4.2 - MATH
1081
 Monday,
February
14
 


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Unformatted text preview: MATH
1081
 Monday,
February
14
 
 Chapter
4
–
Section
2
 
 DERIVATIVES
OF
PRODUCTS
AND
 QUOTIENTS
 
 Homework
#5
(due
2/21):
 Section
4.2
#8,
22,
32,
46,
52
 
 
 
 
 
 
 
 
 Section
4.3
#34,
38,
56,
62
 
 Clicker
Check­in:

Choose
any
letter
to
check
in
now.
 Last
time,
we
saw
the
“Sum
and
Difference”
rule
of
 differentiation.

Because
it
is
so
easy
to
apply,
we
hardly
notice
 when
we
do.
 
 d3 d3 d2 d d 2 ⎡x + x − x − 9⎤ = ⎡ x ⎤ + ⎡ x ⎤ − [ x] − [9] ⎣ ⎦ dx dx ⎣ ⎦ dx ⎣ ⎦ dx dx 
 = 3x 2 + 2x −1 −0 
 However,
there
really
is
a
rule
there
and
it’s
quite
special
that
 it
works
so
nicely
when
we
are
adding/subtracting
terms.
 
 As
we
will
see
today,
when
we
are
multiplying
and
dividing
 terms,
the
rules
are
more
complicated.
 
 Before
we
get
into
those
new
rules,
let’s
do
some
problems
that
 just
require
the
four
rules
we
saw
last
time.
 
 Also,
try
to
keep
in
mind
what
we
are
measuring
with
the
 derivative:
the
rate
of
change
or
the
slope.

When
we
find
 f ' ( x ) 
 whether
it
be
by
the
limit
definition
or
by
the
rules
of
 differentiation,
we
are
finding
a
“slope
function”
that
will
tell
 us
the
slope
of
the
function
 f ( x ) 
at
any
point
in
the
domain
of
 f .
 
 In
some
applications,
there
may
be
even
other
names
to
signify
 this
idea,
like
“velocity”
or
“marginal”.
 Example:

Find
the
equation
of
the
tangent
line
to
the
graph
of

 9 3 
the
function
 g ( x ) = 4 x − 
at
the
point
 x = 27 .
 x 
 Example:

After
severe
conservation
measures
are
 implemented
to
save
a
species
of
turtles
from
 extinction,
it
is
anticipated
that
the
population
at
 the
end
of
the
 t th 
year
will
be
given
by
 
 
 
 N ( t ) = 2t 3 + 3t 2 − 4 t + 1000 
 
 ( 0 ≤ t ≤ 10 ) .
 
 1. What
is
the
initial
population
of
turtles?
 
 
 
 2. Find
 N ' ( 2 )
and
interpret
the
result.
 
 Example:

The
price
(in
dollars)
of
a
stereo
system
is
given
by

 
 1000 
 
 
 
 
 
 p ( q ) = 2 + 1000 
 q 
 
 
 
 
where
 q 
represents
the
demand
for
the
product.
 
 1. According
to
this
model,
what
is
the
price
when
 the
demand
is
10
stereos?
 
 
 
 2. Find
the
marginal
revenue
when
the
demand
is
 10
stereos
and
interpret
that
result.
 To
move
onto
our
two
new
rules,
consider
the
function
 f ( x ) = x 7 .

By
the
Power
Rule,
we
know
 f ' ( x ) = 7 x 6 .
 
 On
the
other
hand,
we
could
imagine
this
function
as
a
product
 of
two
other
functions,
like
 g ( x ) = x 3 
and
 h ( x ) = x 4 .
 
 Then
 f ( x ) = g ( x ) ⋅ h ( x ) .
 
 However,
 g ' ( x ) = 3x 2 
and
 h ' ( x ) = 4 x 3 .
 

 So,
 g ' ( x ) ⋅ h ' ( x ) = 3x 2 ⋅ 4 x 3 = 12 x 5 
and
clearly
this
is
NOT
the
 same
as
 f ' ( x ) . Rules
of
Differentiation
 
 
 5.

Product
Rule
 
 
 
 If
 f ( x ) = u ( x ) ⋅ v ( x ) ,
and
both
 u ' ( x ) 
and
 v ' ( x ) 
exist,
then
 

 f ' ( x ) = u ( x ) ⋅ v ' ( x ) + u ' ( x ) ⋅ v ( x ) .
 
 
 So,
the
derivative
of
a
product
is
NOT
equal
to
the
product
of
 the
derivatives.

Instead,
each
factor
must
“take
its
turn”
at
 being
differentiated
while
the
others
“stay
still”.
 
 It
doesn’t
matter
what
order
you
take
each
derivative,
since
we
 add
the
results.
 
 Rules
of
Differentiation
 
 
 6.

Quotient
Rule
 
 u ( x) 
 
 If
 f ( x ) = 
(with
 v ( x ) ≠ 0 ,
of
course)
and
both
 u ' ( x )

 v( x) v ( x ) ⋅ u '( x ) − u ( x ) ⋅ v '( x ) and
 v ' ( x ) 
exist,
then
 f ' ( x ) = .
 2 ⎡ v ( x )⎤ ⎣ ⎦ 
 Notice
that
just
as
with
the
Product
Rule,
the
numerator
and
 denominator
must
“take
turns”
in
being
differentiated.
 
 Because
there
is
subtraction
in
the
numerator,
the
order
that
 you
do
the
derivatives
does
make
a
difference.
 Example:

Find
the
derivative
of
the
function.
 
 
 
 
 1.

 f ( x ) = 3x 2 − 1 ( 4 x + 2 ) 
 
 
 
 4 − x2 
 
 
 2.

 g ( x ) = 
 5 + 7x 
 
 
 8−x 
 
 
 3.

 h ( x ) = 
 x 
 ( ) Clicker
Check­out:
Choose
any
letter
to
check
out.
 
 
 Next
time
–
Section
4.3
 ...
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This note was uploaded on 05/01/2011 for the course MATH 1081 taught by Professor Johanson during the Spring '08 term at Colorado.

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