SECTION 5-5A
Part I: Exponentials base other than e
Variables in the exponent
Logarithmic differentiation is used to find
derivatives of exponential functions:
1) Which if these is an exponential function?
a)
c)
y2
x
yx
x
b)
y  10
d)
y  sin x 
sin x
x2
Bases other than e
Definition of exponential function with
base a
Let a be a positive real number where a ≠1
and let x be any real number, then the
exponential function to the base a is
defined by:
a e
x
ln a  x
Properties
a)
a 1
b)
a a a
0
x
y
x y
x
c)
d)
a
x y
a
y
a
a 
x y
a
xy
Definition of logarithmic function
with base a
Let a be a positive real number where a ≠1
and let x be any real number, then the
logarithmic function to the base a is defined
by:
log x
log a x 
log a
Change of base formula
Properties
a)
log a 1  0
b)
log a xy  log a x  log a y
c)
log a x  n log a x
d)
x
log a  log a x  log a y
y
n
Differentiate
2)
y6
2x
Derivatives for bases other than e
Let a be a positive real number where a ≠1
and let u be a differentiable function of x
such that:
 
x
I.
da
 a x ln a
dx
II.
 
d au
du
u
 a ln( a)
dx
dx
Derivatives for bases other than e
Let a be a positive real number where a ≠1
and let u be a differentiable function of x
such that:
Proof pg 364
III.
IV.
d
1
log b x 
dx
x ln( b)
d
u'
log b u  
dx
u ln( b)
Differentiate
3)
y  10
sin x
Differentiate
4)
5)
y  x 2 (7 3 x )
 
y 2
x 2
PRODUCT RULE
Differentiate
6)
y  log 10 (3x  1)
Differentiate
7)
yx
x
Careful: not the power rule!
ln y  x ln x
1 dy
1
 1 ln x  x 
y dx
x
dy
 x x ln x  1
dx
Differentiate
8)

y  log 7 x( x 3  4)5


y  log 7 ( x)  5 log 7 x 3  4

Differentiate
9)
 x 
y  log 8  2 
 x 1
y  log 8 ( x)  log 8 ( x 2  1)
1
2x
y 

ln( 8)x ln( 8)(x 2  1)
Through the magic of math:

 x2 1
y  
ln( 8) x( x 2  1)
Assignment
Page 368 # 41-48, 51- 60, 6365, and 67 all