Lesson 9: The Derivative Function

Preview

Here's the on-ramp for this lesson.


What Will We Learn?

  • In this lesson we'll build up the notion of a derivative function. First, we'll discuss when the derivative at a point does not exist. That will help us realize that for most functions the derivative exists at many, many points, and even on specific intervals. By graphing these derivative values on a new set of axes we'll realize that we can construct a derivative function. We'll then learn how to calculate that derivative function using the same limit definition from the previous lesson.


Why Do We Need to Learn This?

  • While knowing how a function changes "instantaneously" has its uses, as we learned in the previous lesson, better yet is to have a broader understanding of how these instantaneous changes themselves change. Are they constant? Do they follow some pattern(s)? These are the questions that are answered by the derivative function.

Review

Learn

The lesson notes below contain a learning plan with three stages -- Learn, Reflect, and Practice -- and guidance for what to do within each stage. Some tips for you as you work through this resource, and those that it points to:

  • 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 9.pdf

Video 1 (Example 3.10)

Video 2 (Example 3.11)

Video 3 (Example 3.13)

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 9 PP.pdf