# Derivatives

### Topics

## Introduction to Derivatives - At A Glance:

From a graph of a function *f*(*x*) we can make a sketched graph of its derivative *f ' *(*x*). To do this, we use some things we talked about earlier.

- If
*f*is decreasing, its slope (and hence its derivative) is negative. If*f*is increasing, its slope (and hence its derivative) is positive.

- From drawing tangent lines to
*f*, we can compare relative values of the derivative and tell where the derivative is greatest.

- If
*f*is a line, its slope is constant.

**Be Careful:** The word "it" is dangerous.

Look at these two sentences:

- It's increasing, so it's positive.

*f*is increasing, so*f '*is positive.

The first sentence is unclear. What does "it" mean? There's no way to know. Whenever we use the words "increasing, decreasing, positive, negative," that we are clear about what (*f*, *f '*, or something else?) is increasing, decreasing, positive, or negative.

#### Example 1

From the graph of |

#### Example 2

From the graph of |

#### Example 3

From the graph of |

#### Example 4

From the graph of |

#### Example 5

From the graph of |

#### Example 6

From the graph of |

#### Exercise 1

For the function *f*(*x*) below, draw a graph of *f'*(*x*). Don't worry too much about whether *f'* is straight or curvy - focus on getting it to cross the *x* axis in the right places.

#### Exercise 2

For the function *f*(*x*) below, draw a graph of *f'*(*x*). Don't worry too much about whether *f'* is straight or curvy - focus on getting it to cross the *x* axis in the right places.

#### Exercise 3

For the function *f*(*x*) below, draw a graph of *f'*(*x*). Don't worry too much about whether *f'* is straight or curvy - focus on getting it to cross the *x* axis in the right places.

#### Exercise 4

*f*(*x*) below, draw a graph of *f'*(*x*). Don't worry too much about whether *f'* is straight or curvy - focus on getting it to cross the *x* axis in the right places.

#### Exercise 5

From the graph of *f'*(*x*), draw a graph of *f*(*x*). Make sure to label each graph.

#### Exercise 6

From the graph of *f'*(*x*), draw a graph of *f*(*x*). Make sure to label each graph.

#### Exercise 7

From the graph of *f'*(*x*), draw a graph of *f*(*x*). Make sure to label each graph.

#### Exercise 8

From the graph of *f'*(*x*), draw a graph of *f*(*x*). Make sure to label each graph.