Transcript Chapter 12 Additional Differentiation Topics

INTRODUCTORY MATHEMATICAL ANALYSIS
For Business, Economics, and the Life and Social Sciences
Chapter 12
Additional Differentiation Topics
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Chapter 12: Additional Differentiation Topics
Chapter Objectives
• 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 discuss the notion of a function defined implicitly.
• 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.
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Chapter 12: Additional Differentiation Topics
Chapter Outline
12.1) Derivatives of Logarithmic Functions
12.2) Derivatives of Exponential Functions
12.3) Elasticity of Demand
12.4) Implicit Differentiation
12.5) Logarithmic Differentiation
12.6) Newton’s Method
12.7) Higher-Order Derivatives
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Chapter 12: Additional Differentiation Topics
12.1 Derivatives of Logarithmic Functions
• The derivatives of log functions are:
x/h

d
1   h  
ln x   ln lim
a.
1  
h

0
dx
x  
x  
d
1
ln x   where x  0
b.
dx
x
d
1 du
lnu    for u  0
c.
dx
u dx
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Chapter 12: Additional Differentiation Topics
12.1 Derivatives of Logarithmic Functions
Example 1 – Differentiating Functions Involving ln x
a. Differentiate f(x) = 5 ln x.
d
5
for x  0
Solution: f ' x   5 ln x  
dx
x
ln x
y

b. Differentiate
.
x2
d 2
2 d
ln x   ln x  x 
x
Solution:
dx
dx
y'
2 2
x 
 1
x 2    (ln x )2 x 
x
   4
x
1  2 ln x

f or x  0
3
x
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Chapter 12: Additional Differentiation Topics
12.1 Derivatives of Logarithmic Functions
Example 3 – Rewriting Logarithmic Functions before Differentiating
a. Find dy/dx if
y  ln2x  5
3
.
dy
6
 1 
 3
for x  5 / 2
2 
Solution:
dx
2x  5
 2x  5 


b. Find f’(p) if f p  ln p  12 p  23 p  34 .
Solution: f ' p   2 1 1  3 1 1  4 1 1
 p  1
 p  2
2
3
4



p 1 p  2 p  3
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 p3
Chapter 12: Additional Differentiation Topics
12.1 Derivatives of Logarithmic Functions
Procedure to Differentiate logbu
lnu
• Convert logbu to
and then differentiate.
ln b
Example 5 – Differentiating a Logarithmic Function to the Base 2
Differentiate y = log2x.
dy
d  ln x 
1
log2 x     
Solution:
dx
dx  ln 2  ln 2x
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Chapter 12: Additional Differentiation Topics
12.2 Derivatives of Exponential Functions
• The derivatives of exponential functions are:
 
d u
u du
a.
e e
dx
dx
 
d x
b.
e  ex
dx
 
d u
du
u
c.
b  b ln b 
dx
dx




d 1
1
1
x   0
d.
f x  
for
f
'
f
1
dx
f ' f x 

dy 1
e.
 dx
dx dy
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
Chapter 12: Additional Differentiation Topics
12.2 Derivatives of Exponential Functions
Example 1 – Differentiating Functions Involving ex
a. Find
 .
d
3e x
dx
Solution:
 
 
d
d x
3e x  3
e  3e x
dx
dx
x
dy
b. If y = e x , find dx .
 
dy
d x 1 x
x d
e
xx
e  x
Solution:
dx
dx
dx
e
c. Find y’ when y  e2  e x  ln 3 .
Solution: y '  0  e x  0  e x
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Chapter 12: Additional Differentiation Topics
12.2 Derivatives of Exponential Functions
Example 3 – The Normal-Distribution Density Function
Determine the rate of change of y with respect to x
when x = μ + σ.
2
1
  21  x    /  
y  f x  
e
 2x
Solution: The rate of change is
dy
dx
x   
2
1
1        1  
  21  x    /   
  2

e
  

 2
   
 2 
1
 2
 2e
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Chapter 12: Additional Differentiation Topics
12.2 Derivatives of Exponential Functions
Example 5 – Differentiating Different Forms

d 2
e  xe  2
Find
dx
Solution:

x
.
d 2
e  xe  2
dx
x
  ex
e 1
 ex

e
e 1
ln 2 
x

 1 
ln 2

2 x 
2 x ln 2

2 x
Example 6 – Differentiating Power Functions Again
Prove d/dx(xa) = axa−1.
 


d a
d a ln x
a
1
a 1
x

e

x
ax

ax
Solution: dx
dx
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Chapter 12: Additional Differentiation Topics
12.3 Elasticity of Demand
• Point elasticity of demand η is
   q  
p
q
dp
dq
where p is price and q is quantity.
Example 1 – Finding Point Elasticity of Demand
Determine the point elasticity of the demand equation
k
p
w herek  0 and q  0
q
Solution: We have  
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p
q
dp
dq

k
q2
k
q2
 1
Chapter 12: Additional Differentiation Topics
12.4 Implicit Differentiation
Implicit Differentiation Procedure
1. Differentiate both sides.
2. Collect all dy/dx terms on one side and other
terms on the other side.
3. Factor dy/dx terms.
4. Solve for dy/dx.
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Chapter 12: Additional Differentiation Topics
12.4 Implicit Differentiation
Example 1 – Implicit Differentiation
Find dy/dx by implicit differentiation if y  y 3  x  7 .


