MA101: Single-Variable Calculus I

Unit 3: Rules for Finding Derivatives   Computing a derivative requires computing a limit.  Because limit computations can be rather involved, we like to minimize the amount of work we have to do in practice.  In this unit, you will build your skill using some rules for differentiation which will speed up your calculations of derivatives.  In particular, you will see how to differentiate the sum, difference, product, quotient, and composition of two or more functions.  You will also learn rules for differentiating power functions, including polynomial and root functions.

This unit should take you approximately 12 hours to complete.

☐    Subunit 3.1: 1 hour

☐    Subunit 3.2: 1.5 hours

☐    Subunit 3.3: 2 hours

☐    Subunit 3.4: 2.5 hours

☐    Subunit 3.5: 5 hours ☐    Reading: 1 hour

☐    Lecture: 1 hour

☐    Assignment: 3 hours

Unit3 Learning Outcomes
Upon successful completion of this unit, the student will be able to:
- Use the power, product, quotient, and chain rules to calculate derivatives.

3.1 The Power Rule   - Reading: Whitman College: David Guichard’s Calculus: Chapter 3: Rules for Finding Derivatives: “Section 3.1: The Power Rule” Link: Whitman College: David Guichard’s Calculus: Chapter 3: Rules for Finding Derivatives: “Section 3.1: The Power Rule” (PDF)

Instructions: Please click on the link above and read Section 3.1 (pages 55-57) in its entirety.  This section will show you a simple rule for how to find the derivative of a power function without explicitly computing a limit.

This reading should take you approximately 30 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.  This text was originally written by Professor David Guichard.  It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills.  You can access the original version here.

• Assignment: Whitman College: David Guichard’s Calculus: Chapter 3: Rules for Finding Derivatives: “Exercises 3.1: Problems 1-6” Link: Whitman College: David Guichard’s Calculus: Chapter 3: Rules for Finding Derivatives: “Exercises 3.1: Problems 1-6” (PDF)

This assignment should take you approximately 30 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.  This text was originally written by Professor David Guichard.  It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here. (PDF)

3.2 Linearity of the Derivative   - Reading: Whitman College: David Guichard’s Calculus: Chapter 3: Rules for Finding Derivatives: “Section 3.2: Linearity of the Derivative” Link: Whitman College: David Guichard’s Calculus: Chapter 3: Rules for Finding Derivatives: “Section 3.2: Linearity of the Derivative” (PDF)

Instructions: Please click on the link above and read Section 3.2 (pages 58-59) in its entirety.  In this reading, you will see how the derivative behaves with regards to addition and subtraction of functions and with scalar multiplication.  That is, you will see that the derivative is a linear operation.

This reading should take you approximately 30 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.  This text was originally written by Professor David Guichard.  It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills.  You can access the original version here. (PDF)

• Assignment: Whitman College: David Guichard’s Calculus: Chapter 3: Rules for Finding Derivatives: “Exercises 3.2: Problems 1-9, 11, and 12” Link: Whitman College: David Guichard’s Calculus: Chapter 3: Rules for Finding Derivatives: “Exercises 3.2: Problems 1-9, 11, and 12” (PDF)

This assignment should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.  This text was originally written by Professor David Guichard.  It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.

3.3 The Product Rule   - Reading: Whitman College: David Guichard’s Calculus: Chapter 3: Rules for Finding Derivatives: “Section 3.3: The Product Rule” Link: Whitman College: David Guichard’s Calculus: Chapter 3: Rules for Finding Derivatives: “Section 3.3: The Product Rule” (PDF)

Instructions: Please click on the link above and read Section 3.3 (pages 60-61) in its entirety.  The naïve assumption is that the derivative of a product of two functions is the product of the derivatives of the two functions.  This assumption is false.  In this reading, you will see that the derivative of a product is slightly more complicated, but that it follows a definite rule called the product rule.

This reading should take you approximately 30 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.  This text was originally written by Professor David Guichard.  It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here. (PDF)

• Assignment: Whitman College: David Guichard’s Calculus: Chapter 3: Rules for Finding Derivatives: “Exercises 3.3: Problems 1-5” Link: Whitman College: David Guichard’s Calculus: Chapter 3: Rules for Finding Derivatives: “Exercises 3.3: Problems 1-5” (PDF)

This assignment should take you approximately 30 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.  This text was originally written by Professor David Guichard.  It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.

• Assignment: Temple University: Gerardo Mendoza and Dan Reich’s Calculus on the Web: Calculus Book I: Dan Reich’s “Derivatives – The Product Rule” Module Link: Temple University: Gerardo Mendoza and Dan Reich’s Calculus on the Web: Calculus Book I: Dan Reich’s “Derivatives – The Product Rule” Module (HTML)

Instructions: Please click on the link above and select the “Index.”  Click on the number 44 next to  “Product Rule” to launch the module and complete problems 1-10.  If at any time the problem set becomes too easy for you, feel free to move forward.

This assignment should take you approximately one hour to complete.

3.4 The Quotient Rule   - Assignment: Temple University: Gerardo Mendoza and Dan Reich's Calculus on the Web: Calculus Book I: Dan Reich's "Derivatives - The Quotient Rule" Module Link: Temple University: Gerardo Mendoza and Dan Reich's Calculus on the Web: Calculus Book I: Dan Reich's "Derivatives - The Quotient Rule" Module (HTML)

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"Index." Click on the number 45 next to "The Quotient Rule" to
launch the module and complete problems 1-10. If at any time the
problem set becomes too easy for you, feel free to move forward.

This assignment should take you approximately one hour to
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• Reading: Whitman College: David Guichard’s Calculus: Chapter 3: Rules for Finding Derivatives: “Section 3.4: The Quotient Rule” Link: Whitman College: David Guichard’s Calculus: Chapter 3: Rules for Finding Derivatives: “Section 3.4: The Quotient Rule” (PDF)

Instructions: Please click on the link above and read Section 3.4 (pages 62-65) in its entirety.  As with product of two functions, the derivative of a quotient of two functions is not simply the quotient of the two derivatives.  This reading will introduce you to the quotient rule for differentiating a quotient of two functions.  In particular, it will allow you to find the derivative of any rational function.

This reading should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.  This text was originally written by Professor David Guichard.  It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.

• Assignment: Whitman College: David Guichard’s Calculus: Chapter 3: Rules for Finding Derivatives: “Exercises 3.4: Problems 5, 6, 8, and 9” Link: Whitman College: David Guichard’s Calculus: Chapter 3: Rules for Finding Derivatives: “Exercises 3.4: Problems 5, 6, 8, and 9” (PDF)

This assignment should take you approximately 30 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.  This text was originally written by Professor David Guichard.  It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills.  You can access the original version here.

3.5 The Chain Rule   - Reading: Whitman College: David Guichard’s Calculus: Chapter 3: Rules for Finding Derivatives: “Section 3.5: The Chain Rule” Link: Whitman College: David Guichard’s Calculus: Chapter 3: Rules for Finding Derivatives: “Section 3.5: The Chain Rule” (PDF)

Instructions: Please click on the link above and read Section 3.5 (pages 65-69) in its entirety.  The chain rule explains how the derivative applies to the composition of functions.  Pay particular attention to Example 3.11, which works through a derivative computation where all of the differentiation rules of this unit are applied in finding the derivative of one function.

This reading should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.  This text was originally written by Professor David Guichard.  It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.

• Lecture: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 4: Chain Rule” Link: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 4: Chain Rule" (YouTube)

Instructions: Please click on the link above and watch the entire video (46:03).  Lecture notes are available here.

Viewing this lecture and taking notes should take you approximately one hour to complete.