46

Chapter 1 Overview: Review of Derivatives The purpose of this chapter is to review the ―how‖ of differentiation. We will review all the derivative rules learned last year in PreCalculus. In the next two chapters, we will review the ―why.‖ As a quick reference, here are those rules:

The Power Rule: - n nd duu nu

dx dx 1

The Product Rule: d dv du

u v u vdx dx dx

The Quotient Rule:

du dv

v uu xd dx dxdx vv x

2

The Chain Rule: ( ( )) '( ( )) '( )

df g x f g x g x

dx

sin cos

cos sin

tan sec

d duu u

dx dx

d duu u

dx dx

d duu u

dx dx

2

csc csc cot

sec sec tan

cot csc

d duu u u

dx dx

d duu u u

dx dx

d duu u

dx dx

2

ln

u u

u u

d due e

dx dx

d dua a a

dx dx

ln

log

ln a

d duu

dx u dx

d duu

dx u a dx

1

1

-

-

-

sin

cos

tan

d duu

dx dxu

d duu

dx dxu

d duu

dx dxu

1

2

1

2

12

1

1

1

1

1

1

-

-

-

csc

sec

cot

d duu

dx dxu u

d duu

dx dxu u

d duu

dx dxu

1

2

1

2

12

1

1

1

1

1

1

47

1.1: The Power Rule and the Exponential Rules In PreCalculus, we developed the idea of the Derivative geometrically. That is, the derivative initially arose from our need to find the slope of the tangent line. In Chapter 2 and 3, that meaning, its link to limits, and other conceptualizations of the Derivative will be Explored. In this Chapter, we are primarily interested in how to find the Derivative and what it is used for.

Derivative—Def’n: ( ) ( )

'( ) limh

f x h f xf x

h

0

—Means: The function that yields the slope of the tangent line.

Numerical Derivative—Def’n: ( ) ( )

'( )  lim x a

f x f af a

x a

—Means: The numerical value of the slope of the tangent line at x = a.

Symbols for the Derivative

dy

dx= "d-y-d-x" f '(x) = "f prime of x" y' = "y prime"

d

dx

= "d-d-x" Dx = "d sub x"

OBJECTIVES Use the Power Rule and Exponential Rules to find Derivatives. Find the equations of tangent and normal lines. Use the equation of a tangent line to approximate a function value.

48

The first and most basic derivative rule is the Power Rule. Among the last rules we learned in PreCalculus were the Exponential Rules. They look similar to one another, therefore it would be a good idea to view them together.

The Power Rule: The Exponential Rules:

- n nd

x n xdx

1

ln

x x

x x

de e

dx

da a a

dx

The difference between these is where the variable is. The Power Rule applies when the variable is in the base, while the Exponential Rules apply when the variable is in the Exponent. The difference between the two Exponential rules is what the base is. e = 2.718281828459…, while a is any positive number other than 1.

Ex 1 Find a) d

xdx

5 and b)

xd

dx 5

The first is a case of the Power Rule while the second is a case of the second Exponential Rule. Therefore,

a)

d

xdx

5 = x45 b)

xd

dx 5 = lnx5 5

There were a few other basic rules that we need to remember.

Dx [constant] is always 0

1n nxD cx cn x

x x xx g x x g xD f D f D

These rules allow us to easily differentiate a polynomial--term by term.

49

Ex 2 y x x 23 5 1; find dy

dx

[ ]

( ) ( ) ( )

dy dx x

dx dx

x x

2

2 1 1 1

3 5 1

3 2 5 1 1 0

x 6 5

Ex 3 ( ) ; find '( ).xf x x x e f x 2 4 3

'( ) xf x x e 2 4

Ex 4 ; find .dy

y x x edxx

43 3 44

y x x ex

y x x x e

43 3 4

33 1442 2

4

4

dy

x x xdx

3 1142 23 3

22 4

Note in Ex 4 that 4e is a constant, therefore, its derivative is 0. Ex 5 Find the equations of the lines tangent and normal to

( )f x x x x 4 3 22 1 at 1x

The slope of the tangent line will be '( )f 1

'( )

'( )

f x x x x

f

3 24 3 4

1 3

[Note that we could have gotten this more easily with the nDeriv function on our calculator.]

