Product and Quotient
Rule
1
Product Rule
The product rule also called Leibniz's law, governs the
differentiation of products of differentiable functions
It is a common error, when studying calculus, to
suppose that the derivative of (uv) equals (u ′)(v ′)
Leibniz himself made this error initially
2
Product Rule
d
f x
dx
g ( x)
f x
Easier to remember as
d
g x
dx
uv
d
f x
dx
g x
u v uv
3
1
Product rule
Credited to Leibniz
His argument follows:
Let u(x) and v(x) be two differentiable functions of x.
Then the differential of uv is
d uv
u du v dv
u dv
v du
uv
du dv
Since (du)(dv) is "negligible" Leibniz concluded
d uv
du v u dv
4
Visual of Product Rule
5
Proof of Product Rule
f x h g x h
d
fg ( x) lim
h 0
dx
h
lim
h
f x h g x h
lim
lim
h
f ( x ) g ( x ) f ( x h) g ( x )
h
g x h g ( x ) g ( x ) f ( x h)
0
h
f x h
f x h
g x h
g ( x)
h
lim f x h lim
0
f ( x h) g ( x )
f ( x)
h
0
0
h
f ( x) g ( x)
h
0
lim
h
g x h
h
f ( x) g ( x) g ( x) f
x
g ( x)
0
g ( x ) f ( x h)
h
f ( x h)
g ( x) lim
h 0
h
f ( x)
f ( x)
6
2
Quotient Rule
d f
dx g
f
x
x g ( x)
f ( x) g ( x)
2
g ( x)
u
v
u v uv
v2
Easier to remember
Let h( x)
We can use the
product rule instead
of the quotient rule
by writing
then
1
,
g ( x)
d f
dx g
d
fh x
dx
x
7
Proof of Quotient Rule
d f
dx g
x
lim
h
0
lim
f x h
g x h
lim
h 0
h
f x
g x
f x h g x f x g x h
g x h g x h
f x h g x
f x g x h f ( x) g ( x) f ( x) g ( x)
g x h g x h
h 0
8
f x h
lim
h
f ( x) g x
f x h
g x h
f ( x) g x
h
g ( x)
g ( x)
g x h g x
f x h
f ( x)
h
0
g x h
h
0
lim
f x
h
lim
h
f x
g x h g x h
0
g x
f x lim
h
0
g x h
g ( x)
h
lim g x h g x
h
f
x g ( x)
g ( x)
0
f ( x) g ( x)
2
9
3
Quotient Rule – Memory Aids
"Lo-dee-hi, hi-dee-lo, draw the line and square below";
Lo being the denominator, Hi being the numerator and
"dee" being the derivative.
Another variation to this mnemonic is given when the
quotient is written with the numerator as Hi the
denominator as Ho: "Ho-dee-Hi minus Hi-dee-Ho all
over Ho-Ho."
A third variation is "Low-dee-high minus high-dee-low,
all over the square of what's below".
