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