This chapter discusses additional topics in differentiation including: - Derivatives of logarithmic and exponential functions, with examples of differentiating functions involving ln(x) and e^x. - Elasticity of demand and how to calculate the elasticity from a demand function. - Logarithmic differentiation, which can be used to differentiate functions involving products or quotients. - Higher-order derivatives, with examples showing how to take second and third derivatives using differentiation rules. The chapter aims to expand students' skills in differentiation and apply it to concepts like elasticity used in economics.

1. INTRODUCTORY MATHEMATICAL ANALYSIS
For Business, Economics, and the Life and Social Sciences
2011 Pearson Education, Inc.
Chapter 12
Additional Differentiation Topics

2. 2011 Pearson Education, Inc.
• To develop a differentiation formula for y = ln u.
• To develop a differentiation formula for y = eu.
• To give a mathematical analysis of the economic
concept of elasticity.
• To show how to differentiate a function of the form uv.
• To approximate real roots of an equation by using
calculus.
• To find higher-order derivatives both directly and
implicitly.
Chapter 12: Additional Differentiation Topics
Chapter Objectives

3. 2011 Pearson Education, Inc.
Derivatives of Logarithmic Functions
Derivatives of Exponential Functions
Elasticity of Demand
Logarithmic Differentiation
Higher-Order Derivatives
12.1)
12.2)
12.3)
Chapter 12: Additional Differentiation Topics
Chapter Outline
12.4)
12.5)

4. 2011 Pearson Education, Inc.
Chapter 12: Additional Differentiation Topics
12.1 Derivatives of Logarithmic Functions
• The derivatives of log functions are:
  0
where
1
ln
b. 
 x
x
x
dx
d

5. 2011 Pearson Education, Inc.
Chapter 12: Additional Differentiation Topics
12.1 Derivatives of Logarithmic Functions
Example 1 – Differentiating Functions Involving ln x
b. Differentiate .
Solution:
2
ln
x
x
y 
     
 
 
0
for
ln
2
1
2
)
(ln
1
ln
ln
'
3
4
2
2
2
2
2













x
x
x
x
x
x
x
x
x
x
dx
d
x
x
dx
d
x
y
a. Differentiate f(x) = 5 ln x.
Solution:     0
for
5
ln
5
' 

 x
x
x
dx
d
x
f

6. 2011 Pearson Education, Inc.
Chapter 12: Additional Differentiation Topics
12.1 Derivatives of Logarithmic Functions
Example 3 – Rewriting Logarithmic Functions before Differentiating
a. Find dy/dx if .
Solution:
b. Find f’(p) if .
Solution:
 3
5
2
ln 
 x
y
  2
/
5
for
5
2
6
2
5
2
1
3 










 x
x
x
dx
dy
       
3
4
2
3
1
2
1
3
1
4
1
2
1
3
1
1
1
2
'




































p
p
p
p
p
p
p
f
       
 
4
3
2
3
2
1
ln 


 p
p
p
p
f

7. 2011 Pearson Education, Inc.
Chapter 12: Additional Differentiation Topics
12.1 Derivatives of Logarithmic Functions
Example 5 – Differentiating a Logarithmic Function to the Base 2
Differentiate y = log2x.
Solution:
Procedure to Differentiate logbu
• Convert logbu to and then differentiate.
b
u
ln
ln
 
 x
x
dx
d
x
dx
dy
2
ln
1
2
ln
ln
log2 








8. 2011 Pearson Education, Inc.
Chapter 12: Additional Differentiation Topics
12.2 Derivatives of Exponential Functions
• The derivatives of exponential functions are:
  dx
du
e
e
dx
d u
u

a.
  x
x
e
e
dx
d

b.
   
dx
du
b
b
b
dx
d u
u
ln
c. 
 
   
   
  0
'
for
'
1
d. 1
1
1

 


x
f
f
x
f
f
x
f
dx
d
dy
dx
dx
dy 1
e. 

