The product rule shows us how to take the derivative of a product of functions. Ditto the quotient rule.

1. Lesson 12 (Section 3.2)
The Product and Quotient Rules
Math 1a
October 22, 2007
Announcements
Midterm I: 10/24, 7-9pm. A-G in Hall A, H-Z in Hall C

2. We have shown that if u and v are functions, that
(u + v ) = u + v
(u − v ) = u − v
What about uv ? Is it u v ?

3. Is the derivative of a product the product of the
derivatives?
NO!

4. Is the derivative of a product the product of the
derivatives?
NO!
Try this with u = x and v = x 2 .

5. Is the derivative of a product the product of the
derivatives?
NO!
Try this with u = x and v = x 2 . Then uv = x 3 , so (uv ) = 3x 2 .
But u v = 1(2x) = 2x.

6. Is the derivative of a product the product of the
derivatives?
NO!
Try this with u = x and v = x 2 . Then uv = x 3 , so (uv ) = 3x 2 .
But u v = 1(2x) = 2x.
So we have to be more careful.

7. Mmm...burgers
Say you work in a fast-food joint. You want to make more money.
What are your choices?

8. Mmm...burgers
Say you work in a fast-food joint. You want to make more money.
What are your choices?
Work longer hours.

9. Mmm...burgers
Say you work in a fast-food joint. You want to make more money.
What are your choices?
Work longer hours.
Get a raise.

10. Mmm...burgers
Say you work in a fast-food joint. You want to make more money.
What are your choices?
Work longer hours.
Get a raise.
Say you get a 25 cent raise in your hourly wages and work 5 hours
more per week. How much extra money do you make?

11. Money money money money
The answer depends on how much you work already and your
current wage. Suppose you work h hours and are paid w . You get
a time increase of ∆h and a wage increase of ∆w . Income is
wages times hours, so
∆I = (w + ∆w )(h + ∆h) − wh
FOIL
= wh + w ∆h + ∆w h + ∆w ∆h − wh
= w ∆h + ∆w h + ∆w ∆h

12. A geometric argument
Draw a box:
∆h w ∆h ∆w ∆h
h wh ∆w h
w ∆w

13. A geometric argument
Draw a box:
∆h w ∆h ∆w ∆h
h wh ∆w h
w ∆w
∆I = w ∆h + h ∆w + ∆w ∆h

14. Supose wages and hours are changing over time. How does income
change?
∆I w ∆h + h ∆w + ∆w ∆h
=
∆t ∆t
∆h ∆w ∆h
=w +h + ∆w
∆t ∆t ∆t
dh dw
→w +h +0
dt dt
as t → 0.

16. Supose wages and hours are changing over time. How does income
change?
∆I w ∆h + h ∆w + ∆w ∆h
=
∆t ∆t
∆h ∆w ∆h
=w +h + ∆w
∆t ∆t ∆t
dh dw
→w +h +0
dt dt
as t → 0.
Theorem (The Product Rule)
Let u and v be differentiable at x. Then
(uv ) (x) = u(x)v (x) + u (x)v (x)

18. Example
Find this derivative two ways: first by FOIL and then by the
product rule:
d
(3 − x 2 )(x 3 − x + 1).
dx

22. The Quotient Rule
What about the derivative of a quotient?

23. The Quotient Rule
What about the derivative of a quotient?
u
Let u and v be differentiable and let Q = . Then u = Qv . If Q
v
is differentiable, we have
u = (Qv ) = Q v + Qv
u − Qv u uv
Q = = − 2
v v v
u v − uv
=
v2

24. The Quotient Rule
What about the derivative of a quotient?
u
Let u and v be differentiable and let Q = . Then u = Qv . If Q
v
is differentiable, we have
u = (Qv ) = Q v + Qv
u − Qv u uv
Q = = − 2
v v v
u v − uv
=
v2
This is called the Quotient Rule.

25. Examples
Example
d 2x + 5
1.
dx 3x − 2
d 2x + 1
2.
dx x 2 − 1
d t −1
3. 2+t +2
.
dt t

26. Examples
Example
d 2x + 5
1.
dx 3x − 2
d 2x + 1
2.
dx x 2 − 1
d t −1
3. 2+t +2
.
dt t
Answers

27. Examples
Example
d 2x + 5
1.
dx 3x − 2
d 2x + 1
2.
dx x 2 − 1
d t −1
3. 2+t +2
.
dt t
Answers
19
1. −
(3x − 2)2

28. Examples
Example
d 2x + 5
1.
dx 3x − 2
d 2x + 1
2.
dx x 2 − 1
d t −1
3. 2+t +2
.
dt t
Answers
19
1. −
(3x − 2)2
2 x2 + x + 1
2. −
(x 2 − 1)2

29. Examples
Example
d 2x + 5
1.
dx 3x − 2
d 2x + 1
2.
dx x 2 − 1
d t −1
3. 2+t +2
.
dt t
Answers
19
1. −
(3x − 2)2
2 x2 + x + 1
2. −
(x 2 − 1)2
−t 2 + 2t + 3
3.
(t 2 + t + 2)2

30. Example (Quadratic Tangent to identity function)
The curve y = ax 2 + bx + c passes through the point (1, 2) and is
tangent to the line y = x at the point (0, 0). Find a, b, and c.

31. Example (Quadratic Tangent to identity function)
The curve y = ax 2 + bx + c passes through the point (1, 2) and is
tangent to the line y = x at the point (0, 0). Find a, b, and c.
Answer
a = 1, b = 1, c = 0.

32. Power Rule for nonnegative integers by induction
Theorem
Let n be a positive integer. Then
d n
x = nx n−1 .
dx

33. Power Rule for nonnegative integers by induction
Theorem
Let n be a positive integer. Then
d n
x = nx n−1 .
dx
Proof.
By induction on n. We have shown it to be true for n = 1.
d n
Suppose for some n that x = nx n−1 . Then
dx
d n+1 d
x = (x · x n )
dx dx
d d n
= x xn + x x
dx dx
= 1 · x n + x · nx n−1 = (n + 1)x n .

34. Power Rule for negative integers
Use the quotient rule to prove
Theorem
d −n
x = (−n)x −n−1 .
dx
for positive integers n.

35. Power Rule for negative integers
Use the quotient rule to prove
Theorem
d −n
x = (−n)x −n−1 .
dx
for positive integers n.
Proof.
d −n d 1
x =
dx dx x n
d d
x n dx 1 − 1 dx x n
=
x 2n
0 − nx n−1
= = −nx −n−1 .
x 2n