Guided Practice 2.7: Derivatives of functions given implicitly

Overview

In this section we take up two problems: (1) finding tangent lines to curves that are not graphs of functions, for example circles or curves that double back on themselves, and (2) finding derivatives of functions for which complete information (such as a formula) is not given. Both of these problems are connect by the same concept, that of an implicit function. The process we develop here, sometimes called implicit differentiation, is a powerful application of the Chain Rule that has some similarities to problems you’ve worked before where the functions involved are given only as data or as graphs.

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).

• Explain what it means for a curve to be an implicit function, as opposed to an explicit function, of x.
• Identify a function as implicit or explicit.
• Take the derivative of a function that is, or that involves, an implicit function. (This is done in the Preview Activity).

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.

• Given an equation in x and y, where y is an implicit function of x, calculate the derivative $\frac{dy}{dx}$.
• Find the slope and equation of a tangent line to a curve that is specified by an equation that is not the graph of a function.
• Find the points on a curve where the tangent line is horizontal and where the tangent line is vertical.

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 2.7. 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).

• Quick Review: Derivatives of functions given implicitly (2:23)
• Derivatives of implicit functions (5:06)
• Finding slope with implicit differentiation (8:21)

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.