Unit 3: Derivatives
In this unit, we start to see calculus become more visible when abstract ideas such as a derivative and a limit appear as parts of slopes, lines, and curves. Then, there are circles, ellipses, and parabolas that are even more geometric, so what was previously an abstract concept can now be something we can see. Nothing makes calculus more tangible than to recognize that the first derivative of an automobile's position is its velocity and the second derivative of that position is its acceleration. We are at the very point that started Isaac Newton on his quest to master this mathematics, what we now call calculus, when he recognized that the second derivative was precisely what he needed to formulate his Second Law of Motion $F = MA$ , where $F$ is the force on any object, $M$ is its mass, and $A$ is the second derivative of its position. Thus, he could connect all the variables of a moving object mathematically, including its acceleration, velocity, and position, and he could explain what really makes motion happen.

Completing this unit should take you approximately 42 hours.
- Upon successful completion of this unit, you will be able to:
  
  state the definition of a derivative of a function $f(x)$ ;
  
  recognize and use the common equivalent notations for the derivative;
  
  state the graph and rate meanings of a derivative;
  
  estimate a tangent line slope and instantaneous rate of change from the graph of a function;
  
  write the equation of the line tangent to the graph of a function $f(x)$ ;
  
  calculate the derivatives of the elementary functions;
  
  calculate second and higher derivatives and state what they measure;
  
  differentiate compositions of functions using the chain rule;
  
  calculate the derivatives of functions given as parametric equations and interpret their meanings geometrically and physically;
  
  state whether a function, given by a graph or formula, is continuous or differentiable at a point or on an interval;
  
  solve related rate problems using derivatives;
  
  approximate the solutions of equations by using derivatives and Newton's method;
  
  approximate the values of difficult functions by using derivatives;
  
  calculate the differential of a function using derivatives and show what the differential represents on a graph; and
  
