1. Warm-Up: October 2, 2017
Find the slope of at

2. Derivative of a Function
Section 3.1

3. Chapter 3 Overview
We learned how to find the slope of a tangent to a curve as the limit of the slopes of secant lines.
This gives us the instantaneous rate of change at a point.
The study of rates of change of functions is called differential calculus.

4. Derivative
The definition of derivative is the same as the function for slope.
Provided the limit exists!!

5. Derivative Notation

6. Derivative Vocabulary
If f’(x) exists, we say that f is differentiable at x. This is the same as saying “f has a derivative at x”
A function that is differentiable at every point of its domain is a differentiable function.
A function must be continuous in order to be differentiable.

7. Example
Differentiate 𝑓 𝑥 = 𝑥 3

8. Definition (Alternate)
The derivative of the function at a point a, is the limit
𝑓 ′ 𝑎 = lim 𝑥→𝑎 𝑓 𝑥 −𝑓 𝑎 𝑥−𝑎
Provided the limit exists.

9. Example
Differentiate 𝑥 .

10. Warm-Up: October 3, 2017
Find the average rate of change of y=3x2 over the interval [-1, 4].

11. Graphing f’ from f
NOTE: The domain of f’ may be smaller than the domain of f.
Calculate/estimate the slope for various values of x.
Plot the (x, y’) points
Connect the points with a smooth curve for x-values where f is smooth and continuous.

12. Example 1

13. Graphing f from f’ You must be given at least one point.
Start your graph at the given point.

14. Example 2 Sketch the graph that has the following properties 𝑓 0 =0
The graph of 𝑓′ is as shown in the figure.
𝑓 is continuous for all x.

15. One-Sided Derivatives
Right-Hand Derivative
Left-Hand Derivative
Derivatives at endpoints are one-sided limits.

16. Example 3
𝑦= 𝑥 2 +1, 𝑥<2 −𝑥+3, 𝑥≥2

17. Derivatives from Data
Open you textbook to page 103
Look at #20