4.4 - MATH
1081
 Monday,
February
21
 


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Unformatted text preview: MATH
1081
 Monday,
February
21
 
 Chapter
4
–
Sections
4
&
5
 
 DERIVATIVES
OF
EXPONENTIAL
AND
 LOGARITHMIC
FUNCTIONS
 
 Homework
#6
(due
2/28):
 Section
4.4
#32,
38,
58
 
 
 
 
 
 
 
 
 Section
4.5
#42,
58
 
 
 
 
 
 
 
 
 Section
5.1
#26,
48,
52
 
 Clicker
Check­in:

Choose
any
letter
to
check
in
now.
 Today
we
will
look
at
our
last
two
rules
of
differentiation:
the
 derivative
of
an
exponential
function
and
the
derivative
of
a
 logarithmic
function.
 
 An
exponential
function
is
a
function
of
the
form
 f ( x ) = b x 
 where
 b > 0 
and
 b ≠ 1.
 
 A
logarithmic
function
is
a
function
of
the
form
 f ( x ) = log b ( x )
 where
 b > 0 
and
 b ≠ 1.
 
 The
“natural”
base
that
is
the
number
 e ≈ 2.71828 .


 In
the
case
of
the
logarithmic
function,
 log e ( x ) = ln ( x ) .
 
 Let’s
start
with
 f ( x ) = e x 
and
the
definition
of
derivative.
 
 dx ex+h − ex ⎡ e ⎤ = lim dx ⎣ ⎦ h→ 0 h e x eh − e x 
 = lim h→ 0 h e x eh − 1 = lim h→ 0 h 
 It
turns
out
that
(although
the
proof
won’t
be
done
here)
the
 eh − 1 limit
 lim = 1.


 h→ 0 h 
 dx So,
 ⎡ e ⎤ = e x .
 dx ⎣ ⎦ ( ) The
function
 f ( x ) = e x 
is
its
own
derivative.


 
 It
is
unchanged
by
the
act
of
differentiation.
 
 The
slope
of
the
exponential
curve
at
any
 x 
is
actually
equal
to
 the
value
of
the
curve
at
that
point.
 
 
 Rules
of
Differentiation
 
 8.

Derivative
of
an
Exponential
Function
 
 If
 f ( x ) = b x ,
then
 f ' ( x ) = b x ⋅ ln ( b ) .


 
 So,
if
 f ( x ) = e x ,
then
 f ' ( x ) = e x .
 
 
 So,
in
general,
the
derivative
of
an
exponential
function
is
just
 that
exponential
function
times
a
special
constant.

In
the
case
 of
base
 e ,
that
special
constant
is
1.
 
 E x a m p l e :
 
 F in d 
 t h e 
 d e r iv a t iv e 
 o f
 t h e 
 fu n c t io n .
 
 e x − xe x 
 f ( x) = 3 1 .

 
 
 g( x ) = 5 x + 4 x −1
 2 .

 
 
 x h( x ) = 5 x 
 e 3 .

 
 
 P( x ) = 9 2 x − 5e7 x 
 4 .

 2 3 R u l e s 
 o f
 D iffe re n t ia t io n 
 
 9 .
 
 D e riv a t iv e 
 o f
 a 
 L o g a rit h m ic 
 F u n c t io n 
 
 
 
 
 
 1 If
 f ( x ) = log b ( x ),
then
 f ' ( x ) = .
 ln ( b ) x 1 So,
if
 f ( x ) = ln ( x ),
then
 f ' ( x ) = .
 x 1 S o ,
 in 
 g e n e ra l ,
 t h e 
 d e riv a t iv e 
 o f
 a 
 l o g a rit h m ic 
 fu n c t io n 
 is 
 
 x t im e s 
 a 
 s p e c ia l 
 c o n s t a n t .
 
 I n 
 t h e 
 c a s e 
 o f
 t h e 
 n a t u ra l 
 l o g ,
 ln ( x ) 
that
special
constant
is
1. E x a m p l e :
 
 F in d 
 t h e 
 d e r iv a t iv e 
 o f
 t h e 
 fu n c t io n .
 
 
 
 
 
 1.

 f ( x ) = ln( x 2 + 3x + 5)
 
 
 g( x ) = e x log 7 ( 3x ) 
 2 .

 
 
 3 .

 
 
 h( x ) = 3 7 
 x ln( x ) ⎛ x + 3⎞ R( x ) = log ⎜ ⎟
 ⎝ x − 5⎠ 
 4 .

 
 
 
 1 . G e n e ra l iz e d 
P o w e r
R u l e 
 d 

 
 f ( x ))n = n ⋅ ( f ( x ))n−1 ⋅ f ' ( x )
 ( dx [ 2 .
 
 T h e 
 G e n e ra l iz e d 
 E x p o n e n t ia l 
 F u n c t io n 
 R u l e 
 ( b a s e 
 e ) 
 d f (x) 

 
 
 e = e f ( x ) ⋅ f ' ( x )
 dx 
 
 3 .
 
 T h e 
 G e n e ra l iz e d 
 L o g a rit h m ic 
 F u n c t io n 
R u l e 
 ( b a s e 
 e ) 
 d 1 

 
 
 ⋅ f ' ( x)
 [ln( f ( x ))] = dx f ( x) [ Clicker
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any
letter
to
check
out
now.
 
 Next
time
–
Section
5.1
 
 ...
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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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