  calculate the derivatives of really difficult functions by using the methods of implicit differentiation and logarithmic differentiation.
- 3.1: Introduction to Derivatives
  - Washington State Board for Community and Technical Colleges: Dale Hoffman's "Contemporary Calculus, Section 3.1: Introduction to Derivatives" URL
    Read this section on pages 1-5 to lay the groundwork for introducing the concept of a derivative. Work through practice problems 1-5. For solutions to these problems, see pages 8-9.
  - 3.1.1: Practice Problems
    Washington State Board for Community and Technical Colleges: Dale Hoffman's "Contemporary Calculus, Section 3.1: Practice Problems" URL
    Work through the odd-numbered problems 1-17 on pages 5-7. Once you have completed the problem set, check your answers for the odd-numbered questions here.
  - 3.1.2: Review
    Before moving on, you should be comfortable with each of these topics:
    Slopes of Tangent Lines; pages 1-2.
    Tangents to $y = x^2$ ; pages 2-5.
- 3.2: The Definition of a Derivative
  - Washington State Board for Community and Technical Colleges: Dale Hoffman's "Contemporary Calculus, Section 3.2: The Definition of a Derivative" URL
    Read this section on pages 1-10 to understand the definition of a derivative. Work through practice problems 1-8. For solutions to these problems, see pages 14-15.
  - 3.2.1: Practice Problems
    Washington State Board for Community and Technical Colleges: Dale Hoffman's "Contemporary Calculus, Section 3.2: Practice Problems" URL
    Work through the odd-numbered problems 1-37 on pages 11-14. Once you have completed the problem set, check your answers for the odd-numbered questions here.
  - 3.2.2: Review
    Before moving on, you should be comfortable with each of these topics:
    Formal Definition of a Derivative; pages 1-2.
    Calculations Using the Definition; pages 2-6.
    Tangent Line Formula; page 4.
    $\sin$ and $\cos$ Examples; pages 4-5.
    Interpretations of the Derivative; pages 6-8.
    A Useful Formula: $D(xn)$ ; pages 8-10.
    Important Definitions, Formulas, and Results for the Derivative, Tangent Line Equation, and Interpretations of $f'(x)$ ; page 10.
- 3.3: Derivatives, Properties and Formulas
  - Washington State Board for Community and Technical Colleges: Dale Hoffman's "Contemporary Calculus, Section 3.3: Derivatives, Properties and Formulas" URL
    Read this section on pages 1-9 to understand the properties of derivatives. Work through practice problems 1-11. For solutions to these problems, see pages 16-17.
  - Khan Academy: "Applying the Product Rule for Derivatives" Page
    Watch this video on the product rule for differentiation.
  - 3.3.1: Practice Problems
    Washington State Board for Community and Technical Colleges: Dale Hoffman's "Contemporary Calculus, Section 3.3: Practice Problems" URL
    Work through the odd-numbered problems 1-55 on pages 10-15. Once you have completed the problem set, check your answers for the odd-numbered questions here.
  - 3.3.2: Review
    Before moving on, you should be comfortable with each of these topics:
    Which Functions Have Derivatives?; pages 1-3.
    Derivatives of Elementary Combination of Functions; pages 3-6.
    Using the Differentiation Rules; pages 7-8.
    Evaluative a Derivative at a Point; page 9.
    Important Results for Differentiability and Continuity; page 9.
- 3.4: Derivative Patterns
  - Washington State Board for Community and Technical Colleges: Dale Hoffman's "Contemporary Calculus, Section 3.4: More Differentiation Problems" URL
    Read this section on pages 1-9 to learn about patterns of derivatives. Work through practice problems 1-8. For solutions to these problems, see pages 12-14.
  - 3.4.1: Practice Problems
    Washington State Board for Community and Technical Colleges: Dale Hoffman's "Contemporary Calculus, Section 3.4: Practice Problems" URL
    Work through the odd-numbered problems 1-47 on pages 9-14. Once you have completed the problem set, check your answers for the odd-numbered questions here.
  - 3.4.2: Review
    Before moving on, you should be comfortable with each of these topics:
    A Power Rule for Functions: $D(f n(x))$ : To review this topic, focus on pages 1 and 2.
    Derivatives of Trigonometric and Exponential Functions: To review this topic, focus on pages 3-6.
    Higher Derivatives - Derivatives of Derivatives: To review this topic, focus on pages 6-7.
    Bent and Twisted Functions: To review this topic, focus on pages 7-8.
    Important Results for Power Rule of Functions and Derivatives of Trigonometric and Exponential Functions; page 9.
- 3.5: The Chain Rule
  - Washington State Board for Community and Technical Colleges: Dale Hoffman's "Contemporary Calculus, Section 3.5: The Chain Rule" URL
    Read this section on pages 1-7 to learn about the Chain Rule. Work through practice problems 1-8. For solutions to these problems, see pages 11-12.
  - Khan Academy: "Chain Rule Introduction" and "Chain Rule Definition and Example" Page
    Watch these videos for an introduction to the chain rule for differentiation and examples of the chain rule for differentiation.
  - 3.5.1: Practice Problems
    Washington State Board for Community and Technical Colleges: Dale Hoffman's "Contemporary Calculus, Section 3.5: Practice Problems" URL
    Work through the odd-numbered problems 1-83 on pages 7-11. Once you have completed the problem set, check your answers for the odd-numbered questions here.
  - 3.5.2: Review
    Before moving on, you should be comfortable with each of these topics:
    Chain Rule for Differentiating a Composition of Functions; page 1.
    The Chain Rule Using Leibnitz Notation Form; page 2.
    The Chain Rule Composition Form; pages 2-5.
    The Chain Rule and Tables of Derivatives; pages 5-6.
    The Power Rule for Functions; page 7.
- Problem Set 4
  - Problem Set 4 Quiz
    Take this assessment to see how well you understood these concepts.
    
    This assessment does not count towards your grade. It is just for practice!
    
    You will see the correct answers when you submit your answers. Use this to help you study for the final exam!
    
