# Ch 9: The Derivative at a Point

### About This Chapter

## The Derivative at a Point - Chapter Summary and Learning Objectives

The generic mathematical expression for a derivative comes from calculating the rate of change of a function. The rate of change allows you to understand a function's behavior around a particular input. This chapter explores how the derivative formula is related to the slope of a function's tangent line. The lessons allow you to form an intuitive understanding of derivatives and functions. Each lesson is taught by subject experts and features a transcript you can use to reference key terms. By the end of the chapter you will have understood:

- How derivative functions represent real life phenomena such as velocity
- How to create a line tangent to a function
- Methods for writing the equation of the tangent line
- How to discover the slopes of tangent lines
- Steps to calculate a function's derivative

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

Velocity and the Rate of Change | Examine the relationship of velocity with acceleration and time. Model velocity as a function of both. |

Slopes and Rate of Change | Discover how the slope of a tangent line can reveal a function's rate of change. |

Equation of a Line Using Point-Slope Formula | Learn how the point-slope formula can reveal a line's equation. |

Interpreting the Slope & Intercept of a Linear Model | Intuitively discover the constant nature of a linear function's slope and y-intercept. |

Slopes and Tangents on a Graph | Examine graphical representations of slopes and tangents. |

Non Differentiable Graphs of Derivatives | Learn the graphical characteristics of non differentiable functions and non differentiable derivatives. |

First Derivative: Function, Examples & Quiz | Explore the formal definition of a derivative and practically use it to discover the derivative of a function. |

Finding Instantaneous Rate of Change of a Function: Formula & Examples | Continue to apply the definition of a derivative to find the rate of change of various functions. |

Calculating & Interpreting a Function's Average Rate of Change | Discover the relationship between derivatives and the average rate of change, and how to calculate the latter. |

### 1. Velocity and the Rate of Change

Running from your little sister or just window-shopping, your speed is just a measure of how fast you move, or how your position is changing over time. In this lesson, learn about how velocity is a rate of change.

### 2. Slopes and Rate of Change

If you throw a ball straight up, there will be a point when it stops moving for an instant before coming back down. Consider this as we study the rate of change of human cannonballs in this lesson.

### 3. Equation of a Line Using Point-Slope Formula

It's time for a road trip to Las Vegas, and after four hours of driving at 60 mph ... Are we there yet? Learn the point-slope form of the equation of a line to help answer this age-old question.

### 4. Interpreting the Slope & Intercept of a Linear Model

You've probably seen slope and intercept in algebra. These concepts can also be used to predict and understand information in statistics. Take a look at this lesson!

### 5. Slopes and Tangents on a Graph

Hit the slopes and learn how the steepness of a line is calculated. Calculate the slopes between points and draw the tangents of curves on graphs in this lesson.

### 6. Non Differentiable Graphs of Derivatives

When I walk along a curve, I stand normal to it. That is, I stand perpendicular to the tangent. Learn how to calculate where I'm standing in this lesson.

### 7. First Derivative: Function & Examples

In this lesson, you will learn about the relationship between the first derivative and rates of change. You will learn how to find the derivative of a polynomial using limits.

### 8. Calculating & Interpreting a Function's Average Rate of Change

Average rates of change have many real-world applications. In this lesson, we'll learn how to calculate and interpret a function's average rate of change with easy-to-follow examples.

### 9. Finding Instantaneous Rate of Change of a Function: Formula & Examples

In this lesson, you will learn about the instantaneous rate of change of a function, or derivative, and how to find one using the concept of limits from Calculus.

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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
- Properties of Derivatives
- 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