Chapter 1: Differentiation
1.6 The Product and Quotient Rules
OBJECTIVE 1: Differentiate using the Product and the Quotient Rules.
THEOREM: The Product Rule
Let J aBb œ 0 aBb † 1aBb. Then,
J w aBb œ
.
c0 aBb † 1aBbd
.B
œ 1aBb †
.
.
c0 aBbd € 0 aBb †
c1aBbd
.B
.B
.
•ˆB# € #B‰a$B € &b‘ using
.B
the Product Rule (Theorem 1).
1. Find
2. Multiply ˆB# € #B‰a$B € &b, then find the
derivative of the resulting product.
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Mr. Getso’s Business Calculus Class Notes
3.
Find
.
•ˆ#B& € B • "‰a$B • #b‘.
.B
THEOREM:
If UaBb œ
5.
.
&
•ˆÈB € "‰ˆÈ
B • B‰‘.
.B
4.
Find
6.
Show that
The Quotient Rule
R a Bb
, then
HaBb
U w a Bb œ
HaBb † R w aBb • R aBb † Hw aBb
cHaBbd#
Show that
. B# • $B
B# • #B € $
œ
.
”
•
.B B • "
a B • "b #
. +B € "
+•,
.
”
•œ
.B ,B € "
a,B € "b#
Chapter 1: Differentiation
27
OBJECTIVE 2: Use the Quotient Rule to differentiate the average cost, revenue, and profit functions.
If G aBb is the cost of producing B items, then the average cost of producing B items is
G a Bb
.
B
If V aBb is the revenue from the sale of B items, then the average revenue from selling B items is
If T aBb is the profit from the sale of B items, then the average profit from selling B items is
V aBb
.
B
T aBb
.
B
Note: Profit is the difference of revenue and cost. That is,
T aBb œ V aBb • G aBb.
7. Paulsen’s Greenhouse finds that the cost,
in dollars, of growing B hundred geraniums is
given by
%
G aBb œ #!! € "!!È
B.
If the revenue from the sale of B hundred
geraniums is given by
V aBb œ "#! € *!ÈB,
find each of the following:
a) The average cost, the average revenue, and
the average profit when B hundred geraniums are
grown and sold.
b) The rate at which average profit is changing
when $!! geraniums are being grown and sold.
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Mr. Getso’s Business Calculus Class Notes
8. The reaction V of the body to a dose U
of a medication is often represented by the
general function
5 U
V aUb œ U# Œ • •,
#
$
where 5 is a constant and V is in millimeters
of mercury (mmHg) if the reaction is a change
in blood pressure or in degrees Fahrenheit (°F)
if the reaction is a change in temperature. The
rate of change .VÎ.U is defined to be the
body’s sensitivity to the medication.
Find a formula for the sensitivity.
Chapter 1: Differentiation
1.7 The Chain Rule
OBJECTIVE 1: Differentiate using the Extended Power Rule.
THEOREM: The Extended Power Rule
Suppose that 1aBb is a differentiable function of B.
Then, for any real number 5 ,
.
.
c1aBbd5 œ 5 c1aBbd5•" †
1aBb.
.B
.B
1. Differentiate 0 aBb œ ˆ" € B$ ‰ # .
2.
Differentiate 0 aBb œ a$B • &b% .
3. Differentiate 0 aBb œ a$B • &b% a( • Bb"! .
4.
Differentiate 0 aBb œ
"
#B# • "
a$B% € #b#
.
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Mr. Getso’s Business Calculus Class Notes
OBJECTIVE 2: Find the composition of two functions.
DEFINITION: Composition of Functions
The composed function 0 ‰ 1, the composition
of 0 and 1, is defined as
0 ‰ 1 œ 0 a1aBbb.
5. For 0 aBb œ B$ and 1aBb œ " € B# , find
a0 ‰ 1baBb.
6. For 0 aBb œ B$ and 1aBb œ " € B# , find
a1 ‰ 0 baBb.
7. For 0 aBb œ ÈB and 1aBb œ B • ", find
a0 ‰ 1baBb.
8. For 0 aBb œ ÈB and 1aBb œ B • ", find
a1 ‰ 0 baBb.
Chapter 1: Differentiation
9. For 0 aBb œ ÈB , find a0 ‰ 0 baBb.
10.
For 1aBb œ B • ", find a1 ‰ 1baBb.
12.
For C œ ?# € ? and ? œ B# € B, find
OBJECTIVE 3: Differentiate using the Chain Rule.
THEOREM: The Chain Rule
The derivative the composition 0 ‰ 1 is given by
.
.
ca0 ‰ 1baBbd œ
c0 a1aBbbd œ 0 w a1aBbb † 1w aBb.
.B
.B
11. For C œ # € È? and ? œ B$ € ", find
.C .?
.C
,
, and
Þ
.? .B
.B
.C
Þ
.B
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Mr. Getso’s Business Calculus Class Notes
13. For C œ ?# • $? and ? œ &> • ", find
.C
Þ
.>
14.
A total cost function is given by
G aBb œ #!!!ˆB# € #‰ € (!!,
where G aBb is the total cost, in thousands of
dollars, of producing B items. Find the rate at
which total cost is changing when #! items
have been produced.
"Î$