1. Find the derivative Find the derivative at the following point.
Warm-up
Find the derivative
Find the derivative at the following point.

2. Table of Contents
10. Section 3.2 The Derivative as a Function

3. The Derivative as a Function
Essential Question – What rules of differentiation will make it easier to calculate derivatives?

4. Notation for Derivative
If derivative exists, we say it is differentiable

5. Power Rule
Power Rule
Bring down the exponent and subtract one from the exponent

6. More notation
means find the derivative of x4 when x = -2

7. 2 more rules
Constant multiple
Sum and Difference

8. Differentiating a polynomial

9. Derivative of ex

10. Example
Find the equation of the tangent line to the graph of f(x) = 3ex -5x2 at x=2

11. What information does the derivative at a point tell us?
Tells us whether the tangent line has a positive or negative slope
Tells us how steep the line is (the larger the derivative, the steeper the line)
Tells us if there is a turning point (slope is 0)

12. Horizontal Tangents
Does y = x4 – 2x2 + 2 have any horizontal tangents?
First find the derivative, then set = 0 (because the slope of a horizontal line is 0)

13. Calculator example
Find the points where horizontal tangents occur.

14. Graphing f’(x) from f(x)
Find slope at each point
Make a new graph using same x points and the slope as the y point
If f is increasing, f ‘ will be positive (above the x axis)
If f has a turning point, f ‘ will be 0
If f is decreasing, f ‘ will be negative (below the x axis)

15. Graph example
Given the graph of f(x), which of A or B is the derivative?

16. Differentiability Differentiability implies continuity
If f is differentiable at x = c, then f is continuous at x = c
The opposite is not true
A function can be continuous at x = c, but not differentiable

17. 4 times a derivative fails to exist
Corner
Cusp

18. 4 times a derivative fails to exist
Vertical tangent
Discontinuity

19. Local Linearity
A function that is differentiable closely resembles its own tangent line when viewed very closely
In other words, when zoomed in on a few times, a curve will look like a straight line.

20. Example

21. Assignment
Pg. 139 #1-11 odd, odd, odd, odd, all, odd