About This Chapter
Calculus: Calculating Derivatives and Derivative Rules
Welcome to this lesson series on calculating derivatives and derivative rules. We've introduced the derivative as being the definitive element to calculus. In this video lesson series, you'll put the derivative to good use by exploring its significant role in calculus. But don't worry about too much confusion; we've got you covered with several relatable examples and quirky animations (like our trusty human cannonball, Super C) to help illustrate these concepts with ease.
Building on your knowledge of derivatives (how functions change as inputs change), you'll learn to use limits to calculate derivatives and learn the linear properties of derivatives. You'll also work outside in by using the chain rule to differentiate complex functions. You'll then see how tangent lines are related to derivatives and how you can 'divide and conquer' to calculate limits.
Additionally, you'll learn how to calculate the derivatives of trigonometric and implicit functions, as well as polynomial and exponential equations. You'll also get plenty of practice graphing these concepts, learn how to calculate higher-order derivatives and use the quotient rule for differentiation. You can test your knowledge of derivatives and derivative rules through short, multiple-choice quizzes, as well as a culminating chapter exam.
Sound like a lot to take in? Well, we won't go off on too many tangents. We want you to learn all you can about the definitive derivative!
1. 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.
2. 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.
3. 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.
4. 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.
5. 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.
6. 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.
7. 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.
8. 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.
9. 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.
10. 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.
11. 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)!
12. 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.
13. 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.
14. Optimization Problems in Calculus: Examples & Explanation
In this lesson, we'll take a step-by-step approach to learning how to use calculus to solve problems where a parameter, such as area or volume, needs to be optimized for a given set of constraints.
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Other chapters within the Math 104: Calculus course
- Graphing and Functions
- Vectors in Calculus
- Geometry and Trigonometry
- How to Use a Scientific Calculator
- Rate of Change
- Graphing Derivatives and L'Hopital's Rule
- Applications of Derivatives
- Area Under the Curve and Integrals
- Integration and Integration Techniques
- Integration Applications
- Differential Equations
- Studying for Math 104