1. Derivative of a Function
Chapter 3.1

2. Definition of the Derivative
In the previous chapter, we defined the slope of the tangent line to a curve 𝑦=𝑓(𝑥) at a point 𝑎 as
𝑚= lim ℎ→0 𝑓 𝑎+ℎ −𝑓 𝑎 ℎ
When this limit exists, it is called the derivative of 𝒇 at 𝒂 and is a number
In this chapter we study the derivative as a function derived from 𝑓 by considering the limit at each point in the domain of 𝑓

3. Definition of the Derivative
The derivative of the function 𝑓 with respect to the variable 𝑥 is the function 𝑓′ (“f prime”) whose value is
𝑓 ′ 𝑥 = lim ℎ→0 𝑓 𝑥+ℎ −𝑓 𝑥 ℎ
provided the limit exists. Note that 𝑥 is any value in the domain of 𝑓 where the derivative exists (that is, the limit exists).

4. Definition of the Derivative
It is important to remember that 𝑓′(𝑥) is a function
The domain of this function may be the same as that of 𝑓, or it may be smaller than that of 𝑓; it cannot be larger than that of 𝑓
If 𝑓′ exists, we say that 𝑓 is differentiable at 𝑥 (or that 𝑓 has a derivative at 𝑥)
A function that is differentiable at every point of its domain is a differentiable function

5. Example 1: Applying the Definition
Find the derivative of 𝑓 𝑥 = 𝑥 3 .

6. Example 1: Applying the Definition
Apply the definition: 𝑓 ′ 𝑥 = lim ℎ→0 𝑥+ℎ 3 − 𝑥 3 ℎ = lim ℎ→0 𝑥 3 +3 𝑥 2 ℎ+3𝑥 ℎ 2 + ℎ 3 − 𝑥 3 ℎ = lim ℎ→0 3 𝑥 2 ℎ+3𝑥 ℎ 2 + ℎ 3 ℎ = lim ℎ→0 3 𝑥 2 +3𝑥ℎ+ ℎ 2 =3 𝑥 2 +3𝑥⋅ =3 𝑥 2

7. Example 1: Applying the Definition
The derivative of 𝑓 𝑥 = 𝑥 3 is 𝑓 ′ 𝑥 =3 𝑥 2
IMPORTANT: remember that the value of 𝑓′ at any value of 𝑥 is the slope of the tangent line at 𝑥!
If we need to find the slope of the tangent line at, say, 𝑥=2, then 𝑓 ′ 2 = =12
That is, the slope of the tangent line at 𝑥=2 is 12

8. Example 1: Applying the Definition

9. Alternate Definition DEFINITION:
The derivative of the function 𝑓 at the point 𝒙=𝒂 is the limit
𝑓 ′ 𝑎 = lim 𝑥→𝑎 𝑓 𝑥 −𝑓 𝑎 𝑥−𝑎
provided the limit exists.

10. Example 2: Applying the Alternate Definition
Differentiate (i.e., find the derivative) 𝑓 𝑥 = 𝑥 using the alternate definition.

11. Example 2: Applying the Alternate Definition
Apply the alternate definition: 𝑓 ′ 𝑥 = lim 𝑥→𝑎 𝑥 − 𝑎 𝑥−𝑎 = lim 𝑥→𝑎 𝑥 − 𝑎 𝑥−𝑎 ⋅ 𝑥 + 𝑎 𝑥 + 𝑎 = lim 𝑥→𝑎 𝑥−𝑎 𝑥−𝑎 𝑥 + 𝑎 = lim 𝑥→𝑎 1 𝑥 + 𝑎 = 1 𝑎 + 𝑎 = 1 2 𝑎

12. Example 2: Applying the Alternate Definition
The result, 𝑓 ′ 𝑎 = 1 2 𝑎 is the derivative at some point 𝑥=𝑎. We can now let 𝑎 be any number in the domain of 𝑓 (i.e., use 𝑥 in place of 𝑎) at which the limit exists and write 𝑓 ′ 𝑥 = 1 2 𝑥 Note that, although 𝑓 0 =0, 𝑓′(0) is not defined. So the domain of 𝑓 is 𝑥≥0, but the domain of 𝑓′ is 𝑥>0.

