Calculus Practice Makes Perfect Ch2

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15

DIFFERENTIATION

Diferentiation is the process o determining the derivative o a unction. Part IIbegins with the ormal denition o the derivative o a unction and shows how the denition is used to nd the derivative. However, the material swily moveson to nding derivatives using standard ormulas or diferentiation o certainbasic unction types. Properties o derivatives, numerical derivatives, implicit di-erentiation, and higher-order derivatives are also presented.

·I I ·



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17

Definition of the derivativeTe derivative f ` (read “ f prime”) o the unction f at the number x is dened as

`

l f x

f x h f x

hh( ) lim

( ) ( ),

i this limit exists. I this limit does not exist, then f does

not have a derivative at x. Tis limit may also be written ` l

f cf x f c

x cx c( ) lim

( ) ( )

or the derivative at c.

PROBLEM Given the unction f dened by f x x ( ) , use the

denition o the derivative to nd ` f x ( ).

SOLUTION By denition, `

l f x

f x h f x

hh( ) lim

( ) ( )

l llim

( ( ) ) ( )lim

(h h

x h x

h

x h

)) x

h

l l l

lim lim lim( )h h h

x h x

h

h

h

.

PROBLEM Given the unction f dened by f x x x ( ) , use the denition

o the derivative to nd ` f x ( ).

SOLUTION By denition, `

l f x

f x h f x

hh( ) lim

( ) ( )

llim

(( ) ( )) ( )h

x h x h x x

h

llim

( )h

x xh h x h x x

h

l llim limh h

x xh h x h x x

h

xh h

h

h

l llim ( ) lim( ) .h h

h x hh

x h x

Various symbols are used to represent the derivative o a unction f . I you

use the notation y f (x ), then the derivative o f can be symbolized by

` ` f x y D f x D y dy

dx x x ( ), , ( ), , , or

d

dx f x ( ).

Definition of the derivativeand derivatives of some

simple functions·4 ·



18 Differentiation

Note: Hereaer, you should assume that any value or which a unction is undened is excluded.

4 ·1

EXERCISE

Use the defnition o the derivative to fnd ` x( ).

1. x ( ) 4 6. x x x ( ) 5 32

2. x x ( ) 7 2 7. x x x ( ) 3 13

3. x x ( ) 3 9 8. x x ( ) 2 153

4. x x ( ) 10 3 9. x x

( ) 1

5. x x ( ) 3

410. x

x ( )

1

Derivative of a constant functionFortunately, you do not have to resort to nding the derivative o a unction directly rom thedenition o a derivative. Instead, you can memorize standard ormulas or diferentiating cer-tain basic unctions. For instance, the derivative o a constant unction is always zero. In other

words, i f x c( ) is a constant unction, then ` f x ( ) ; that is, i c is any constant,d

dx c( ) .

Te ollowing examples illustrate the use o this ormula:

Ud

dx ( )

Ud

dx ( )

4 ·2

EXERCISE

Find the derivative o the given unction.

1. x ( ) 7 6. g x ( ) 25

2. y 5 7. s t ( ) 100

3. x ( ) 0 8. z x ( ) 23

4. t ( ) 3 9. y 1

2

5. x ( ) P 10. x ( ) 41



Definition of the derivative and derivatives of some simple functions 1

Derivative of a linear functionTe derivative o a linear unction is the slope o its graph. Tus, i f x mx b( ) is a linear unc-

tion, then ` f x m( ) ; that is,d

dx mx b m( ) .


U I f x x ( ) , then ` f x ( )

U I y 2x + 5, then ` y

Ud

dx x

¤ ¦ ¥

³ µ ´

4 ·3

EXERCISE


1 x x ( ) 9 6. x x ( ) P 25

2. g x x ( ) 75 7. x x ( ) 3

4

3. x x ( ) 1 8. s t t ( ) 100 45

4. y 50 x + 30 9. z x x ( ) . 0 08 400

5. t t ( ) 2 5 10. x x ( ) 41 1

Derivative of a power functionTe unction f (x ) x n is called a power unction. Te ollowing ormula or nding the derivativeo a power unction is one you will use requently in calculus:

I n is a real number, thend

dx x nx n n( ) .