10
Compare
Product Rule
uv
u v uv
Quotient Rule
u
v
u v uv
v2
11
Let u
3x 2
2 x, v
u
6 x 2,
f
uv
f
3x
3x 2
f ( x)
Example
v
2x x 2x2
x 2x2
1 4x
uv
2
2x 1 4x
12 x 2 5 x 2
2x
6x 2 x 2x2
12 x 3 2 x 2
2x
24 x3 3x 2 4 x
12
4
Example
2 3x 2
6x
f ( x)
2 3x2 and v 6 x
Let u
u
6x
and v
6x 6x
u v uv
v2
2 3x
6
2
6
2
6x
36 x 2 12 18 x 2
36 x 2
2
3x 2
6 x2
We took this derivative in 4.1 without using
the quotient rule
x 3x3
f ( x)
Example
Let u
u
x1 2 and v 3x 2 2 x
1
and v 6 x 2
2 x
2 x 6x 2
x
3x 2 2 x +
2x
2
x 3x 2
x 12 x 4
+
2
2
x 15 x 6
3 x 5x 2
2
2
Practice
x
u
f
2
g x
14
Find the derivative
(a) f x
2 and v 3x 2
2x
x
2x
1
3x 2 2 x + x 6 x 2
2 x
f
Let u
13
and v
3
( x2
2) 3 x 2
2
(b) g x
u v uv
2 x 3x 2
9 x2 4 x 6
x2
x 2
3x 2
2 3
2 x 3x 2
x2 2 3
u v uv
2
2
v
3x 2
2
3x 4 x 6
3x 2
2
15
5
Example
2
2 x3 3x 2 x3 3x
f ( x)
Let u
u
f
2 x3 3x
f ( x)
2 x3 3x and v 2 x3 3x
6 x2 3 and v 6 x2 3
6 x 2 3 2 x3 3x
2 x3 3x 6 x 2 3
2 6 x 2 3 2 x3 3x
6 x 2 x2 1 2 x2 3
16
Practice
1
x3 3
f ( x)
Let u 1 and v x3 3
u 0 and v 3x2
f
0 x3 3
3x 2
1 3x 2
x3 3
2
x3 3
2
17
Remarks
We can use the product rule to verify the
constant multiple rule cf ( x)
Let u
Let u
u v uv
c and v
cf ( x)
f x
0 and v
f
x
0 f x
cf
x
cf
x
18
6
Practice
Use the product rule to verify
xf ( x)
Let u x and v
u 1 and v
u v uv
1 f x
xf
f x
xf ( x)
f x
f x
x
f x
xf
x
19
Costs
Discretionary costs are not strictly necessary for
current production but correspond to strategic goals;
example: advertising
Fixed costs are simply not responsive to production
levels, for instance, the cost of renting an office is a
fixed cost
Variable costs grow with higher levels of
production; example: raw materials
20
Marginal Average Cost
We defined Marginal f(x) as the derivative of f(x)
We defined Average f(x) as f(x) / x
Marginal Average f(x) is the derivative of Average f(x)
So Marginal Average Cost is (C(x) / x)’
(the derivative of the average cost)
21
7
Example
C(x) = 20,000 + 10x
C(x) is the total cost (in $) of printing x dictionaries
Find the average cost per unit if 1,000 books are printed.
Find marginal average cost at a production level of 1,000
units, and interpret.
Since C(x) is the total cost, and x is the number of
units, the average cost per unit is C(x) divided by x
Average Cost per Unit
AC
C x
x
Total Cost
Number of Units
22
Example
C(x) = 20,000 + 10x
We have the notation
In this case AC x
AC x
C x
x
20, 000 10x
x
if 1000 units are produced the average cost per unit is
AC 1, 000
20, 000 10 1, 000
x
$30
23
Example
C(x) = 20,000 + 10x
Marginal average cost means the derivative of
average cost. We take the derivative of the average
cost function from the previous step, using the
quotient rule.
MAC x
x 10
20, 000 10 x 1
x2
20, 000
x2
Now we evaluate marginal average cost when x = 1,000
MAC 1, 000
20, 000
1, 0002
.02 or -2¢ per dictionary
24
8
Example
C(x) = 20,000 + 10x
This says that at a production level of 1,000 books, the
average cost per book is decreasing at a rate of 2 cents
per book.
In other words, producing the 1001st book will cause
the average cost per book to decrease by about 2
cents.
This means that at a production level of 1001 books,
the average cost per book will be roughly $29.98
25
Example
Suppose you inherit an apartment complex with 100
units each renting for $500 each, so your income is
(100)(500) = $50,000.
You want to maximize your rental income and raise the
rent $50. Five renters move out, making your rental
income (95)(550) = $52,250
If every time you raise the rent $50 an additional five
renters leave, what is your maximum potential rental
income?
