Lesson 6: Partial Derivatives; The Chain Rule

Preview

This lesson begins the generalization of the derivative concept to a multivariable context.

In Module 1 we’ll learn about partial derivatives, which hold all but one variable in a multivariable function constant and differentiate with respect to the only remaining variable. This will represent perhaps the simplest first step in generalizing the derivative concept from single-variable calculus to a multivariable context.
In Module 2 we’ll generalize the Chain Rule to the multivariable context. We’ll learn that there two different types of multivariable Chain Rules, depending on how many independent variables we are considering. We’ll also learn what it means for a multivariable function to be differentiable (in the simplest case).
Finally, in Module 3 we’ll revisit two of the interpretations of derivatives from single-variable calculus—the rate of change interpretation and the slope of the tangent line one—and discuss how those carry over to the multivariable context.

Review

Lesson 5

Learn

Work through the lesson notes below, consulting the videos below it when you get to the "See Class Notes" boxes. For your records, the annotated lesson notes are below the videos. Some tips for you as you work through these resources:

I recommend using Cornell Notes (or a modification of it; see this video starting at the 1:05 mark) to take notes on the lesson and the videos. This note-taking method balances detail with big-picture thinking to help you summarize and retain what you are learning. See this other video for additional note-taking techniques you might want to experiment with.

Lesson Notes

Lesson 6.pdf

Module 1 Video

Module 2 Video

Module 3 Video

Reflect

If you are currently enrolled in this course with me, submit the written reflections Google Form I have emailed you after working through the lesson notes and videos. Some tips:

Submit substantive, but concise, answers to each question; you will be doing the future you a big favor by taking time now to accurately and succinctly summarize what you have learned from the lesson.
Send yourself a copy of your reflections; they will come in handy later when you start preparing for quizzes and other assessments.

If you are not currently enrolled in this course with me, those written reflections ask three reflective questions designed to help you retain what you've learned and pinpoint any remaining areas of confusion. Those questions are:

Please summarize the main mathematical takeaways from the lesson notes.
What was the most interesting part of what you learned, and why?
What, if anything, do you still find confusing?

Practice

Work through the practice problems suggested below to see how much of this lesson you've understood.

Lesson 6 PP.pdf

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