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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 Probability10. Limits and Continuity11. Differentiation
12. Additional Differentiation Topics13. Curve Sketching14. Integration15. Methods and Applications of Integration16. Continuous Random Variables17. 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 xh
xx
dxd /
01limln1ln a.
0 where1ln b. xx
xdxd
0 for 1ln c. udxdu
uu
dxd
2007 Pearson Education Asia
Chapter 12: Additional Differentiation Topics12.1 Derivatives of Logarithmic Functions
Example 1 – Differentiating Functions Involving ln x
b. Differentiate .Solution:
2
lnx
xy
0 for ln21
2)(ln1
lnln'
3
4
2
22
22
xx
xx
xxx
x
x
xdxdxx
dxdx
y
a. Differentiate f(x) = 5 ln x.Solution: 0 for 5ln5' x
xx
dxdxf
2007 Pearson Education Asia
Chapter 12: Additional Differentiation Topics12.1 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
6252
13
x
xxdxdy
34
23
12
13
1412
1311
12'
ppp
ppppf
432 321ln ppppf
2007 Pearson Education Asia
Chapter 12: Additional Differentiation Topics12.1 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
ulnln
xx
dxdx
dxdy
2ln1
2lnlnlog2
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:
dxduee
dxd uu a.
xx eedxd
b.
dxdubbb
dxd uu ln c.
0' for '
1 d. 11
1
xffxff
xfdxd
dydxdx
dy 1 e.
2007 Pearson Education Asia
Chapter 12: Additional Differentiation Topics12.2 Derivatives of Exponential Functions
Example 1 – Differentiating Functions Involving ex
a.Find .
Solution:
b. If y = , find .
Solution:
c. Find y’ when .Solution:
xex
xxx
exe
dxdxx
dxde
dxdy
1
3ln2 xeeyxx eey 00'
xedxd 3
xxx eedxde
dxd 333
dxdy
2007 Pearson Education Asia
Chapter 12: Additional Differentiation Topics12.2 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 /
21
xe
xxfy
Solution: The rate of change is
e
edxdy x
x
21
1221
21
2
/ 221
2007 Pearson Education Asia
Chapter 12: Additional Differentiation Topics12.2 Derivatives of Exponential Functions
Example 5 – Differentiating Different Forms
Example 7 – Differentiating Power Functions Again
Find .
Solution:
xexedxd 22
xex
xeexxe
dxd
xe
xexe
22ln2
212ln2
1
2ln12
Prove d/dx(xa) = axa−1.
Solution: 11ln aaxaa axaxxedxdx
dxd
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.
dqdpqp
q
Determine the point elasticity of the demand equation
Solution: We have
0 and 0 where qkqkp
12
2
q
kqk
dqdpqp
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 Topics12.4 Implicit Differentiation
Example 1 – Implicit DifferentiationFind dy/dx by implicit differentiation if .
Solution:
73 xyy
2
2
3
311
013
7
ydxdy
dxdyy
dxdy
dxdxyy
dxd
2007 Pearson Education Asia
Chapter 12: Additional Differentiation Topics12.4 Implicit Differentiation
Example 3 – Implicit DifferentiationFind the slope of the curve at (1,2).
Solution:
223 xyx
27
2443
223
2,1
2
32
22
223
dxdy
xyxxyx
dxdy
xdxdyxy
dxdyx
xydxdx
dxd
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 Topics12.5 Logarithmic Differentiation
Example 1 – Logarithmic Differentiation
Find y’ if .
Solution:
4 22
3
1
52
xxxy
xx
xx
xxxy
xxxy
21
141ln252ln3
1ln52lnln
1
52lnln
2
4 223
4 22
3
2007 Pearson Education Asia
Chapter 12: Additional Differentiation Topics12.5 Logarithmic DifferentiationExample 1 – Logarithmic Differentiation
)1(2
526
1
)52('
)1(22
526
)2)(1
1(41)1(2)2)(
521(3'
24 22
3
2
2
xxx
xxxxxy
xx
xx
xxxxy
y
Solution (continued):
2007 Pearson Education Asia
Chapter 12: Additional Differentiation Topics12.5 Logarithmic Differentiation
Example 3 – Relative Rate of Change of a ProductShow 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'
'''
pp
rr
pp
rr
pp
rr
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 nxfxfxx
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 Topics12.6 Newton’s MethodExample 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
xxfxfxx
25099.0 ,3 When25099.0 ,2 When25.0 ,1 When
0 ,0 When
4
3
2
1
xnxnxnxn
144 xxxf
21411
11000
ff
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 Topics12.7 Higher-Order Derivatives
Example 1 – Finding Higher-Order Derivativesa. 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
xfxf
xxfxxxf
0''
0'
xfxf
2007 Pearson Education Asia
Chapter 12: Additional Differentiation Topics12.7 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
ydx
xf
32
2
2
432
416
xdx
yd
xdxdy
161
42
2
xdx
yd
Solution:
yx
dxdydxdyyx
4
082
.44 if Find 222
2
yxdx
yd
2007 Pearson Education Asia
Chapter 12: Additional Differentiation Topics12.7 Higher-Order DerivativesExample 5 – Higher-Order Implicit Differentiation
Solution (continued):
32
2
3
22
2
2
41
164
get to 4
ateDifferenti
ydxyd
yxy
dxyd
yx
dxdy