• ### 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 $F = MA$, 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 13 hours.

• ### 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.

• ### 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.
• ### 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.

• ### 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.

• ### 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.

• ### 3.6: Some Applications of the Chain Rule

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

• ### 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.

• ### 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.

• ### 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.

• ### 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.