50

'( )f 1 1, so the tangent line will be

( )

or

y x

y x

1 3 1

3 2

The normal line is perpendicular to the tangent line and, therefore, has the negative reciprocal slope = 1/3. The normal line is

( )y x 1

1 13

One of the uses of the tangent line is based on the idea of Local Linearity. This means that in small areas, algebraic curves act like lines—namely their tangent lines. Therefore, one can get an approximate y-value for points near the point of tangency by plugging x-values into the equation of the tangent line. As we have seen, when the variable is in the Exponent, we use the Exponential Rules. When the variable was in the base, we used the Power Rule. But what if

the variable is in both places, such as xd

xdx

2

2 1 ? It is definitely an

Exponential problem, but the base is not a constant as the rules above have. The Change of Base Rule allows us to clarify the problem:

ln x xxd d

x edx dx

22 2 12 1

but we will need the Product Rule for this derivative. Therefore, we will save this for section 1-4.

51

1.1 Homework Set A Differentiate.

1. ( )f x x x 2 3 4 2. ( )f t t 418

4

3. y x

2

3 4. xy e 35

5. ( )v r r 34

3 6. ( )g x x

x 2

2

1

7. x x

yx

2 4 3 8. u t t 3 2 32

9. yAz y Be

y 10 10. xy e 1 1

11. Find an equation of the tangent to the curve xy x e 4 2 at the point ( ,  )0 2 .

12. Find the points on the curve y x x x 3 22 3 12 1 where the tangent is

horizontal.

13. Find the equation of the tangent line to 5( ) 5 1 at 2f x x x x .

14. Find the tangent line equation to 2

5( ) at 1F x x x

x .

15. Find the equation of all tangent lines that have a slope of 5 for the function

2

42

xy x

16. Find the equation of the tangent line for 2 5 6f x x x that has a slope

of 4. 17. Find the equations of the lines tangent and normal to the function

35f x x x at the point (1,5)

52

1.2: Composite Functions and the Chain Rule

Composite Function--A function made of two other functions, one within the

other. For Example, y x x 316 , sin y x 3 , cosy x 3 , and ( )y x x 2 32 5 .

The general symbols are ( ( ))f g x or [ ]( )f g x .

Ex 1 Given cos xf x 1 , xg x 2 1, and xh x 21 , find (a) f g 2 ,

(b) h g 1 , and (c) f h g 1 .

(a) g 2

2 2 1 1, so cosf g f 12 1 1 0 .

(b) g 01 , so h g h 21 0 1 0 1.

(c) g 01 and h g h 21 0 1 0 1, so

cosf h g f 11 1 1 0

So how do we take the derivative of a composite function? There are two (or more) functions that must be differentiated, but, since one is inside the other, the derivatives cannot be taken at the same time. Just as a radical cannot be distributed over addition, a derivative cannot be distributed concentrically. The composite function is like a matryoshka (Russian doll) that has a doll inside a doll. The derivative is akin to opening them. They cannot both be opened at the same time and, when one is opened, there is an unopened one within. You end up with two open dolls next to each other.

The Chain Rule: ( ( )) '( ( )) '( )

df g x f g x g x

dx

If you think of the inside function (the g x ) as equaling u, we could write The

Chain Rule like this: ( ) '

d duf u f u

dx dx . This is the way that most of the

derivatives are written with The Chain Rule.

53

The Chain Rule is one of the cornerstones of Calculus. It can be embedded within each of the other Rules, as it was in the introduction to this chapter. So the Power Rule and Exponential Rules in the last section really should have been stated as:

The Power Rule: The Exponential Rules:

- n nd du

u n udx dx

1

ln

u u

u u

d due e

dx dx

d dua a a

dx dx

where u is a function of x.