    You can take this assessment as many times as you want, whenever you want.
- 3.6: Some Applications of the Chain Rule
  - Washington State Board for Community and Technical Colleges: Dale Hoffman's "Contemporary Calculus, Section 3.6: Some Applications of the Chain Rule" URL
    Read this section on pages 1-8 to learn how to apply the Chain Rule. Work through practice problems 1-8. For solutions to these problems, see pages 13-14.
  - 3.6.1: Practice Problems
    Washington State Board for Community and Technical Colleges: Dale Hoffman's "Contemporary Calculus, Section 3.6: Practice Problems" URL
    Work through the odd-numbered problems 1-49 on pages 9-11. Once you have completed the problem set, check your answers for the odd-numbered questions here.
  - 3.6.2: Review
    Before moving on, you should be comfortable with each of these topics:
    Derivatives of Logarithms; pages 1-2.
    Derivative of $ax$ ; pages 2-3.
    Applied Problems; pages 3-5.
    Parametric Equations; pages 5-6.
    Speed; page 8.
- 3.7: Related Rates
  - Washington State Board for Community and Technical Colleges: Dale Hoffman's "Contemporary Calculus, Section 3.7: Related Rates: An Application of Derivatives" URL
    Read this section on pages 1-7 to learn to connect derivatives to the concept of the rate at which things change. Work through practice problems 1-3. For solutions to these problems, see pages 12-13.
  - 3.7.1: Practice Problems
    Washington State Board for Community and Technical Colleges: Dale Hoffman's "Contemporary Calculus, Section 3.7: Practice Problems" URL
    Work through the odd-numbered problems 1-21 on pages 8-12. Once you have completed the problem set, check your answers for the odd-numbered questions here.
  - 3.7.2: Review
    Before moving on, you should be comfortable with each of these topics:
    The Derivative as a Rate of Change; pages 1-7.
- 3.8: Newton's Method for Finding Roots
  - Washington State Board for Community and Technical Colleges: Dale Hoffman's Contemporary Calculus: "Section 3.8: Newton's Method for Finding Roots" URL
    Read this section on pages 1-8. Work through practice problems 1-6. For solutions to these problems, see pages 10-11.
  - Massachusetts Institute of Technology: Christine Breiner's "Using Newton's Method" Page
    Watch this video on Newton's method.
  - 3.8.1: Practice Problems
    Washington State Board for Community and Technical Colleges: Dale Hoffman's "Contemporary Calculus, Section 3.8: Practice Problems" URL
    Work through the odd-numbered problems 1-21 on pages 8-9. Once you have completed the problem set, check your answers for the odd-numbered questions here.
  - 3.8.2: Review
    Before moving on, you should be comfortable with each of these topics:
    Newton's Method Using the Tangent Line; pages 1-3.
    The Algorithm for Newton's Method; pages 3-5.
    Iteration; page 5.
    What Can Go Wrong with Newton's Method?; pages 5-6.
    Chaotic Behavior and Newton's Method; pages 6-8.
- 3.9: Linear Approximation and Differentials
  - Washington State Board for Community and Technical Colleges: Dale Hoffman's "Contemporary Calculus, Section 3.9: Linear Approximation and Differentials" URL
    Read this section on pages 1-10 to learn how linear approximation and differentials are connected. Work through practice problems 1-10. For the solutions to these problems, see pages 14-15.
  - RootMath: "Linear Approximation" Page
    Watch this video on linear approximation and differentials.
  - 3.9.1: Practice Problems
    Washington State Board for Community and Technical Colleges: Dale Hoffman's "Contemporary Calculus, Section 3.9: Practice Problems" URL
    Work through the odd-numbered problems 1-19 on pages 10-13. Once you have completed the problem set, check your answers for the odd-numbered questions here.
  - 3.9.2: Review
    Before moving on, you should be comfortable with each of these topics:
    Linear Approximation and Its Process; pages 1-4.
    Applications of Linear Approximation to Measurement Error; pages 4-6.
    Relative Error and Percentage Error; pages 6-7.
    The Differential of a Function; pages 7-8.
    The Linear Approximation Error; pages 8-10.
- 3.10: Implicit and Logarithmic Differentiation
  - Washington State Board for Community and Technical Colleges: Dale Hoffman's "Contemporary Calculus, Section 3.10: Implicit and Logarithmic Differentiation" URL
    Read this section on pages 1-5 to learn about implicit and logarithmic differentiation. Work through practice problems 1-6. For solutions to these problems, see pages 8-9.
  - Khan Academy: "Implicit Differentiation" and "Derivative of x^(x^x)" Page
    Watch these videos on implicit and logarithmic differentiation.
  - 3.10.1: Practice Problems
    Washington State Board for Community and Technical Colleges: Dale Hoffman's "Contemporary Calculus, Section 3.10: Practice Problems" URL
    Work through the odd-numbered problems 1-55 on pages 5-8. Once you have completed the problem set, check your answers for the odd-numbered questions here.
  - 3.10.2: Review
    Before moving on, you should be comfortable with each of these topics:
    Implicit Differentiation; pages 1-3.
    Logarithmic Differentiation; pages 3-5.
- Problem Set 5
  - Problem Set 5 Quiz
    Take this assessment to see how well you understood these concepts.
    
    This assessment does not count towards your grade. It is just for practice!
    
    You will see the correct answers when you submit your answers. Use this to help you study for the final exam!
    
    You can take this assessment as many times as you want, whenever you want.

Unit 3: Derivatives