d
d
3
Solution:
7
yy x 
dx
dx
dy
2 dy
 3y
1 0
dx
dx
dy
1

dx 1  3 y 2
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Chapter 12: Additional Differentiation Topics
12.4 Implicit Differentiation
Example 3 – Implicit Differentiation
Find the slope of the curve x  y  x
3
Solution:

 

d 3
d
2 2
x 
yx
dx
dx

2 dy
2  dy
3x
2yx 
 2x 
dx
 dx



dy 3 x 2  4 xy  4 x 3

dx
2 y  x2

dy
dx
1,2 
7

2
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

2 2
at (1,2).
Chapter 12: Additional Differentiation Topics
12.5 Logarithmic Differentiation
Logarithmic Differentiation Procedure
1. Take the natural logarithm of both sides which
gives ln y  lnf x  .
2. Simplify In (f(x))by using properties of logarithms.
3. Differentiate both sides with respect to x.
4. Solve for dy/dx.
5. Express the answer in terms of x only.
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Chapter 12: Additional Differentiation Topics
12.5 Logarithmic Differentiation
Example 1 – Logarithmic Differentiation
3

2x  5
Find y’ if y  2 4 2
.
x x 1
Solution:
2x  5
3
ln y  ln
x2 4 x2  1
ln y  ln2x  5  ln x 2  4 x 2  1
3
1


 3 ln 2x  5  2 ln x  
1 
2x 
2
4  x  1
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Chapter 12: Additional Differentiation Topics
12.5 Logarithmic Differentiation
Example 1 – Logarithmic Differentiation
Solution (continued):
y'
1
1 1
1
 3(
)(2)  2( )  ( 2 )(2 x )
y
2x  5
x 4 x 1
6
2
x

 
2 x  5 x 2( x 2  1)

( 2 x  5 )3  6
2
x
y'
 2 x  5  x  x ( x 2  1) 
24
2
x x 1 

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Chapter 12: Additional Differentiation Topics
12.5 Logarithmic Differentiation
Example 3 – Relative Rate of Change of a Product
Show that the relative rate of change of a product is
the sum of the relative rates of change of its factors.
Use this result to express the percentage rate of
change in revenue in terms of the percentage rate of
change in price.
Solution: Rate of change of a function r is
r ' p' q '
 
r
p q
r'
p'
q'
100%  100%  100%
r
p
q
r'
p'
100%  1    100%
r
p
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Chapter 12: Additional Differentiation Topics
12.6 Newton’s Method
Newton’s method:
f xn 
xn 1  xn 
f ' xn 
n  1,2,3,...
Example 1 – Approximating a Root by Newton’s Method
Approximate the root of x4 − 4x + 1 = 0 that lies
between 0 and 1. Continue the approximation
procedure until two successive approximations differ
by less than 0.0001.
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Chapter 12: Additional Differentiation Topics
12.6 Newton’s Method
Example 1 – Approximating a Root by Newton’s Method
4


f
x

x
 4x  1, we have
Solution: Letting
f 0  0  0  1  1
f 1  1  4  1  2
Since f (0) is closer to 0, we choose 0 to be our first x1.
f x n   x n4  4 x n  1
f ' x n   4 x n3  4
Thus,
f xn 
3 xn4  1
xn 1  xn 
 3
f ' xn  4 xn  4
When n  0, x1  0
When n  1, x2  0.25
When n  2, x3  0.25099
When n  3, x 4  0.25099
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Chapter 12: Additional Differentiation Topics
12.7 Higher-Order Derivatives
For higher-order derivatives:
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Chapter 12: Additional Differentiation Topics
12.7 Higher-Order Derivatives
Example 1 – Finding Higher-Order Derivatives
a. If f x   6x 3  12x 2  6x  2 , find all higher-order
derivatives.
Solution:
f ' x   18 x 2  24 x  6
f ' ' x   36 x  24
f ' ' ' x   36
f  4  x   0
b. If f(x) = 7, find f(x).
Solution:
f ' x   0
f ' ' x   0
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Chapter 12: Additional Differentiation Topics
12.7 Higher-Order Derivatives
Example 3 – Evaluating a Second-Order Derivative
16
d 2y
If f x  
, find 2 w henx  4.
x4
dx
Solution: dy  16x  42
dx
d 2y
3



32
x

4
dx 2
d 2y
dx 2
x 4
1

16
Example 5 – Higher-Order Implicit Differentiation
d 2y
Find 2 if x 2  4 y 2  4.
dx
Solution: 2x  8y dy  0
dx
dy  x

dx 4 y
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Chapter 12: Additional Differentiation Topics
12.7 Higher-Order Derivatives
Example 5 – Higher-Order Implicit Differentiation
Solution (continued):
dy  x
Diff erentiate

to get
dx 4 y
d 2y
4y 2  x 2

2
dx
16 y 3
d 2y
1
 3
2
dx
4y
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