9. 2011 Pearson Education, Inc.
Chapter 12: Additional Differentiation Topics
12.2 Derivatives of Exponential Functions
Example 1 – Differentiating Functions Involving ex
a.Find .
Solution:
b. If y = , find .
Solution:
c. Find y’ when .
Solution:
x
e
x
  x
x
x
e
x
e
dx
d
x
x
dx
d
e
dx
dy 


 
 1
3
ln
2


 x
e
e
y
x
x
e
e
y 


 0
0
'
 
x
e
dx
d
3
    x
x
x
e
e
dx
d
e
dx
d
3
3
3 

dx
dy

10. 2011 Pearson Education, Inc.
Chapter 12: Additional Differentiation Topics
12.2 Derivatives of Exponential Functions
Example 5 – Differentiating Different Forms
Example 6 – Differentiating Power Functions Again
Find .
Solution:
 
x
e
x
e
dx
d
2
2


   
  
x
ex
x
e
ex
x
e
dx
d
x
e
x
e
x
e
2
2
ln
2
2
1
2
ln
2
1
2
ln
1
2














Prove d/dx(xa) = axa−1.
Solution:     1
1
ln 



 a
a
x
a
a
ax
ax
x
e
dx
d
x
dx
d

11. 2011 Pearson Education, Inc.
Chapter 12: Additional Differentiation Topics
12.3 Elasticity of Demand
• Which of the following seem more serious:
An increase of 50 cents or an increase of 50% in the
price of a hamburger.
An increase of $100 or an increase of 1% in the
price of a new car.
• Percentage changes are often more important than the
amount of change
Therefore economists often use elasticities to
examine percentage change or responsiveness
• Price Elasticity of Demand (Ep)
The responsiveness of quantity demanded of a
commodity to changes in its price

12. 2011 Pearson Education, Inc.
Chapter 12: Additional Differentiation Topics
12.3 Elasticity of Demand
Given a demand function that gives q in terms of p (so q=D(p)), the
elasticity of demand is
Note that since demand is [normally] a decreasing function of p, the
derivative is [normally] negative. That's why we have the absolute
value: so that E will always be positive.
 If E<1, we say demand is inelastic. In this case, raising prices increases revenue.
 If E>1, we say demand is elastic. In this case, raising prices decreases revenue.
 If E=1, we say demand is unitary. E=1 at critical points of the revenue function.

13. 2011 Pearson Education, Inc.
Chapter 12: Additional Differentiation Topics
12.3 Elasticity of Demand
Determine the point elasticity of the demand equation
Solution
From the definition, we have q = k/p, then
E=1, we say demand is unitary.

14. 2011 Pearson Education, Inc.
Chapter 12: Additional Differentiation Topics
12.3 Elasticity of Demand
Example 2:
A company sells q ribbon winders per year at $p per ribbon
winder. The demand function for ribbon winders is given by
p=300−0.02q. Find the elasticity of demand when the price is $70
a piece. Will an increase in price lead to an increase in revenue?

15. 2011 Pearson Education, Inc.
Chapter 12: Additional Differentiation Topics
12.7 Higher-Order Derivatives
For higher-order derivatives:

16. 2011 Pearson Education, Inc.
Chapter 12: Additional Differentiation Topics
12.7 Higher-Order Derivatives
Example 1 – Finding Higher-Order Derivatives
a. If , find all higher-order
derivatives.
Solution:
b. If f(x) = 7, find f(x).
Solution:
  2
6
12
6 2
3



 x
x
x
x
f
 
 
 
 
  0
36
'
'
'
24
36
'
'
6
24
18
'
4
2







x
f
x
f
x
x
f
x
x
x
f
 
  0
'
'
0
'


x
f
x
f

17. 2011 Pearson Education, Inc.
Chapter 12: Additional Differentiation Topics
12.7 Higher-Order Derivatives
Example 3 – Evaluating a Second-Order Derivative
Solution:
  .
4
when
find
,
4
16
If 2
2


 x
dx
y
d
x
x
f
 
  3
2
2
2
4
32
4
16







x
dx
y
d
x
dx
dy
16
1
4
2
2


x
dx
y
d