U I f x x ( ) , then ` f x x ( )

U I y x , then `

y x

Ud

dx x x ( )



20 Differentiation

4 ·4

EXERCISE


1. x x ( ) 3 6. x x ( ) P

2. g x x ( ) 100 7. x x

( ) 1

5

3. x x ( ) 1

4 8. s t t ( ) . 0 6

4. y x 9. h s s( ) 4

5

5. t t ( ) 1 10. x x

( ) 1

23

Numerical derivatives

In many applications derivatives need to be computed numerically. Te term numerical derivativereers to the numerical value o the derivative o a given unction at a given point, provided theunction has a derivative at the given point.

Suppose k is a real number and the unction f is diferentiable at k, then the numerical de-rivative o f at the point k is the value o ` f x ( ) when x k. o nd the numerical derivative o aunction at a given point , rst nd the derivative o the unction, and then evaluate the derivativeat the given point. Proper notation to represent the value o the derivative o a unction f at a point

k includes `

f kdy

dx x k

( ), , and dy

dx k

.

PROBLEM I f x x ( ) , nd ` f ( ).

SOLUTION For f x x f x x ( ) , ( ) ; ` thus, ` f ( ) ( )

PROBLEM I y x , nd

dy

dx x 9

.

SOLUTION For y x y dy

dx x `

, ; thus,

dy

dx x

9

9

6

( )

PROBLEM Findd

dx x ( ) at x 25.

SOLUTIONd

dx x x ( ) ; at x x

6 , ( )

Note the ollowing two special situations:

1. I f x c( ) is a constant unction, then ` f x ( ) , or every real number x ; and

2. I f x mx b( ) is a linear unction, then ` f x m( ) , or every real number x.



Definition of the derivative and derivatives of some simple functions 2

Numerical derivatives o these unctions are illustrated in the ollowing examples:

U I f x ( ) , then ` f ( )

U I y 2x + 5, thendy

dx x

9

4 ·5EXERCISE

Evaluate the ollowing.

1. I x x ( ) , 3 fnd ` ( ).5 6. I x x ( ) , P fnd ` ( ).10

2. I g x ( ) , 100 fnd `g ( ).25 7. I x x

( ) ,1

5fnd ` ( ).2

3. I x x ( ) ,1

4 fnd ` ( ).81 8. I s t t ( ) ,. 0 6 fnd `s ( ).32

4. I y x , fnddy

dx x 49

. 9. I h s s( ) ,4

5 fnd `h ( ).32

5. I t t ( ) , fnd ` ( ).19 10. I y x

1

23, fnd

dy

dx 64

.






23

Constant multiple of a function ruleSuppose f is any diferentiable unction and k is any real number, then kf is alsodiferentiable with its derivative given by

d

dx kf x k

d

dx f x kf x ( ( )) ( ( )) ( ) `

Tus, the derivative o a constant times a diferentiable unction is the prod-

uct o the constant times the derivative o the unction. Tis rule allows you toactor out constants when you are nding a derivative. Te rule applies even whenthe constant is in the denominator as shown here:

d

dx

f x

k

d

dx k f x

k

d

dx f x

( )( ) ( ( ))

¤ ¦ ¥

³ µ ´

¤ ¦ ¥

³ µ ´

k f x `( )

U I f (x ) 5x 2, then ` f x d

dx x x x ( ) ( ) ( )

U I y x 6 , then `

¤ ¦ ¥

³ µ ´

y dy

dx

d

dx x

d

dx x x 6 6 6

x x

Ud

dx x

d

dx x x ( ) ( )

5·1

EXERCISE

For problems 1–10, use the constant multiple o a unction rule to fnd the

derivative o the given unction.

1. x x ( ) 2 3 6. x x

( ) P

P 2

2. g x x

( ) 100

257. x

x ( )

105

3. x x ( ) 201

4 8. s t t ( ) . 100 0 6

4. y x 16 9. h s s( ) 254

5

5. t t

( ) 2

310. x

x ( )

1

4 23

·5·

Rules of differentiation ·5·



24 Differentiation

For problems 11–15, fnd the indicated numerical derivative.

11. ` ( )3 when ( x ) 2 x 3 14.dy

dx 25

when y x 16

12. `g ( )1 when g x x

( ) 100

2515. ` ( )200 when t

t ( )

2

3

13. ` ( )81 when x x ( ) 201

4

Rule for sums and differencesFor all x where both f and g are diferentiable unctions, the unction ( f + g ) is diferentiable withits derivative given by

d

dx f x g x f x g x ( ( ) ( )) ( ) ( ) ` `

Similarly, or all x where both f and g are diferentiable unctions, the unction ( f g ) is di-erentiable with its derivative given by

d

dx f x g x f x g x ( ( ) ( )) ( ) ( ) ` `

Tus, the derivative o the sum (or diference) o two diferentiable unctions is equal to thesum (or diference) o the derivatives o the individual unctions.

