Guided Practice 2.5: The Chain Rule

Overview

This section introduces a very important rule: The Chain Rule, which is used to differentiate composite functions. Composite functions are functions composed (hence the name) of two or more functions “chained” together in sequence, such as $f(x) = (x^2 + x + 1)^{1/2}$. Many functions in applications are composites and so this rule gives us a much broader range of functions we can work with in calculus. The Chain Rule is also used as a basis for several upcoming concepts involving derivatives in which we have incomplete information about the functions involved.

The Chain Rule is a Big Idea, and so we will actually be spending the entire week on jsut this one section.

Learning objectives

Basic objectives: Each student is responsible for gaining proficiency with each of these tasks prior to engaging in class discussions, through the use of the learning resources (below) and through the working of exercises (also below).

• State the Chain Rule and explain how it works in English.
• Identify situations where the Chain Rule should be used when taking a derivative.
• Use the Chain Rule to differentiate a simple composite function, for example a composition of a polynomial and a power function (e.g., $f(x) = (x^2 + x + 1)^{1/2}$).

Advanced objectives: The following objectives are the subject of class discussion and further work; they should be mastered by each student during and following class discussions.

• Use the Chain Rule to differentiate a composite function involving two functions.
• Use the Chain Rule to differentiate a composite function involving three or more functions.
• Use the Chain Rule in combination with other rules from earlier in this chapter.
• Use the Chain Rule to differentiate a composite function in which at least one of the functions in the composite is given as a graph or a table of values.

Learning resources

To gain proficiency in the learning objectives, use the following resources. You may include other resources if you wish, in addition to or in replacement of the following.

Textbook: In Active Calculus, read Section 2.5. Make sure to read actively, working through examples and activities as you go.

Video: Watch the following videos at the MTH 201 YouTube playlist (http://bit.ly/GVSUCalculus). This is a considerably longer list this time, because (1) many students find the Chain Rule hard to grasp, and so we’ve prepared lots of examples; and (2) this one section occupies two class periods. These have a total running time of 41 minutes, 38 seconds:

• Quick Review: The Chain Rule (2:21)
• Example of the Chain Rule: Polynomials (4:33)
• Example of the Chain Rule: Radicals (7:20)
• Chain Rule Examples: Trigonometric functions (5:15)
• Chain Rule Examples: Exponential functions (6:01)
• Chain Rule Examples: Mixing rules (10:30)
• Chain Rule examples: Graphs only (5:38)

Here are two optional videos if you need a refresher on the concept of a composite function:

• Composition of functions part 1 (13:07)
• Composition of functions part 2 (6:23)

Activities

The following activity is to be done during and following your reading and viewing of the resources. Go to student.desmos.com and join using your name in the format Last, First. For example, I would enter my name as Ballif, Serge. Complete each part of the activity. Some of these problems will require you to work them out on paper before entering your answer. Practice producing high quality work so that your work is readable and meaningful. You will receive a mark of Pass if each item response shows a good-faith effort to be right and is submitted prior to the deadline. Remember to use the Piazza discussion board to ask about any questions you have.