chapter 12 - additional differentiation topics

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INTRODUCTORY MATHEMATICAL ANALYSIS INTRODUCTORY MATHEMATICAL ANALYSIS For Business, Economics, and the Life and Social Sciences 2007 Pearson Education Asia Chapter 12 Chapter 12 Additional Differentiation Topics Additional Differentiation Topics

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Page 1: Chapter 12 - Additional Differentiation Topics

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

Page 2: Chapter 12 - Additional 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

Page 3: Chapter 12 - Additional Differentiation Topics

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

Page 4: Chapter 12 - Additional Differentiation Topics

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

Page 5: Chapter 12 - Additional Differentiation Topics

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)

Page 6: Chapter 12 - Additional Differentiation Topics

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

Page 7: Chapter 12 - Additional Differentiation Topics

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

Page 8: Chapter 12 - Additional Differentiation Topics

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

Page 9: Chapter 12 - Additional Differentiation Topics

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

Page 10: Chapter 12 - Additional Differentiation Topics

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.

Page 11: Chapter 12 - Additional Differentiation Topics

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

Page 12: Chapter 12 - Additional Differentiation Topics

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

Page 13: Chapter 12 - Additional Differentiation Topics

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

Page 14: Chapter 12 - Additional Differentiation Topics

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

Page 15: Chapter 12 - Additional Differentiation Topics

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.

Page 16: Chapter 12 - Additional Differentiation Topics

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

Page 17: Chapter 12 - Additional Differentiation Topics

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

Page 18: Chapter 12 - Additional Differentiation Topics

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

Page 19: Chapter 12 - Additional Differentiation Topics

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

Page 20: Chapter 12 - Additional Differentiation Topics

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):

Page 21: Chapter 12 - Additional Differentiation Topics

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

qq

pp

rr

qq

pp

rr

Page 22: Chapter 12 - Additional Differentiation Topics

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.

Page 23: Chapter 12 - Additional Differentiation Topics

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

Page 24: Chapter 12 - Additional Differentiation Topics

2007 Pearson Education Asia

Chapter 12: Additional Differentiation Topics

12.7 Higher-Order Derivatives12.7 Higher-Order DerivativesFor higher-order derivatives:

Page 25: Chapter 12 - Additional Differentiation Topics

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

Page 26: Chapter 12 - Additional Differentiation Topics

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

Page 27: Chapter 12 - Additional Differentiation Topics

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