U I h x x x ( ) , then ` h x d

dx x

d

dx x x ( ) ( ) ( )

U I y x x x , then ` y d

dx x

d

dx x

d

dx x

d

dx

6 6

x x x x

U

d

dx x x

d

dx x

d

dx x x ( ) ( ) ( )

5·2

EXERCISE

For problems 1–10, use the rule or sums and dierences to fnd the derivative o the given

unction.

1. x x x ( ) 7 102 4. C x x x ( ) 1000 200 40 2

2. h x x ( ) 30 5 2 5. y x

15

25

3. g x x x ( ) 100 540 6. s t t t

( ) 162

3102



Rules of differentiation 2

7. g x x

x ( ) 100

2520 9. q v v v ( )

2

5

3

57 15

8. y x x 12 0 450 2. . 10. x x x

( )

5

2

5

2

5

22 2


11. `¤ ¦ ¥

³ µ ´ h

1

2when h x x ( ) 30 5 2

14. `q ( )32 when q v v v ( ) 2

5

3

57 15

12. `C ( )300 when C x x x ( ) 1000 200 40 2 15. ` ( )6 when x x x

( )

5

2

5

2

5

22 2

13. `s ( )0 when s t t t

( ) 162

3102

Product ruleFor all x where both f and g are diferentiable unctions, the unction ( fg ) is diferentiable with itsderivative given by

d

dx f x g x f x g x g x f x ( ( ) ( )) ( ) ( ) ( ) ( ) ` `

Tus, the derivative o the product o two diferentiable unctions is equal to the rst unc-tion times the derivative o the second unction plus the second unction times the derivative o the rst unction.

U I h x x x ( ) ( )( ),

then ` h x x d

dx x x

d

dx x ( ) ( ) ( ) ( ) ( ) ( )( ) ( )( )x x x

6 6 6 x x x x x

U I y x x x ( )( ),

then ` y x d

dx x x x x

d

dx x ( ) ( ) ( ) ( )

( )( ) ( )( ) 6 x x x x x

( ) ( ) 6 6 x x x x x x

6 x x x x

Notice in the ollowing example that converting to negative and ractional exponents makesdiferentiating easier.



26 Differentiation

U

d

dx x

x x x

d

dx x x ( ) ( )

¤ ¦ ¥

³ µ ´

§

©¨

¶

¸·

x x

d

dx x ( )

( ) ( )x x x x x x

6

x x x x x x

x x x .

You might choose to write answers without negative or ractional exponents.

5·3

EXERCISE

For problems 1–10, use the product rule to fnd the derivative o the given unction.

1. x x x ( ) ( )( ) 2 3 2 32 6. s t t t ( ) ¤ ¦ ¥

³ µ ´

¤ ¦ ¥

³ µ ´ 4

1

25

3

4

2. h x x x x ( ) ( )( ) 4 1 2 53 2 7. g x x x x ( ) ( )( ) 2 2 23 2 3

3. g x x x

( ) ( ) ¤ ¦ ¥ ³ µ ´ 2 5 3 8. x

x x ( ) 10 1

55

3

4. C x x x ( ) ( )( ) 50 20 100 2 9. q v v v ( ) ( )( ) 2 27 5 2

5. y x

x

¤

¦ ¥³

µ ´ 15

25 5( ) 10. x x x ( ) ( )( ) 2 3 33 23


11. ` ( . )1 5 when x x x ( ) ( )( ) 2 3 2 32

12. `g ( )10 when g x x x ( ) ( )

¤

¦ ¥

³

µ ´ 2

5

3

13. `C ( )150 when C x x x ( ) ( )( ) 50 20 100 2

14.dy

dx x 25

when y x

x

¤

¦ ¥³

µ ´ 15

25 5( )

15. ` ( )2 when x x

x ( )

10 1

55

3

Quotient ruleFor all x where both f and g are diferentiable unctions and g x ( ) ,w the unction

f

g

¤ ¦ ¥

³ µ ´ is di-

erentiable with its derivative given by

d

dx

f x

g x

g x f x f x g x

g x

( )

( )

( ) ( ) ( ) ( )

( ( ))

¤ ¦ ¥

³ µ ´

` `

, ( ) g x w




Tus, the derivative o the quotient o two diferentiable unctions is equal to the denomina-tor unction times the derivative o the numerator unction minus the numerator unction timesthe derivative o the denominator unction all divided by the square o the denominator unction,or all real numbers x or which the denominator unction is not equal to zero.

