About This Chapter
Rate of Change & Calculating Derivatives - Chapter Summary
The objective of this chapter is to help you improve your skills calculating rate of change and derivatives by improving your understanding of the different properties of derivatives and the rules used to calculate derivatives. The lessons in this chapter have been assembled by our professional instructors to provide you with an effective review of:
- Formal definition and graphical representations of derivatives
- Limits and how they can be used to calculate derivatives
- Linear properties of derivatives
- Procedures used to calculate derivatives of trigonometric functions, polynomial equations and exponential equations
- The chain rule, quotient rule and differentiating complex functions
- Calculations for higher order derivatives
- Finding derivatives of implicit functions and derivatives of inverse trigonometric functions
- Using the rule of differentiation to calculate derivatives
After each lesson, use the lesson quiz to test your understanding of the material covered in it. Then, use the provided links to return to the parts of the lesson that explained the topics you missed. After you've completed all the lessons and quizzes in this chapter, be sure to complete the practice chapter exam to test your overall understanding of the material.
1. Derivatives: The Formal Definition
The derivative defines calculus. In this lesson, learn how the derivative is related to the instantaneous rate of change with Super C, the cannonball man.
2. Derivatives: Graphical Representations
Take a graphical look at the definitive element of calculus: the derivative. The slope of a function is the derivative, as you will see in this lesson.
3. Using Limits to Calculate the Derivative
If you know the position of someone as a function of time, you can calculate the derivative -- the velocity of that person -- as a function of time as well. Use the definition of the derivative and your knowledge of limits to do just that in this lesson.
4. The Linear Properties of a Derivative
In this lesson, learn two key properties of derivatives: constant multiples and additions. You will 'divide and conquer' in your approach to calculating the limits used to find derivatives.
5. Calculating Derivatives of Trigonometric Functions
The trigonometric functions show up almost everywhere that you have a repeating pattern. In this lesson, learn how to find the derivatives of the trigonometric functions.
6. Calculating Derivatives of Polynomial Equations
Polynomials can describe just about anything and are especially common in describing motion. Learn the tricks to quickly finding the derivatives of these ubiquitous functions.
7. Calculating Derivatives of Exponential Equations
Earth's population is booming! But why is the population increasing so much more drastically than it did many years ago? In this lesson, learn how to calculate the rates of change in exponentials by discovering the rules of derivatives with exponents.
8. Using the Chain Rule to Differentiate Complex Functions
If you've ever seen a complicated function, this lesson is for you. Most functions that we want to differentiate are complicated functions, for which no single derivative rule will work. In this lesson, learn how to use the chain rule to simplify nesting equations.
9. Differentiating Factored Polynomials: Product Rule and Expansion
Most functions that we want to differentiate are complicated functions for which no single derivative rule will work. In this lesson, learn what happens to derivatives when you multiply functions together.
10. When to Use the Quotient Rule for Differentiation
Lo D Hi minus Hi D Lo, all over the square of what's below! Learn the quotient rule chant for differentiating functions that take the form of fractions in this lesson.
11. Understanding Higher Order Derivatives Using Graphs
The derivative is a rate of change, like velocity. What happens, though, when your velocity - that is, your rate of change - is changing? Explore the changing changes in this lesson.
12. Calculating Higher Order Derivatives
Differentiating functions doesn't have to stop with the first or even second derivative. Learn what a mathematical jerk is as you calculate derivatives of any order in this lesson.
13. How to Find Derivatives of Implicit Functions
How do you define the rate of change when your function has variables that cannot be separated? Learn how implicit differentiation can be used to find dy/dx even when you don't have y=f(x)!
14. How to Calculate Derivatives of Inverse Trigonometric Functions
Like a metronome, trigonometric functions are regular. Even predictable. In this lesson, you will learn how to use this predictability to remember the derivative formulas for these common functions.
15. Applying the Rules of Differentiation to Calculate Derivatives
In this lesson, we'll review common derivatives and their rules, including the product, quotient and chain rules. We'll also examine how to solve derivative problems through several examples.
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