OBJECTIVE Find the Derivative of Composite Functions.

Ex 2

dx x

dx

10

24 2 1

dx x x x x

dx

10

2 29

4 2 1 10 4 2 1 8 2

x x x 29

20 4 2 1 4 1

Ex 3 If y x x 316 , find dy

dx.

y x x x x 123 316 16

dyx x x

dx

123 21

16 16 32

x

x x

2

123

16 3

2 16

54

In this case, the is the f function and the polynomial x x 316 is the g. Each

derivative is found by the Power Rule, but, as x x 316 is inside the , it is inside

the derivative of the .

Ex 4 xd

dxe

24

xd

dxe

24 x xe x xe

2 24 48 8

Ex 5 Given this table of values, find df g x

dx

and dg f x

dx

at 1x .

x f x g x 'f x 'g x

1 3 2 4 6

2 1 8 5 7

3 7 2 7 9

' 'd

f g x f g x g xdx

and ' 'd

g f x g f x f xdx

.

At 1x , At 1x ,

' 1 ' 1

' 2 6

5 6

30

df g x f g g

dx

f

' 1 ' 1

' 3 4

9 4

36

dg f x g f f

dx

g

55

1.2 Homework Set A

1. 7

3 4d

x xdx

2. xy e , find dy

dx.

3. 4 3( ) 1 2f x x x , find '( ).f x

4. Given the following table of values, find the indicated derivatives.

( ) ( )x f x f x

2 1 7

8 5 3

a. ( )g 2 , where ( ) ( )g x f x

3 b. '( )h 2 , where ( ) ( )h x f x 3

5. Given this graph and u f g x , v g f x , and w f f x , find

(a) ' 4.5u (b) ' 1v (c) ' 1w

x

y

f(x)

g(x)

56

6. If ( )f x x x 37

3 2 , find '( )f x .

7. If ( )g 2 3 and ( )g 2 4 , find ( )f 2 if ( )

( )g x

f x e .

8. 244 ; find ' 5

9f x x f

9. 2 3 4 9

dx x

dx

10. 7 3 2 ; find dy

y x xdx

1.2 Homework Set B

1. Given the table of values below, find ' 2g if f h x

g x e

x f x 'f x h x 'h x

2 ln 4 1 3 -7 3 ln9 2 -9 11 -7 ln49 3 8 -1

2. If 2g and ' 2 2g , find ' 2f if 2 2 2

g xf x e g x x

3. If 1 5h and ' 1 3h , find ' 1f if 3 h xf x h x e

4. Find 3df g h x

dx

5. If 0 5g and ' 0 3g , find ' 0f if 2

3 6f x g x g x g x

57

1.3: Trig and Log Rules Trigonometric--Defn: "A function (sin, cos, tan, sec, csc, or cot) whose independent variable represents an angle measure." Means: an equation with sine, cosine, tangent, secant, cosecant, or cotangent in it. Logarithmic--Defn: "The inverse of an Exponential function." Means: there is a Log or Ln in the equation.

sin cos

cos sin

tan sec

d duu u

dx dx

d duu u

dx dx

d duu u

dx dx

2

csc csc cot

sec sec tan

cot csc

d duu u u

dx dx

d duu u u

dx dx

d duu u

dx dx

2

a

d duLn u

dx u dx

d duLog u

dx u Ln a dx

1

1

Note that all these Rules are expressed in terms of the Chain Rule.

OBJECTIVES Find Derivatives involving Trig and Logarithmic Functions.

