Chapter12 (More Differentiation).ppt

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

• 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

Derivatives of Logarithmic Functions
Derivatives of Exponential Functions
Elasticity of Demand
Logarithmic Differentiation
Higher-Order Derivatives
12.1)
12.2)
12.3)
Chapter Outline
12.4)
12.5)

12.1 Derivatives of Logarithmic Functions
• The derivatives of log functions are:
  0
where
1
ln
b. 
 x
x
x
dx
d

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

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

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 








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. 

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

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

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

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.

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.

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?

12.7 Higher-Order Derivatives
For 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

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

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