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Matematika Bisnis
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INTRODUCTORY MATHEMATICAL INTRODUCTORY MATHEMATICAL ANALYSISANALYSISFor Business, Economics, and the Life and Social Sciences
2007 Pearson Education Asia
Chapter 12 Chapter 12 Additional Differentiation TopicsAdditional Differentiation Topics
2007 Pearson Education Asia
INTRODUCTORY MATHEMATICAL ANALYSIS
0. Review of Algebra
1. Applications and More Algebra
2. Functions and Graphs
3. Lines, Parabolas, and Systems
4. Exponential and Logarithmic Functions
5. Mathematics of Finance
6. Matrix Algebra
7. Linear Programming
8. Introduction to Probability and Statistics
2007 Pearson Education Asia
9. Additional Topics in Probability
10. Limits and Continuity
11. Differentiation
12. Additional Differentiation Topics
13. Curve Sketching
14. Integration
15. Methods and Applications of Integration
16. Continuous Random Variables
17. Multivariable Calculus
INTRODUCTORY MATHEMATICAL ANALYSIS
2007 Pearson Education Asia
• 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.
Chapter 12: Additional Differentiation Topics
Chapter ObjectivesChapter Objectives
2007 Pearson Education Asia
Derivatives of Logarithmic Functions
Derivatives of Exponential Functions
Elasticity of Demand
Implicit Differentiation
Logarithmic Differentiation
Newton’s Method
Higher-Order Derivatives
12.1)
12.2)
12.3)
Chapter 12: Additional Differentiation Topics
Chapter OutlineChapter Outline
12.4)
12.5)
12.6)
12.7)
2007 Pearson Education Asia
Chapter 12: Additional Differentiation Topics
12.1 Derivatives of Logarithmic Functions12.1 Derivatives of Logarithmic Functions• The derivatives of log functions are:
hx
h x
h
xx
dx
d/
01limln
1ln a.
0 where1
ln b. xx
xdx
d
0 for 1
ln c. udx
du
uu
dx
d
2007 Pearson Education Asia
Chapter 12: Additional Differentiation Topics
12.1 Derivatives of Logarithmic Functions
Example 1 – Differentiating Functions Involving ln x
b. Differentiate .
Solution:2
ln
x
xy
0 for ln21
2)(ln1
lnln'
3
4
2
22
22
xx
xx
xxx
x
x
xdxd
xxdxd
xy
a. Differentiate f(x) = 5 ln x.
Solution: 0 for 5
ln5' xx
xdx
dxf
2007 Pearson Education Asia
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:
352ln xy
2/5 for 52
62
52
13
x
xxdx
dy
3
4
2
3
1
2
13
141
2
131
1
12'
ppp
ppppf
432 321ln ppppf
2007 Pearson Education Asia
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
xx
dx
dx
dx
dy
2ln
1
2ln
lnlog2
2007 Pearson Education Asia
Chapter 12: Additional Differentiation Topics
12.2 Derivatives of Exponential Functions12.2 Derivatives of Exponential Functions• The derivatives of exponential functions are:
dx
duee
dx
d uu a.
xx eedx
d b.
dx
dubbb
dx
d uu ln c.
0' for '
1 d. 1
11
xff
xffxf
dx
d
dydxdx
dy 1 e.
2007 Pearson Education Asia
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:
xe
x
x
xx
e
xe
dx
dxx
dx
de
dx
dy 1
3ln2 xeeyxx eey 00'
xedx
d3
xxx eedx
de
dx
d333
dx
dy
2007 Pearson Education Asia
Chapter 12: Additional Differentiation Topics
12.2 Derivatives of Exponential Functions
Example 3 – The Normal-Distribution Density FunctionDetermine the rate of change of y with respect to x when x = μ + σ.
221 /
2
1
xe
xxfy
Solution: The rate of change is
e
edx
dy x
x
2
1
12
2
1
2
1
2
/ 221
2007 Pearson Education Asia
Chapter 12: Additional Differentiation Topics
12.2 Derivatives of Exponential Functions
Example 5 – Differentiating Different Forms
Example 7 – Differentiating Power Functions Again
Find .
Solution:
xexedx
d22
xex
xeexxe
dx
d
xe
xexe
2
2ln2
2
12ln2
1
2ln12
Prove d/dx(xa) = axa−1.
