Power Point:

Overview

In single-variable calculus, we found that one of the most useful differentiation rules is the chain rule, which allows us to find the derivative of the composition of two functions. The same thing is true for multivariable calculus, but this time we have to deal with more than one form of the chain rule. Now that we have functions of more than one variable, how do we apply the chain rule when we have composed functions, such as f (g (x, y)) or g(x(t), y(t))? We will learn how to compute the derivative of each of these, but we will need to pay attention to the needs of the domains and ranges of the composed functions. In this section, we study extensions of the chain rule and learn how to take derivatives of compositions of functions of more than one variable. We will learn tree diagrams to help us parse out the derivative of a multivariable function. Here is an example of a tree diagram.

A diagram that starts with z = f(x, y). Along the first branch, it is written ∂z/∂x, then x = x(t), then dx/dt, then t, and finally it says ∂z/∂x dx/dt. The branch along y is similar.

Tree diagram for the case $\frac{d z}{d t} = \frac{\partial z}{\partial x} \frac{d x}{d t} + \frac{\partial z}{\partial y} \frac{d y}{d t} .$

Section Learning Objectives

The Module Learning Objectives (MLO) are given in parentheses at the end of each learning objective.

4.5.1 State the chain rules for one or two independent variables. (MLO 10)
4.5.2 Use tree diagrams as an aid to understanding the chain rule for several independent and intermediate variables. (MLO 10)
4.5.3 Perform implicit differentiation of a function of two or more variables. (MLO 11)

Read, Watch, Do

Watch the Lecture Video and Complete the Lecture Notes

Read through the course lecture notes, while watching the lecture video. Pause the video frequently to take additional notes. Pause the video to complete the Examples on your own, then play the video to check your work.

Be sure to write out the steps so that you can use your notes several weeks later to study.

Section Lecture Notes (pdf)

Accessible Section Lecture Notes (Word document)

Section Video

Course Video Playlist

Section 4.5 The Chain Rule pdf version

Section 4.5 The Chain Rule Accessible Word version

Section 4.5 Video

Calculus 3 Playlist

The URL for the Calculus 3 playlist is https://www.youtube.com/playlist?list=PLssGY9fxBrDKptvi-0-LajyUpOGmTCVck.

Read through the Worked-out Examples in the etextbook

You may find it helpful to also read the OpenStax etextbook. However, since all the information is in the lecture notes as well, it is not necessary. Nevertheless, there are worked out examples in the etextbook that are not in the lecture notes. Read through these additional examples to be sure you understand.

Answers to some problems and Checkpoint examples can be found by clicking on the links in the numbered homework problems at the end of each section (the number itself is the link).

From OpenStax Calculus: Volume 3, see Section 4.5 The Chain Rule.

Watch Optional Topic Videos

If you still have questions after the lecture video, consider watching one or more of these topic videos.

Explore

Learn more about these concepts and explore the interactive applets from MathInsight.org.

Introduction to multivariable chain rule: https://mathinsight.org/chain_rule_multivariable_introduction

Special case of multivariable chain rule: https://mathinsight.org/chain_rule_multivariable_special_cases

Examples: https://mathinsight.org/chain_rule_multivariable_examples

Practice

Complete the homework set below from MyOpenMath (MOM). It will open directly into D2L.

Homework 4.5 Chain Rule

Remember that many exercises also have attached videos or worked-out solutions specific to that problem type. These are optional to watch, but may help when you are stuck.

Discuss

Each week has a Discussion Forum, and you are required to post your work a minimum of 7 times throughout the semester (in different weeks). Your options for posts are to post one worked-out exercise you are confident in, one worked-on exercise you are not confident in, or respond to a classmate to help with a particular problem (showing work on how to solve it) or explain a concept in detail.

See the Discussion tab on Grading and Assessment for more information. See the Course Schedule (15 week) for due dates.

Weekly Discussion Boards

Reflect

Reflection is an important part of the learning process. It allows us to make personal connections to course content and create neurological connections to ideas and concepts. The more you reflect on what you have learned, the more likely the content will be stored in long-term memory. See Grading and Assessment for Reflection Guidelines.

Don't forget to complete the first reflection. The deadline is approaching: Reflection 1.

Assess

Quiz

When you have completed your work on this section, you are ready to take a Quiz over Sections 4.4 - 4.5.

Quiz #5 4.4-4.5

Program

It is time to start working on the second Python project, where we look at differentiation of multivariable functions. Python Project Lab 2: Differentiation in Multivariable Functions