About This Chapter
Properties of Derivatives - Chapter Summary and Learning Objectives
Derivatives are a very important part of calculus. They stem naturally from observing a function's slope, tangent lines, and limits. These three concepts examine the behavior of a function between or near certain inputs. The derivative is more general and useful. It is a mathematical expression describing how a function changes as it crosses through all inputs. This chapter discusses the definition of a derivative, a derivative's relationship to limits, and how to represent a derivative graphically. By the end of the chapter, you will have mastered:
- Calculating the derivative
- The meaning of differentiability
- How limits lead to derivatives
- Using the quotient rule
- What continuity means
|Derivatives: The Formal Definition||Define a derivative and identify its various components.|
|Derivatives: Graphical Representations||Learn the representations of derivatives in a Cartesian coordinate system.|
|What It Means To Be 'Differentiable'||Discuss the meaning of differentiability through an examination of the derivative's formula.|
|Using Limits To Calculate the Derivative||Explore the relationship between limits and derivatives.|
|The Linear Properties of a Derivative||Identify the various properties of derivatives and their corresponding equations.|
|When to Use the Quotient Rule for Differentiation||Learn to use the quotient rule to find the derivative of two separate functions dividing into each other.|
|The Relationship Between Continuity & Differentiability||Define a continuous function and consider it in relation to differentiable functions.|
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.
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Other chapters within the AP Calculus AB: Exam Prep course
- Graph Basics
- The Basics of Functions
- How to Graph Functions
- Limits of Functions
- Continuity of Functions
- Understanding Exponentials & Logarithms
- Using Exponents and Polynomials
- The Derivative at a Point
- The Derivative as a Function
- Second Derivatives
- Using Derivatives
- Computing Derivatives
- Properties of Definite Integrals
- Applications of Integrals
- Using the Fundamental Theorem of Calculus
- Understanding & Applying Integration Techniques
- Approximating Definite Integrals
- Using a Scientific Calculator for Calculus
- About the AP Calculus Exam
- AP Calculus AB Flashcards