13. Notation The following notations can be used to represent a derivative
𝑦′ “y prime”
𝑑𝑦 𝑑𝑥 “dy dx” or “the derivative of y with respect to x”
𝑑𝑓 𝑑𝑥 “df dx” or “the derivative of f with respect to x”
𝑑 𝑑𝑥 [𝑓 𝑥 ] “d dx of f at x” or “the derivative of f at x”

14. Example 3: Graphing 𝑓′ from 𝑓
Graph the derivative of the function 𝑓 whose graph is shown in the next slide. Discuss the behavior of 𝑓 in terms of the signs and values of 𝑓′.

15. Example 3: Graphing 𝑓′ from 𝑓

16. Example 3: Graphing 𝑓′ from 𝑓

17. Example 4: Graphing 𝑓 from 𝑓′
Sketch the graph of a function 𝑓 that has the following properties:
𝑓 0 =0
The graph of 𝑓′, the derivative of 𝑓, is as shown on the next slide
𝑓 is continuous for all 𝑥

18. Example 4: Graphing 𝑓 from 𝑓′

19. Example 4: Graphing 𝑓 from 𝑓′

20. Example 5: Downhill Skier (Prob. 29)
The table below gives the approximate distance traveled by a downhill skier after 𝑡 seconds for 0≤𝑡≤10. Sketch a graph of the derivative then answer the following questions.
What does the derivative represent?
In what units would the derivative be measured?
Guess an equation of the derivative by considering its graph.

21. Example 5: Downhill Skier (Prob. 29)
Time (seconds)
Distance (feet) 1
3.3 2
13.3 3
29.9 4
53.2 5
83.2 6
119.8 7
163.0 8
212.9 9
269.5 10
332.7

22. Example 5: Downhill Skier (Prob. 29)

23. Example 5: Downhill Skier (Prob. 29)
The table below gives the approximate distance traveled by a downhill skier after 𝑡 seconds for 0≤𝑡≤10. Sketch a graph of the derivative then answer the following questions.
What does the derivative represent? Speed of skier
In what units would the derivative be measured? Feet per second
Guess an equation of the derivative by considering its graph. 𝐷= 6.65𝑡

24. Example 5: Downhill Skier (Prob. 29)

25. One-Sided Derivatives
DEFINITION:
A function 𝑦=𝑓(𝑥) is differentiable on a closed interval [𝒂,𝒃] if it has a derivative at every interior point of the interval, and if the limits
lim ℎ→ 𝑓 𝑥+ℎ −𝑓 𝑥 ℎ
lim ℎ→ 0 − 𝑓 𝑥+ℎ −𝑓 𝑥 ℎ
exist at the endpoints.

26. One-Sided Derivatives
At the left endpoint of a closed interval, ℎ is positive and approaches zero from the right (i.e., from values that are greater than zero)
At the right endpoint, ℎ is negative and approaches zero from the left (i.e., from values that are less than zero)
We may determine a closed interval (and, thus, right- and left-hand derivatives) at any two distinct points of a function’s domain

27. One-Sided Derivatives
Recall Theorem 3 from section 2.1:
A function 𝑓(𝑥) has a limit as 𝑥 approaches 𝑐 if and only if the right-hand and left-hand limits exist at 𝑐 and are equal.
Since the derivative is defined as a kind of limit, then this theorem applies generally
This means that if, at a value 𝑐 of a function’s domain, lim ℎ→ 𝑓 𝑐+ℎ −𝑓 𝑐 ℎ ≠ lim ℎ→ 0 − 𝑓 𝑐+ℎ −𝑓 𝑐 ℎ , then lim ℎ→0 𝑓 𝑐+ℎ −𝑓 𝑐 ℎ does not exist
Thus, the function is not differentiable at 𝒄

28. Example 6: One-Sided Derivatives Can Differ at a Point
Show that the following function has left-hand and right-hand derivatives at 𝑥=0, but no derivative there. 𝑦= 𝑥 2 , if 𝑥≤0 2𝑥, if 𝑥>0

29. Example 6: One-Sided Derivatives Can Differ at a Point
For the left-hand limit (for 𝑥≤0) use the function piece 𝑥 2 lim ℎ→ 0 − 0+ℎ 2 − 0 2 ℎ = lim ℎ→ 0 − ℎ 2 ℎ = lim ℎ→ 0 − ℎ =0 For the right-had limit (for 𝑥>0) use the function piece 2𝑥 lim ℎ→ ℎ −2 0 ℎ = lim ℎ→ 0 + 2ℎ ℎ = lim ℎ→ =2 The left- and right-had derivatives are different, so the function is not differentiable at 𝑥=0.

30. Example 6: One-Sided Derivatives Can Differ at a Point

31. Exercise 3.1
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