### 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. Circles, ellipses, and parabolas are even more geometric; what were abstract concepts are now something we can see. Nothing makes calculus more tangible than recognizing 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 math that we now call calculus. He recognized that the second derivative was just 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 13 hours.**

Upon successful completion of this unit, you should be able to:

- find the derivative of a function using the definition;
- recognize 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;
- use the chain rule to solve applied questions;
- 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

Now that we've laid the foundation for learning calculus by learning about functions, graphs, lines, and limits, we will move on to derivatives. You will use derivatives to solve many application problems, such as optimization.

Read this section to lay the groundwork for introducing the concept of a derivative. Work through practice problems 1-5.

- Watch this video on the slope of secant lines converging to tangent line slopes.
Work through the odd-numbered problems 1-17. Once you have completed the problem set, check your answers.

### 3.2: The Definition of a Derivative

As was discussed in the last unit, knowing the definition of a concept is important. The definition of a derivative will help you calculate the derivative of functions.Read this section to understand the definition of a derivative. Work through practice problems 1-8.

Watch this video on how to find a derivative from the definition.

Work through the odd-numbered problems 1-37. Once you have completed the problem set, check your answers.

### 3.3: Derivatives, Properties, and Formulas

Properties and formulas of derivatives are essential in helping you calculate the derivatives of a variety of functions. In this section, you will learn how to apply rules to calculate the derivatives of various combinations of functions.

Read this section to understand the properties of derivatives. Work through practice problems 1-11.

Watch this video on the product rule for differentiation.

Work through the odd-numbered problems 1-55. Once you have completed the problem set, check your answers.

### 3.4: Derivative Patterns

In this section, you will use your knowledge of properties and formulas of derivates to develop techniques for solving derivates of certain functions.

Read this section to learn about patterns of derivatives. Work through practice problems 1-8.

Watch this video on how to find derivatives of polynomial functions using the power rule.

Watch this video on derivatives of exponential functions.

Watch this video on how to find the derivatives of trig functions.

Watch this video on how to calculate and intepret higher ordered derivatives.

Work through the odd-numbered problems 1-47. Once you have completed the problem set, check your answers.

### 3.5: The Chain Rule

The Chain Rule is commonly used in derivate patterns. You will use the Chain Rule to determine the derivative of functions such as logarithm and inverse trig functions.

Read this section to learn about the Chain Rule. Work through practice problems 1-8.

Watch these videos for an introduction to the chain rule for differentiation and examples of the chain rule for differentiation.

Work through the odd-numbered problems 1-83. Once you have completed the problem set, check your answers.

### 3.6: Some Applications of the Chain Rule

In this section, you will learn how to use the Chain Rule in various applications.

Read this section to learn how to apply the Chain Rule. Work through practice problems 1-8.

- Watch this video on using the chain rule to solve rates problems.
- Work through the odd-numbered problems 1-49. Once you have completed the problem set, check your answers.

### 3.7: Related Rates

Related rates are an application of derivatives. To solve these application problems, you will need to apply the derivate techniques you learned in previous sections.

Read this section to learn to connect derivatives to the concept of the rate at which things change. Work through practice problems 1-3.

- Watch this video on application of derivatives
- Watch this video on differentiating parametric functions.
Work through the odd-numbered problems 1-21. Once you have completed the problem set, check your answers.

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

In this section, you will learn how to use Newton's Method to find roots. Newton's Method is important in finding the roots of functions with graphs that cross.

- Read this section. Work through practice problems 1-6.
Watch this video on Newton's method.

Watch this video on how to determine differentiability graphically.

Watch this video on continuity of intervals and continuous functions.

- Work through the odd-numbered problems 1-21. Once you have completed the problem set, check your answers.

### 3.9: Linear Approximation and Differentials

You just learned about how to use Newton's Method to find the root of a function. In this section, you will learn about another characteristic of tangent lines.

Read this section to learn how linear approximation and differentials are connected. Work through practice problems 1-10.

Watch this video on linear approximation and differentials.

- Watch this video on how to calculate the differential of a function using derivatives.
Work through the odd-numbered problems 1-19. Once you have completed the problem set, check your answers.

### 3.10: Implicit and Logarithmic Differentiation

In this section, you will learn about two specialized differentiation techniques you encountered earlier. These specialized techniques are important to know for solving certain functions.

Read this section to learn about implicit and logarithmic differentiation. Work through practice problems 1-6.

- Watch these videos on implicit and logarithmic differentiation.
- Work through the odd-numbered problems 1-55. Once you have completed the problem set, check your answers.

### Unit 3 Assessment

- Receive a grade
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