About This Chapter
Overview of Properties of Derivatives - Chapter Summary
Review the lessons in this chapter for an overview of the basic principles behind using derivatives in calculus. You can gain a better understanding of what it means to be 'differentiable' and study how to calculate these derivatives. You'll also see examples of the linear properties of derivatives, review when to use the quotient rule and explore optimization and differentiation.
After you complete the chapter, you should be able to:
- Explain what derivatives are and how they define calculus
- Understand and create a graphical representation of the slope of a function
- Describe the meaning of the term 'differentiable' and how it applies to calculus
- Use limits, such as position, to calculate velocity as a function of time
- Define constant multiples and additions and demonstrate their application in calculating limits
- Identify a continuous function and the relationship between continuity and differentiability
Our lessons are accessible from your computer and mobile device so you can study anywhere, at your convenience. Take advantage of the tools offered in this chapter, including video lessons with full transcripts and lesson quizzes, to track your progress and help the information stick.
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. What It Means To Be 'Differentiable'
Lots of jets can go from zero to 300 mph quickly, but super-jets can do this instantaneously. In this lesson, learn what that means for differentiability.
4. 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.
5. 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.
6. 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.
7. The Relationship Between Continuity & Differentiability
Why is it that all differentiable functions are continuous but not all continuous functions are differentiable? Learn why in this video lesson. Also see what a continuous function looks like versus one that isn't.
8. Optimization and Differentiation
In this lesson, you can learn what optimization means from a mathematical standpoint. Using the techniques taught in this lesson, you can use the five steps to optimization to figure out practical things, like how much sleep you need to get before an exam.
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Other chapters within the AP Calculus AB & BC: Exam Prep course
- Graph Basics
- The Basics of Functions
- How to Graph Functions
- Overview of Limits of Functions
- Overview of Function Continuity
- Understanding Exponentials & Logarithms
- Using Exponents and Polynomials
- Parametric, Polar and Vector Functions
- The Derivative at a Point
- The Derivative as a Function
- Second Derivatives
- Derivative Applications
- Finding Derivatives
- Properties of Definite Integrals
- Applications of Integrals
- Using the Fundamental Theorem of Calculus
- Applying Integration Techniques
- Approximation of Definite Integrals
- Understanding Sequences & Series
- Series of Constants
- Taylor Series
- Using a Scientific Calculator for Calculus
- AP Calculus AB & BC Flashcards