Guided Practice 2.3: The Product and Quotient Rules

Overview

This section introduces two new derivative-taking rules: the Product Rule for differentiating functions of the form $f(x) \cdot g(x)$, and the Quotient Rule for differentiating functions of the form $\dfrac{f(x)}{g(x)}$. We’ll learn these rules and a little bit about why they work, and then put them to work for us in doing computations.

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 why the derivative of a product of two functions, $f(x) \cdot g(x)$, is not just the product of the derivatives, $f’(x) \cdot g’(x)$.
• Explain why the derivative of a quotient of two functions, $f(x)/g(x)$, is not just the quotient of the derivatives, $f’(x)/g’(x)$.
• State the Product Rule and use it in a simple situation.
• State the Quotient Rule and use it in a simple situation.

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.

• Differentiate a function for which the derivative involves a combination of the Product Rule, Quotient Rule, and other rules we’ve learned.
• Use the Product and Quotient rules in the context of a real-world problem to find the slope of a tangent line, the instantaneous rate of change in a function, or the instantaneous velocity of an object.

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 2.3. 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: The Product and Quotient Rules (2:00)
• Product Rule examples (6:48)
• Quotient Rule examples (10:38)
• Combining the Product and Quotient Rules (9:01)

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.