

Instructor: Prof. Jason Starr
Prior Knowledge: None

#### Continuously Compounded Interest

PDF
Problem 3 (page 2)

Two part question involving a bank's liability from a loan and a savings account, each with continuously compounded interest.

Instructor: Prof. Jason Starr
Prior Knowledge: None

#### Derivatives of Exponential and Trigonometric Functions

PDF
Problem 4 (page 2 to page 3)

Taking the first and second derivatives of a function involving an exponential and a cosine.

Instructor: Prof. Jason Starr
Prior Knowledge: None

## Exam Questions

#### Implicit Differentiation and Tangent Line to a Graph

PDF
Problem 3 (page 4)

Finding the equation of a tangent line to the graph of a function that is defined with an implicit equation.

Instructor: Prof. Jason Starr
Prior Knowledge: None

#### Evaluating Derivatives

PDF
Problem 1 (page 1) to problem 2 (page 1)

Two questions finding the derivatives of functions.

Instructor: Prof. David Jerison
Prior Knowledge: None

#### Implicit Differentiation

PDF
Problem 4 (page 1)

Finding the derivative of the inverse sine function using implicit differentiation.

Instructor: Prof. David Jerison
Prior Knowledge: None

#### Implicit Differentiation

PDF
Problem 3 (page 1)

Finding the derivative of an implicitly defined function.

Instructor: Prof. David Jerison
Prior Knowledge: None