### 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 , where is the force on any object, is its mass, and 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 ;
- 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 ;
- 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

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

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.

### 3.2: The Definition of a Derivative

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

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.
- and Examples; pages 4-5.
- Interpretations of the Derivative; pages 6-8.
- A Useful Formula: ; pages 8-10.
- Important Definitions, Formulas, and Results for the Derivative, Tangent Line Equation, and Interpretations of ; page 10.

- Formal Definition of a Derivative; pages 1-2.

### 3.3: Derivatives, Properties and Formulas

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.

Watch this video on the product rule for differentiation.

### 3.3.1: Practice Problems

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.

- Which Functions Have Derivatives?; pages 1-3.

### 3.4: Derivative Patterns

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

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: : 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.

- A Power Rule for Functions: : To review this topic, focus on pages 1 and 2.

### 3.5: The Chain Rule

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.

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

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.

- Chain Rule for Differentiating a Composition of Functions; page 1.

### Problem Set 4

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.

- This assessment

### 3.6: Some Applications of the Chain Rule

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

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.

### 3.7: Related Rates

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

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.

- The Derivative as a Rate of Change; pages 1-7.

### 3.8: Newton's Method for Finding Roots

Read this section on pages 1-8. Work through practice problems 1-6. For solutions to these problems, see pages 10-11.

Watch this video on Newton's method.

### 3.8.1: Practice Problems

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.

- Newton's Method Using the Tangent Line; pages 1-3.

### 3.9: Linear Approximation and Differentials

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.

Watch this video on linear approximation and differentials.

### 3.9.1: Practice Problems

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.

- Linear Approximation and Its Process; pages 1-4.

### 3.10: Implicit and Logarithmic Differentiation

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.

Watch these videos on implicit and logarithmic differentiation.

### 3.10.1: Practice Problems

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.

- Implicit Differentiation; pages 1-3.

### Problem Set 5

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.

- This assessment