Solution: 11ln aaxaa axaxxedx
dx
dx
d
2007 Pearson Education Asia
Chapter 12: Additional Differentiation Topics
12.3 Elasticity of Demand12.3 Elasticity of Demand
Example 1 – Finding Point Elasticity of Demand
• Point elasticity of demand η is
where p is price and q is quantity.
dqdp
qp
q
Determine the point elasticity of the demand equation
Solution: We have
0 and 0 where qkq
kp
12
2
q
k
qk
dqdp
qp
2007 Pearson Education Asia
Chapter 12: Additional Differentiation Topics
12.4 Implicit Differentiation12.4 Implicit DifferentiationImplicit 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.
2007 Pearson Education Asia
Chapter 12: Additional Differentiation Topics
12.4 Implicit Differentiation
Example 1 – Implicit Differentiation
Find dy/dx by implicit differentiation if .
Solution:
73 xyy
2
2
3
31
1
013
7
ydx
dydx
dyy
dx
dydx
dxyy
dx
d
2007 Pearson Education Asia
Chapter 12: Additional Differentiation Topics
12.4 Implicit Differentiation
Example 3 – Implicit Differentiation
Find the slope of the curve at (1,2).
Solution:
223 xyx
2
7
2
443
223
2,1
2
32
22
223
dx
dy
xy
xxyx
dx
dy
xdx
dyxy
dx
dyx
xydx
dx
dx
d
2007 Pearson Education Asia
Chapter 12: Additional Differentiation Topics
12.5 Logarithmic Differentiation12.5 Logarithmic DifferentiationLogarithmic Differentiation Procedure
1. Take the natural logarithm of both sides which gives .
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.
xfy lnln
2007 Pearson Education Asia
Chapter 12: Additional Differentiation Topics
12.5 Logarithmic Differentiation
Example 1 – Logarithmic Differentiation
Find y’ if .
Solution:
4 22
3
1
52
xx
xy
xx
xx
xxxy
xx
xy
21
1
4
1ln252ln3
1ln52lnln
1
52lnln
2
4 223
4 22
3
2007 Pearson Education Asia
Chapter 12: Additional Differentiation Topics
12.5 Logarithmic Differentiation
Example 1 – Logarithmic Differentiation
)1(
2
52
6
1
)52('
)1(2
2
52
6
)2)(1
1(
4
1)
1(2)2)(
52
1(3
'
24 22
3
2
2
xx
x
xxxx
xy
x
x
xx
xxxxy
y
Solution (continued):
2007 Pearson Education Asia
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
%100'
1%100'
%100'
%100'
%100'
'''
p
p
r
r
q
q
p
p
r
r
q
q
p
p
r
r
2007 Pearson Education Asia
Chapter 12: Additional Differentiation Topics
12.6 Newton’s Method12.6 Newton’s Method
Example 1 – Approximating a Root by Newton’s Method
Newton’s method: ,...3,2,1
'1 nxf
xfxx
n
nnn
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.
2007 Pearson Education Asia
Chapter 12: Additional Differentiation Topics
12.6 Newton’s Method
Example 1 – Approximating a Root by Newton’s Method
Solution: Letting , we have
Since f (0) is closer to 0, we choose 0 to be our first x1.
Thus, 44
13
' 3
4
1
n
n
n
nnn x
x
xf
xfxx
25099.0 ,3 When
25099.0 ,2 When
25.0 ,1 When
0 ,0 When
4
3
2
1
xn
xn
xn
xn
144 xxxf
21411
11000
f
f
44'
143
4
nn
nnn
xxf
xxxf
2007 Pearson Education Asia
Chapter 12: Additional Differentiation Topics
12.7 Higher-Order Derivatives12.7 Higher-Order DerivativesFor higher-order derivatives:
2007 Pearson Education Asia
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:
26126 23 xxxxf
0
36'''
2436''
62418'
4
2
xf
xf
xxf
xxxf
0''
0'
xf
xf
2007 Pearson Education Asia
Chapter 12: Additional Differentiation Topics
12.7 Higher-Order Derivatives
Example 3 – Evaluating a Second-Order Derivative
Example 5 – Higher-Order Implicit Differentiation
Solution:
.4 when find ,4
16 If
2
2
xdx
yd
xxf
3
2
2
2
432
416
xdx
yd
xdx
dy
16
1
4
2
2
x
dx
yd
Solution:
y
x
dx
dydx
dyyx
4
082
.44 if Find 222
2
yxdx
yd
2007 Pearson Education Asia
Chapter 12: Additional Differentiation Topics
12.7 Higher-Order Derivatives
Example 5 – Higher-Order Implicit Differentiation
Solution (continued):
32
2
3
22
2
2
4
1
16
4
get to 4
ateDifferenti
ydx
yd
y
xy
dx
yd
y
x
dx
dy