Chapter 5 Derivatives of
Transcendental Functions
5-1 The Natural Logarithmic Function
5-4 Exponential Functions
5-5 Bases other than e and Applications
5-3 Inverse Functions and their Derivatives
5-8 Inverse Trig Functions
7-7 Indeterminate Forms and L’Hopital’s Rule
Sect. 5-5 (Derivatives)
Bases Other than e and Applications
Goals:
• Differentiate exponential & logarithmic functions that
have bases other than e.
Review
Definition: The logarithmic function with base a, where a > 0 and a 1 is
LOGS = EXPONENTS
y = logax if and only if x = a y
logarithmic form
exponential form
Convert to exponential form:
1
log 2 3
8
Convert to log form:
16 4
2
2
3
1
8
log 416 2
YOU TRY…
Write the equivalent logarithmic equation.
Exponential Equation
3 27
3
Logarithmic Equation
log3 27 3
log5 1 0form.
2. Write 5 1 in logarithmic
0
1
2
. Finally, write 16 4
in logarithmic form.
1
log16 4
2
YOU TRY…
Write the equivalent exponential equation and
solve for y.
Logarithmic Equation
Exponential
Equation
Solution
y = log2 ¼
¼ = 2y
y = -2
y = log3 3
3 = 3y
y=1
y = log416
16 = 4y
y=2
y = log1
1 = 10y
y=0
y = log255
5 = 25y
y=½
y = log1/3 9
9 = (1/3)y
y = -2
Review
logb1
= 0 (because b0 = 1)
logbb = 1 (because b1 = b)
logbbr = r (because br = br)
blog b M = M (because logbM = logbM)
logbx = logax (a can be any base > 0)
logab
log 4 4 = 1
log 8 1 = 0
3 log 3 6 = 6
log 5 5 3 = 3
2 log 2 7 = 7
log712 = log 12 = ln 12
log 7
ln 7
= logπ12 = log10012
logπ7
log1007
Use the properties of logarithms to simplify:
1. e
ln 5
3. log10
2. ln1
7.1
4. log1000
5. 2log4 1
6. 10log e
7. log 2 64
8. 5log5 3
1
9. ln 2
e
11. log 6 6
10. 3ln e
12. log 3 7
Review
Ex: Graph f (x) = log
2x 2 x
x x
2log
2
x
–2 1
–2 1
4
4
–1 1
–1 1
2
2
1
0
10
2
1
21
4
2
42
8
3
83
y
y-intercept
horizontal
asymptote y = 0
y = 2x
y=x
y = log2 x
x-intercept
x
vertical asymptote
x=0
Review
Characteristics about the graph of an
exponential function f x a x a > 1
Characteristics about the graph of a
log function f x log a x where a > 1
1. Domain: all real numbers
1. Range: all real numbers
2. Range: y > 0
2. Domain: x > 0
3. No x intercepts
3. No y intercepts
4. y-intercept: (0,1)
4. x-intercept: (1,0)
5. The graph is always increasing
5. The graph is always increasing
6. HA: y = 0
6. VA: x = 0
Review
logam
logbm = -------logab
log712 =
log 12
log 7
OR
log712 =
ln 12
ln 7
Derivatives of au
Challenge: Since f(x) = ax, find f’(x).
d x
a
dx
d ln a x
e
dx
d x ln a
e
dx
d
x ln a
e x ln a
dx
1.
d [ax ] = (ln a)ax
dx
2. d [au ] = (ln a)au du
dx
dx
dy
Find
:
dx
a. y 2
x
y ' ln 2 2x
x3 cos x
y ' ln 4 4
b. y 4
x3 cos x
2
3
x
sin x
c. y 33 x 6 tan x
But WAIT!! This rule will ONLY work if the base is a
constant and the exponent contains variables.
Differentiate.
d e
e 0
dx
d x
x
x
e e ln e e
dx
d e
e 1
Power Rule
x
ex
dx
d x
( x ) x x 1 ln x
dx
Logarithmic Differentiation
You Try…
Find the derivative of each function.
1. y 4
x 2 1
2. g 5
3
2t 3
9
3. f t
6t
4. y x ln x
5
2
sec
4
Challenge: If f(x) = logax, find f’(x).
Hint: Use the change of base formula.
ln x
Since log a x
, we can find the following result.
ln a
3. d [loga x ] =
1
dx
(ln a)x
4. d [loga u ] =
dx
1
du
(ln a)u dx
Differentiate :
log 3 x 6 x 1
1. y
2
2
2. g ( x) log 5 x sin x
2
a 4a 6
3. h(a ) log 2
5
ln x 2 sin x
d
d
2
(log5 ( x sin x)
dx
dx
ln 5
1
1
2 x cos x
* 2
(2 x cos x)
ln 5 ( x sin x)
ln 5( x 2 sin x)
1. y log cos x
2.
y'
1
sin x
sin
x
ln10 cos x
ln 10 cos x
tan x
ln 10
x2
y log2
2log2 x log2 x 1
x 1
dy
1 1
1
1
2
1
1
dx
ln2 x
ln2 x 1
2
1
1
x ln 2 ln 2 x 1
1 2
1
ln2 x x 1
1 x 2
ln 2 x x 1
d u
u du
e e
dx
dx
d
du
u
u
a a ln a
dx
dx
d
1 du
ln u
dx
u dx
d
1 du
log a u
dx
u ln a dx
Explain the difference in differentiating each
function in the following forms:
x2
vs.
2x
ln x
vs.
log 3 x
ex
vs.
e2
0
