5.1 The Natural Logarithmic
Function: Differentiation
AB and BC
2015
Calculus Warm-Up 5.1
Find the derivative with respect to x:
y 3  y 2  5 y  x 2  4
Calculus Warm-Up 5.1
Find the indefinite integral:
1
dx
x
Expanding Logarithmic Expressions
10
ln
9
 ln10  ln 9
ln 3x  2
6x
ln
5
x

ln
2
 3
2
x 3 x2 1
e
1
Because  dt  1,
t
1
We can use that to define e, and
To give a new way to define the
natural log function.
x
d 1 1

Based on the relationship above, we can see that

dt 1 t x
Therefore :
d
1
ln x 
dt
x
Proof that d ln x  1
dt
x
ln  x  x   ln x
d
ln x  lim
x 0
dt
x
 lim
x 0
ln
x  x
x
x
 1  x  
 lim  ln 1 

x 0
x 
 x 
 x 
 lim ln 1  
x 0
x 

1
x
x
let u 
x
1
1

, xu  x ,
x xu
 lim ln 1  u 
1
xu
u 0
1


u
 lim ln  1  u  
u 0


1
x
1
1 

 lim ln  1  u  u 
u 0 x


let u 
1
1 

 ln  lim 1  u  u 
x  u 0

u0
1
1
 ln e 
x
x
1
n
as n  
 1
1  
 n
n
Example 1 – Differentiation of Logarithmic Functions
Example 1 – Differentiation of Logarithmic Functions
You Try:
d 

e.
ln
x

1

dx 
1
2x  2
Example 2 – Differentiation of Logarithmic Functions
 x( x 2  1) 2 
Differentiate f ( x)  ln 

3
 2x  1 
 ln x( x  1)  ln  2 x  1
1
2
3
 ln x  2 ln( x  1)  ln  2 x  1
2
2

 Be sure you see the benefit
1
2
x
1
6
x


f '( x)   2  2    3 
x
 x  1  2  2 x  1  of applying logarithmic properties
before you differentiate.
2
2
1
4x
3x
  2
 3
x x  1 2x  1
2
3
1/ 2
Example 3 – You Try
Find the derivative of the function:
x 1
f  x   ln
x 1
3
f  x 
2
3  x 2  1
The natural log function is undefined for negative numbers
If u > 0,
d
u
u  u , and
ln u  .
dx
u
d
d
u u 
ln u  ln  u  
 .
If u < 0, u  u , and
dx
dx
u u
Example 4 – Derivative Involving Absolute Value
You Try: Find the derivative of
f(x) = ln|cosx|.
Solution:
Example 5 – Derivative Involving Absolute Value
You Try: Find the derivative of
f(x) = ln|secx+tanx|.
y  sec x
Finding Relative Extrema
• Locate the Relative Extrema of:
y  ln  x 2  2x  3
Day 2 Warm-up
Find the derivative :
ln


x x 1
2
2
2x 1
3
Proof that d ln x  1
dt
x
ln  x  x   ln x
d
ln x  lim
x 0
dt
x
 lim
x 0
ln
x  x
x
x
 1  x  
 lim  ln 1 

x 0
x 
 x 
 x 
 lim ln 1  
x 0
x 

1
x
x
let u 
x
1
1

, xu  x ,
x xu
 lim ln 1  u 
1
xu
u 0
1


u
 lim ln  1  u  
u 0


1
x
1
1 

 lim ln  1  u  u 
u 0 x


let u 
1
1 

 ln  lim 1  u  u 
x  u 0

u0
1
1
 ln e 
x
x
1
n
as n  
 1
1  
 n
n
Day 2 Warm-up #2
Find the derivative :
Logarithmic Differentiation (Optional)
Sometimes, it is convenient to use logarithms as
aids in differentiating nonlogarithmic functions.
This procedure is called logarithmic
differentiation.
Example 6 – Logarithmic Differentiation
Find the derivative of
Solution:
Note that y > 0 for all x ≠ 2. So, ln y is defined.
Begin by taking the natural logarithm of each side of the equation.
Then apply logarithmic properties and differentiate implicitly.
Finally, solve for y'.
Example 6 – Solution
cont’d
5.1 BC Homework
• Day 1: Pg. 330 45-59 odd, 63, 71, 77, 79, 83,
85, 93-97 odd
• Day 2: MMM pgs.191-192
Derivative of ln x
y  ln  x 
The natural log function as follows:
Rewrite in exponential form:
Now differentiate implicitly
with respect to x:
Derivative of ln x:
ey  x
e y y  1
d
1
ln x 
dx
x
1 1
y  y 
e
x
 x  0