58

Ex 1 sin

dx

dx3

sin sin cos

dx x x

dx3 23

Ex 2 sin

dx

dx

3

sin cos

dx x x

dx

3 3 23

cos x x 2 33

Ex 3

d

Ln xdx

34

lnd

x xdx x

3 23

14 12

4

x

3

We could have also simplified algebraically before taking the derivative:

ln ln lnd d

x xdx dx

34 4 3

x

3

Oftentimes it is significantly easier to take a derivative if we simplify the function first – in the above case, it eliminated the need for The Chain Rule. Of course, composites can involve more than two functions. The Chain Rule has as many derivatives in the chain as there are functions.

Ex 4 sec

dx

dx5 43

sec sec sec tan

dx x x x x

dx5 4 4 4 4 4 33 5 3 3 3 12

sec tan x x x 3 5 4 460 3 3

59

Ex 5 ln cos

dx

dx

ln cos sin

cos

tan

tan

dx x x

dx x

x x

x

x

1 12 2

12

1 12 2

12

12

1 1

2

1

2

2

60

1.3 Homework Set A For #1-12, find the derivatives of the given functions.

1. sin4y x 2. 54secy x

3. 3 3cosa xy 4. 2cot (sin )y

5. 3( ) 1 tanf t t 6. ( ) ln cosf

7. 3 3cosy a x 8. 2tan 3y

9. ( ) cos lnf x x 10. 5( ) lnf x x

11. 10( ) log 2 sinf x x 12. 2( ) log 1 3f x x

13. Find the equation of the tangent line to cosy x x at the point 0,1 .

14. Find the equation of the tangent line to sec 2cosy x x at the point ,13

.

15. Find all points on the graph of 2( ) 2sin sinf x x x at which the tangent line

is horizontal. 16. Find the equations of the lines tangent and normal to the function

2 2ln 4 5xf x e x x at the point (0,1+ln5)

17. Find the equation of the line tangent to 2

cos 4y x x

when 2

x

.

18. Find the equation of the line tangent to sec 2 cot 2y x x through the

point ,1  28

. Use exact values in your answers.

61

1.3 Homework Set B

1. Given the table of values below, find ' 3g if g x f h x

x f x 'f x h x 'h x

2 2

1

2

-7

3 4

2

3

1

3

2 3

4

-1

2. csc 2ln cot secd

ed

3. If 32

g

, ' 34

g

, and sin

3cos3

g xf x g x g x e

, find

' 3f

4. If ln cot sec lnz , find dz

d

5. 3

3ln sec 5ln 7d

x xdx

6. If 2ln cos sec 7tz t e , find dz

dt

7. If 2ln tan sin 7tz t e , find dz

dt

8. If ln cos sin lnz , find dz

d

9. Find '6

f

when 3cos 3f x x

10. Find ' 2g when 2ln 3g x x

11. 2ln 4 5d

x xdx

12. 5sin ln 7 3d

tdt

13. 2csc ln 7d

x xdx

14. 24 6ln td

edt

62

15. sec 5 cot 10lnxdx e x

dx

For problems 16 to 18, find the first derivative for the following functions

16. 10

yAz y Be

y 17. sinrA

f r Ber

18. 3 2

2

3 12x xy

x

19. 23

59 27

d dx x

dx dx x

20. 23

59 27

d dx x

dx dx x

21. 3

2ln tan 5 7xdx e

dx

22.

2

2

cos ln 5

sin ln 5

xd

dx x

63

1.4: Product and Quotient Rules Remember:

The Product Rule: '( )dv du

f x U Vdx dx

The Quotient Rule: '( )

du dvV U

dx dxf xV

2

OBJECTIVE Find the Derivative of a product or quotient of two functions.

Ex 1 sin

dx x

dx2

sin cos sin

cos sin

dx x x x x x

dx

x x x x

2 2

2

2

2

Ex 2

d x x

dx x

2 2 3

4

, so

, so

( ) ( ) ( )’( )

( )

( )

                                    ( )

duU x x x

dx

dvV x

dx

du dvV U x x x xdx dxf x

V x

x x x x

x

x x

x

2

2

2 2

2 2

2

2

2 3 2 2

4 1

4 2 2 2 3 1

4

2 6 8 2 3

4

8 5

4 2

64

Ex 3 d x x

dx x x

2

2

4 3

2 5 3

Notice that this problem becomes much easier if we simplify before applying the Quotient Rule.

