AP CALCULUS - AB
Section Number:
LECTURE NOTES
Topics: The Natural Logarithm Function: Differentiation
MR. RECORD
Day: 1 of 1
5.1
x n 1
C has a very important disclaimer. It does not apply when
n 1
1
n 1 . Consequently we yet do not know how to take the integral of . The answer turns out to be
x
logarithmic.
Recall: The General Power Rule
n
x dx
Definition of the Natural Logarithmic Function
The natural logarithmic function is defined by
x
1
ln x dt , x 0.
t
1
Note: The domain of the natural logarithmic function is the set of all positive real numbers
y
y
1
t
y
y
x
1
1 t dt 0 when x 1
1
t
x
1
t dt 0 when x 1
1
x
x
x
x
The graph of lnx KNOW THIS!
y ln x
y
x
Properties of the Natural Logarithmic Function
The natural logarithmic function has the following properties.
1. The domain is (0, ) and the range is ( , ) .
2. The function is continuous, increasing, and one-to-one.
3. The graph is concave downward.
From all the above, it then follows that
d[ln x ] 1
d[lnu] 1 du
u
. The Chain Rule version is
or
dx
x
dx
u dx
u
Logarithmic Properties
If a and b are positive numbers and n is rational, then the following properties are true.
1. ln(1) 0
2. ln(ab) ln a ln b
3. ln(an ) nlna
a
4. ln ln a ln b
b
Example 1: Rewrite each using the logarithm rules.
10
a. ln
9
c. ln
6x
5
Example 2: Find the derivative of each.
a. y ln(2 x)
c. y x ln x
b. ln 3x 2
d. ln
(x 2 3)2
x 3 x2 1
b. y ln(x 2 1)
d. y (ln x)3
Example 3:
x(x 2 1)2
Find the derivative of y ln
.
3
2x 1
Logarithmic Differentiation
Example 4:
Find the derivative of y
(x 2)2
x2 1
.
Example 5: Find an equation of the tangent line to the graph of f ( x)
Example 6: Find
1
x ln x 2 at the point 1, 0 .
2
dy
for 4 xy ln x 2 y 7 .
dx
Example 7: Locate any relative extrema and inflection points for y x 2 ln
x
4
0
