# Ch 8: Properties of Derivatives

### 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

Video | Objective |
---|---|

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

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