( )( )

( )( )

( )( ) ( )( )

( )

( )

d x x d x x

dx dx x xx x

d x

dx x

x x

x

x

2

2

2

2

4 3 1 3

2 1 32 5 3

1

2 1

2 1 1 1 2

2 1

3

2 1

Ex 4 cot 3d x

dx x

2 1

csc cot cot

csc csc cot

x x x xd x

dx x x

x x x x x

x

2 2

22

2 2 2

2

2

2

1 3 3 3 23

1 1

3 3 3 3 2 3

1

Ex 5

cos xdx

dx5

cos sin cos ln

ln cos sin

x x x

x

dx x x

dx

x x

5 5 5 5

5 5

65

Remember that, in section 1.1, we said we would need the Product Rule to deal with the derivative of a function where the variable is in both the base and the Exponent. We can now address that situation.

Ex 6

cos xd

xdx

2

ln

ln cos

cos

cos

sin ln cos cos

cos ln cos tan

x x x

x x

x

d dx e

dx dx

e x x x xx

x x x x x

2 2

22

22

12

2

66

1.4 Homework Set A Find the derivative of the following functions.

1. 3 cosy t t 2. 85 2( ) 1 4 3g x x x x

3. tan 1

sec

xy

x

4.

34 22 5 8 5y x x

5. 2xy xe 6.

2

sin xy

x

7. cosx xy e 8. 2 1

ry

r

9. 1

siny xx

10. 5 cos3xy e x

11. ( ) lnf x x x 12. ln x xy e xe

13. Find the equation of the line tangent to 2 xy x e at the point 11, e .

14. If ( )ln

xf x

x , find '( )f e . 15. ( )

xh t

x

172

2

1

1, find '( )h t

16. y x x 2 25 , find 'y 1

17. ( ) sin tanf x x x x

5

4 72 , find 'f x .

18. Find the equation of the lines tangent and normal to sin ln2

y x x

when

x e

19. Find the equation of the line tangent to sin 4

2x x

y e when 0x

67

20. Find the equation of the lines tangent and normal to 1

siny xx

when

4x

.

1.4 Homework Set B

1. Find the first derivative for the following function: 2 2 4sin 5tx t e t t

2. Find the first derivative for the following function: 5 4tan 3tx t e t

3. Find the first derivative for the following function: 2 2 15

3

x xy

x

4. Find the first derivative for the following function: 2 45tx t e t t

5. 2

3

7 5

sin

xd e x

dx x

6.

sinln cot

y yde e

dy

7. 2 2 1sin

ln

d xx x

dx x

8. If 1 5h and ' 1 3h , find ' 1f if 4

lnf x h x x h x

9. Find 'g z if 118

5

1 ln

zeg z

z

10. 2 2cosxd e

x xdx x

11. 2 2 3

4

d x x

dx x

12.

2

5

cos 3

x

xd

dx e

13. 5 ln 5 4ln

d xx x

dx x

14. Find 'g t if 15

2

2

4

1

tg t

t

68

15.

2

cosxde x

dx

16. 1 tan

ln 4

d x

dx x

17. sin tand

t tdt

18.

1 ln

csc

d x

dx x

19. 45 ln sinxd

e xdx

20. 2 22

5 sin ln 3 11

pd pp p e p

dp p

21. 4 3tan 5xde x x x

dx

22. 5 2

ln 3 7

d x

dx x

23. 2

4

9 20

d x

dx x x

24. 2sin 4 2

d dx

dx dx

25. 5 3

3

12 19

3

d c c c

dc c

26. Given the table of values below, find ' 3g if sing x f h x h x

x f x 'f x h x 'h x

2 2

1

2

-7

3 4

2

3

1

3

2 3

4

-1

27. If 32

g

, ' 34

g

, and sin3 3cos

3

g xf x x g x g x e

, find

' 3f

69

1.5: Higher Order Derivatives What we have been calling the Derivative is actually the First Derivative. There can be successive uses of the derivative rules, and they have meanings other than the slope of the tangent line. In this section, we will explore the process of finding the higher order derivatives.