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We can build a table
500
Number
Renters
100
$50,000
550
95
$52,250
2
600
90
$54,000
3
650
85
$55,250
4
700
80
$56,000
5
750
75
$56,250
6
800
70
$56,000
7
850
65
$55,250
Cycle
Rent
0
1
Income
27
9
Example
Find where the slope of the following
revenue function is zero
R x
500 50 x 100 5x
Let u 500 50x
then u
R x
50
u v uv
and v 100 5x
and v
50 100 5 x
5
5 500 5x
2500 250x 5000 250x
2500 500x
28
Example
Find where the slope of the following
revenue function is zero
R x
R x
500 50 x 100 5x
2500 500 x
0 2500 500x
2500 500x
2500
500
x 5
x
Our max income is $56,250
with 75 renters and 25 empty units
The max occurs where the slope is zero
29
Example
The total cost in hundreds of dollars to
produce x units is C x
3x 2
x 4
Find the average cost for
(a) 10 units
(b) 20 units
(c) x units
(d) Find the marginal cost function
30
10
Example
The total cost in hundreds of dollars to
3x 2
x 4
produce x units is C x
AC x
C x
x
3x 2
x2 4 x
Find the average cost for
(a) 10 units
3x 2
AC 10
x2 4 x
3 10
10
2
2
32
140
4 10
0.2286 dollars per hundred units
or $22.86 dollars per unit
31
Example
The total cost in hundreds of dollars to
3x 2
x 4
produce x units is C x
Find the average cost for
(b) 20 units
3x 2
x2 4 x
AC 20
3 20
20
2
2
4 20
62
480
0.1292 dollars per hundred units
or $12.92 dollars per unit
32
Example
The total cost in hundreds of dollars to
produce x units is C x
3x 2
x 4
Find the average cost for
(c) x units
AC x
3x 2
dollars per hundred units
x2 4x
33
11
Example
The total cost in hundreds of dollars to
3x 2
x 4
produce x units is C x
(d) Find the marginal cost function
3x 2
and C x
x 4
We have C x
3x 2
x2 4 x
To find the Marginal Average Cost we must
3x 2
x2 4x
take the derivative of AC x
34
Example
The total cost in hundreds of dollars to
3x 2
x 4
produce x units is C x
(d) Find the marginal cost function
AC x
3x 2
Let u
x 2 4 x then u
vu
MAC x
x2 4 x 3
uv
v
3x 2 and v
3
and v
2
x
3x 2 2 x 4
2
4x
2
3x 2 4 x 8
3x 2 12 x 6 x 2 12 x 4 x 8
x2 4 x
x2 4x
2x 4
2
x2 4 x
2
35
Example
The total cost to produce x units of paint
is C(x)=(5x+3)(4x+2)
Find the marginal average cost function.
5x 3 4x 2
x
5x 3 4 x 2 and v
AC x
Let u
u
u
MAC
40 x 22 x
x
5 4 x 2 + 5 x 3 4 and v
40x 22
and v
1
1
5x 3 4 x 2 1
x2
40 x 2 22 x 20 x 2 22 x 6
x2
20 x 2 6
x2
36
12
Example
Find the equation of the tangent line to
y = (2x3+3x)(x-4) at x =2
Let u 2 x3 +3x and v
u 6 x2 +3 and v
6x2 3 x 4
y
m
y 2
6 2
27
y2
2 2
3
2
3
22
3 2
2
y
y1
2 x3 3x 1
2
2
x 4
1
4
2 2
3
3 2
32
22
4
2
44
m x x1
37
Find the equation of the tangent line to
y = (2x3+3x)(x-4) at x =2
Example
y
x1
y1
2
y
m x x1
y1
44
44
m
32
32 x 2
y 32 x 20
38
TI Solution
39
13
Practice
g( 2) = -1
f( 2) = -1,
g’( 2) = 5
f’( 2) = 3
What is the value of h’( 2) where h(x) = f(x) g(x)?
h
fg
fg
3
1
1 5
8
40
Practice
g( 2) = -1
f( 2) = -1,
g’( 2) = 5
f’( 2) = 3
What is the value of h’( 2) where h(x) = f(x)/g(x)?
h
fg
fg
g2
3
1
1 5
1
2
3 5
1
2
41
Remarks
Profit = Revenue - Cost
Profit increases if
• Revenue increases
• Cost decreases
Marginal Profit = Marginal Revenue – Marginal Cost
Profit is at a maximum when Marginal Profit = 0
That is Profit is a maximum when the ARC is zero
This happens when Marginal Revenue = Marginal Cost
42
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