1. The Derivative Function
Objective: To define and use the derivative function

2. Definition 2.2.1 The function defined by the formula
is called the derivative of f with respect to x. The domain of consists of all x in the domain of for which the limit exists.
Remember, this is called the difference quotient.

3. Example 1 Find the derivative with respect to x of
and use it to find the equation of the tangent line to at
Note: The independent variable is x. This is very important to state. Later, we will be taking derivatives with respect to other independent variables.

4. Example 1 Find the derivative with respect to x of
and use it to find the equation of the tangent line to at

5. Example 1 Find the derivative with respect to x of
and use it to find the equation of the tangent line to at

6. Example 1 The slope of the tangent line to at
is When , so the equation of the tangent line at is

7. Example 1
We can also use the other formula to find the derivative of

8. Example 2
a) Find the derivative with respect to x of

9. Example 2
a) Find the derivative with respect to x of

10. Example 2
a) Find the derivative with respect to x of

11. Example 2
We can use the other formula to find the derivative of

12. Example 2
Lets look at the two graphs together and discuss the relationship between them.

13. Example 2
Since can be interpreted as the slope of the tangent line to the graph at it follows that
is positive where the tangent line has positive slope, is negative where the tangent line has negative slope, and zero where the tangent line is horizontal.

14. Example 3
At each value of x, the tangent line to a line is the line itself, and hence all tangent lines have slope m. This is confirmed by:

15. Example 4 Find the derivative with respect to x of
Recall from example 4, section 2.1 we found the slope of the tangent line of was , thus,
Memorize this!!!!

16. Example 4 Find the derivative with respect to x of
Find the slope of the tangent line to
at x = 9.
The slope of the tangent line at x = 9 is

17. Example 4 Find the derivative with respect to x of
Find the slope of the tangent line to
at x = 9.
Find the limits of as and as
and explain what those limits say about the graph of

18. Example 4 Find the limits of as and as
and explain what the limits say about the graph of
The graphs of f(x) and f /(x) are shown. Observe that
if , which means that all tangent lines to the graph of have positive slopes, meaning that the graph becomes more and more vertical as
and more and more horizontal as

19. Instantaneous Velocity
We saw in section 2.1 that instantaneous velocity was defined as
Since the right side of this equation is also the definition of the derivative, we can say
This is called the instantaneous velocity function, or just the velocity function of the particle.

20. Example 5
Recall the particle from Ex 5 of section 2.1 with position function Here f(t) is measured in meters and t is measured in seconds. Find the velocity function of the particle.

21. Example 5
Recall the particle from Ex 5 of section 2.1 with position function Here f(t) is measured in meters and t is measured in seconds. Find the velocity function of the particle.

22. Differentiability
Definition A function is said to be differentiable at x0 if the limit
exists. If f is differentiable at each point in the open interval (a, b) , then we say that is differentiable on (a, b), and similarly for open intervals of the form In the last case, we say that it is differentiable everywhere.

23. Differentiability
Definition A function is said to be differentiable at x0 if the limit
exists. When they ask you if a function is differentiable on the AP Exam, this is what they want you to reference.

24. Differentiability
Geometrically, a function f is differentiable at x if the graph of f has a tangent line at x. There are two cases we will look at where a function is non-differentiable.
Corner points
Points of vertical tangency

25. Corner points
At a corner point, the slopes of the secant lines have different limits from the left and from the right, and hence the two-sided limit that defines the derivative does not exist.

26. Vertical tangents
We know that the slope of a vertical line is undefined, so the derivative makes no sense at a place with a vertical tangent, since it is defined as the slope of the line.

27. Differentiability and Continuity
Theorem If a function f is differentiable at x, then f is continuous at x.
The inverse of this is not true. If it is continuous, that does not mean it is differentiable (corner points, vertical tangents).

28. Differentiability and Continuity
Theorem If a function f is differentiable at x, then f is continuous at x.
Since the conditional statement is true, so is the contrapositive:
If a function is not continuous at x, then it is not differentiable at x.

29. Other Derivative notations
We can express the derivative in many different ways.
Please note that these expressions all mean the derivative of y with respect to x.

30. Other formulas to use
There are several different formulas you can use to find the derivative of a function. The only ones we will use are:

31. Homework Section 2.2 Page 152-153 1-25 odd, 31
For numbers 15,17,19, use formula 13, not formula 12.

32. Example
Evaluate:

33. Example Evaluate: This is the definition of the derivative of .
The answer is You are not supposed to do any work, just recognize this!