Second Derivative--Defn: The derivative of the derivative. Just as with the First Derivative, there are several symbols for the 2nd Derivative:

Higher Order Derivative Symbols

Liebnitz: d y

dx

2

2 = d squared y, d x squared;d y

dx

3

3;…

n

nd ydx

Function: "f x = f double prime of x; '''f x ; IVf x ;… nf x

Combination: y'' = y double prime

OBJECTIVE Find higher order derivatives.

Ex 1 d

x x x xdx

2

4 3 22 7 3 2 5

d d dx x x x x x x x

dx dxdx

dx x x

dx

24 3 2 4 3 2

2

3 2

7 3 2 5 7 3 2 5

4 21 6 2

x x 212 42 6

70

Ex 2 Find d y

dx

3

3 if sin y x 3

sin

cos cos

sin sin

cos cos

y x

dyx x

dx

d yx x

dx

d yx x

dx

2

2

3

3

3

3 3 3 3

3 3 3 9 3

9 3 3 27 3

More complicated functions, in particular Composite Functions, have a complicated process. When the Chain Rule is applied, the answer becomes a product or quotient. Therefore, the 2nd Derivative will require the Product or Quotient Rules as well as, possibly, the Chain Rule again.

Ex 3 xy e23 , find y''.

x x

x x

x x

x

dye x xe

dx

d yx e x e

dx

x e e

e x

2

2

2

2

2 2

2 2

2 2

2

3 3

3 3

3 3

3

6 6

6 6 6

36 6

6 6 1

Ex 4 siny x 3 , find y''

' sin cos

'' sin sin cos sin cos

sin cos sin

y x x

y x x x x x

x x x

2

2

2 2

3

3 6

3 2

71

Ex 5 lnf x x x 2 3 1 , find ’’f x .

’

’’

xf x x

x x x x

x x x xf x

x x

x x x x

x x

x x

x x

2 2

2

22

2 2

22

2

22

1 2 32 3

3 1 3 1

3 1 2 2 3 2 3

3 1

2 6 2 4 12 9

3 1

2 6 11

3 1

Ex 6 g x x 24 1 , find ’’g x .

’

’’

xg x x x

x

x x x x

g x

x

xx

x

x

x x

x

x

122

122

1 12 22 2

122

2122

122

2

2 2

322

322

1 44 1 8

24 1

14 1 4 4 4 1 8

2

24 1

164 1 4

4 1

4 1

4 1 4 16

4 1

4

4 1

72

1.5 Homework Set A In #1-5, find the first and second derivatives of the function.

1. 5 2( ) 6 7f x x x x 2. 2( ) 1h x x

3. 2

33 1y x 4. ( ) tan3H t t

5. 3 5( ) tg t t e 6. If 23xy e , find ''y .

7. If 3siny x , find ''y . 8. If ( ) cosf t t t , find '''(0).f

1.5 Homework Set B

1. Find 'f x , ''f x , and '"f x if ln secf x x .

2. Find dy

d and

2

2

d y

d for 10cot 2 1y

3. For the function tan2y x , show that 2

22

8sec 2 tan 2d y

x xdx

.

4. Find "f x for 2ln 3 1f x x x

5. Find the first, second, and third derivative for 2 5 6 xf x x x e

6. Find the first, second, and third derivative for tan 3 2f

7. A fourth differentiable function is defined for all real numbers and satisfies each of the following:

2 5g , ' 2 2g , and " 2 3g

If the function f is given by 1

2k x

f x e g x

, where k is a constant.

a. Find 1f , ' 1f , '' 1f

b. Show that the fourth derivative of f is

14 16 '''' 2k x

k e g x

73

1.6: Derivatives of Inverse Functions We already know something about inverse functions. Exponential and Logarithmic functions are inverses of each other, as are Radicals and Powers. Inverse Functions—Defn: Two functions wherein the domain of one serves as the range of the other and vice versa. --Means: Two functions which cancel (or undo) each other. The definition gives us a way to find the inverse for any function, the symbol for

which is f 1. You just switch the x and y variables and isolate y.

Ex 1 Find f 1 if f x x 3

f x x y x 3 3

So, for f 1, x y 3

y x

f x

3

1 3

This is little more interesting with a Rational Function.

Ex 2 Find f 1 if x

f xx

1

2

x x

f x yx x

1 1

2 2

So, for f 1,

yx

y

y x y

x xy y

x y xy

x y x

xy

xx

f xx

1

1

2

2 1

2 1

2 1

2 1 1

2 1

12 1

1

74

Note that - f f x x1 . For Example, f 1

02

and - f

1 10

2.

General inverses are not all that interesting. We are more interested in particular inverse functions, like the Ln. Another particular kind of inverse function that bears more study is the Trig Inverse Function. Interestingly, as with the Log functions, the derivatives of these Transcendental Functions become Algebraic Functions.

Inverse Trig Derivative Rules

-

-

-

sin

cos

tan

u

u

u

du D

dx u

du D

dx u

du D

dx u

1

2

1

2

12

1

1

1

1

1

1

-

-

-

csc

sec

cot

u

u

u

du D

dx u u

du D

dx u u

du D

dx u

1

2

1

2

12

1

1

1

1

1

1

OBJECTIVE Find the derivatives of inverse trig functions.

Proof that sin

dx

dx x

1

2

1

1.

sin sin y x y x 1

sin

cos

cos

xD y x

dyy

dx

dy

dx y

1

1

75

But this derivative is not in terms of x, so we are not done. Consider the right triangle that would yield this SOHCAHTOA relationship:

1-x 2

1x

y

Note that for sin y to equal x, x must be the opposite leg and the hypotenuse is 1 (SOH). The Pythagorean Theorem gives us the adjacent leg. By CAH,

cos y x 21

Therefore,

sin

cos

dx

dx y

x

1

2

1

1

1

Note that in the proof above, when we wanted to take the derivative of sin y, we had to use The Chain Rule (since the y is a function other than x, and the

derivative of y is dy

dx.

Ex 4 - tan

dx

dx

1 43

-

tan d

x xdx

x

x

x

1 4 32

4

3

8

13 12

3 1

12

9 1

76

Ex 5 - sec

dx

dx

1 2

- sec

dx x

dxx x

x

x x

x x

1 2

22 2

22 2

4

12

1

2

1

2

1

77

1.6 Homework Set A Find the derivative of the function. Simplify where possible.

1. 1sin xy e 2. 1tany x

3. 1sin 2 1y x 4. 2 1( ) 1 tan ( )H x x x

5. 1 2cos 1y x x x 6. 2( ) arctanxf x e x x

7. 1sin 2y x 8. 21csc 1y x

9. 1 11cot tany x

x

10. 21cos 1y x x x

11. 1sec

0x

y for xx

12. 2 1ln 4 tan2

xy x x

1.6 Homework Set B

1. 1 3cos zy e 2. 1 2tan 1y x

3. 1 214sin 4

2y x x x

4. 1 1cos

1

xy

x

5. 1 1sec 4 csc 4y x x 6. 1 2 2( ) sint

f t c c tc

7. 2( ) arccosf x x x 8. 1( ) ln tan 5f x x

9. 1 1sin 5 cos 5g w w w 10. 1 2( ) sec 9f t t

11. 2 1ln 1 coty u u u 12. 2

1 2tan

1

x

x

ey

e

78

1.7: Local Linearity and Approximations Before calculators, one of the most valuable uses of the derivative was to find approximate function values from a tangent line. Since the tangent line only shares one point on the function, y-values on the line are very close to y-values on

the function. This idea is called local linearity—near the point of tangency, the function curve appears to be a line. This can be easily demonstrated with the graphing calculator by zooming in on the point of tangency. Consider the graphs

of .y x 425 and its tangent line at x 1 , .y x 75 .

x

y

x The closer you zoom in, the more the line and the curve become one. The y-values on the line are good approximations of the y-values on the curve. For a good animation of this concept, see

http://www.ima.umn.edu/~arnold/tangent/tangent.mpg

Since it is easier to find the y-value of a line arithmetically than for other

functions—especially transcendental functions—the tangent line

approximation is useful if you have no calculator.

OBJECTIVES Use the equation of a tangent line to approximate function values.

79

Ex 1 Find the tangent line equation to ( )f x x x x 4 3 22 1 at x = –1 and use it

to approximate value of ( . )f 0 9 .

The slope of the tangent line will be '( )f 1

'( )

'( )

f x x x x

f

3 24 3 4

1 3

[Note that we could have gotten this more easily with the nDeriv function on our calculator.]

( )f 1 1, so the tangent line will be

( )

or

y x

y x

1 3 1

3 2

While we can find the exact value of ( . )f 0 9 with a calculator, we can get a

quick approximation from the tangent line. If x = –0.9 on the tangent line, then

0.9 0.9 3 0.9 2 .7f y

This last example is somewhat trite in that we could have just plugged –0.9 into

( )f x x x x 4 3 22 1 and figured out the exact value even without a calculator.

It would have been a pain, but it is doable. Consider the next example, though.

Ex 2 Find the tangent line equation to xf x e 2 at x 0 and use it to

approximate value of .e0 2 .

Without a calculator, we could not find the exact value of .e0 2 . In fact, even the calculator only gives an approximate value.

' xf x e 22 and ' xf e 20 2 2

f e 00 1

So the tangent line equation is y x 1 2 0 or y x 2 1

In order to find .e0 2 , we are looking for .f 0 1 . Therefore, we plug in

x = 0.1

. . .e 0 2 2 0 1 1 1 2

80

1.7 Homework Set A

1. Find the equation of the tangent line to 5( ) 5 1 at 2f x x x x and use it

to get an approximate value of 1.9f .

2. Find the tangent line equation to 2

5( ) at 1F x x x

x and use it to get an

approximate value of 1.1F .

3. Find all points on the graph of 22sin siny x x where the tangent line is

horizontal.

4. Find the equation of the tangent line at x = 2 for 2ln 3g x x . Use this

to approximate 2.1g .

5. Find the equation of the line tangent to 25 2 tan 1g x x x when

1x . Use this tangent line to find an approximation for 1.1g .

6. Use a tangent line to find the approximate the value of .02f if

sin2 x xf x e x . Then use your calculator to find an actual value for .02f .

7. Find the equation of the tangent line for 2ln sin2

f x x x

for x = 1.

Plug in x = 1.1 into both the original function and the tangent line. Explain why the values are so similar.

81

Chapter 1 Test

1.   

xdx x

dx xx

45 333 4

1 13 2 3

56

2. tan cot x

yx

1 1 3

3; find y' .

3. xD x x

3 5

4 77 5 2 1

4. Find the equation of the line tangent to 3 4

2 1f x x at x = 2 and use it to

estimate f (1.98).

5. ln z

g zz

2

32

2; Find g'(w) and g'(4).

6. Find the equation of the line tangent to lny x 5

24 3 at x 1 .

7. Find  "f

6 if sin 3f t t 27 .

8. cosxdx e

xdx

2

7 3 52

12