U I h x x

x ( ) ,

then `

h x x

d

dx x x

d

dx x

x ( )

( ) ( ) ( ) ( )

( )

( )( ) ( )( )

( )

9

x x x

x

x x

x x

9

x

x

x

x

U I y x

, then `

y x

d

dx

d

dx x

x

x d

dx x ( ) ( ) ( ) ( )

( )

( )( ) ( )

( )x

( )

x

x x

Ud

dx

x

x

x d

dx x x

d

6

6

¤

¦ ¥

³

µ ´

( )d dx

x

x

x x x x ( )

( )

( ) ( 6

6

6

6

)

( )x

6 6

6

x x x

x x

x 66 6

6

6

x x

x x

x x

x x

9

5·4

EXERCISE

For problems 1–10, use the quotient rule to fnd the derivative o the given unction.

1. x x

x ( )

5 2

3 16. s t

t

t ( )

2 3

4 6

3

2

1

2

2. h x x

x ( )

4 5

8

2

7. g x x

x ( )

100

5 10

3. g x x

( ) 5

8. y x

x

4 5

8 7

3

2

4. x x x

( )

3 12 6

3

2

1

2

9. q v v

v v

( )

3

2

3

21

5. y x

15

10. x x

x

( )

4

48

2

2



28 Differentiation


11. ` ( )25 when x x

x ( )

5 2

3 114.

dy

dx 10

when y x

15

12. `h ( . )0 2 when h x x

x ( )

4 5

8

2

15. `g ( )1 when g x x

x ( )

100

5 10

13. `g ( . )0 25 when g x

x

( ) 5

Chain ruleI y f (u) and u g (x ) are diferentiable unctions o u and x , respectively, then the compositiono f and g , dened by y f ( g (x )), is diferentiable with its derivative given by

dy

dx

dy

du

du

dx

or equivalently,

d

dx f g x f g x g x [ ( ( ))] ( ( )) ( ) ` `

Notice that y f g x ( ( )) is a “unction o a unction o x ”; that is, f ’s argument is the unction

denoted by g x ( ), which itsel is a unction o x. Tus, to ndd

dx f g x [ ( ( ))], you must diferentiate

f with respect to g x ( ) rst, and then multiply the result by the derivative o g x ( ) with respect to x.

Te examples that ollow illustrate the chain rule.

U Find ` y , when y x x x ; let u x x x ,

then ` y dy

dx

dy

du

du

dx

d

duu

d

dx x x x ( ) (

)) ( )

6

u x x

6

6

( ) ( )x x x x x

x x

x x x

U Find ` f x ( ), when f x x ( ) ( ) ; let g x x ( ) ,

then d dx

f g x d dx

x f g x g x [ ( ( ))] [( ) ] ( ( )) ( ) ` `

` 6 ( ( )) ( ) ( ) ( ) g x g x x x x x

U

d

dx x x

d

dx x x x ( ) ( ) ( ) ( )

¤ ¦ ¥

³

µ µ ´ ( )x

x




5·5

EXERCISE

For problems 1–10, use the chain rule to fnd the derivative o the given unction.

1. x x ( ) ( ) 3 102 3 6. y x

1

82 3( )

2. g x x ( ) ( ) 40 3 102 3 7. y x x 2 5 13

3. h x x ( ) ( ) 10 3 102 3 8. s t t t ( ) ( ) 2 531

3

4. h x x ( ) ( ) 3 2 9. x x

( )( )

10

2 6 5

5. uu

u( ) ¤ ¦ ¥

³ µ ´

12

3

10. C t t

( )

50

15 120


11. ` ( )10 when x x ( ) ( ) 3 102 3 14. ` ( )2 when u

u

u( ) ¤

¦

¥³

µ

´ 1

2

3

12. `h ( )3 when h x x ( ) ( ) 10 3 102 3 15.dy

dx 4

when y x

1

82 3( )

13. ` ( )144 when x x ( ) ( ) 3 2

Implicit differentiationTus ar, you’ve seen how to nd the derivative o a unction only i the unction is expressed in what

is called explicit orm. A unction in explicit orm is dened by an equation o the type y f (x ), where y is on one side o the equation and all the terms containing x are on the other side. For example, theunction f dened by y f (x ) x 3 + 5 is expressed in explicit orm. For this unction the variable y is dened explicitly as a unction o the variable x.

On the other hand, or equations in which the variables x and y appear on the same side o theequation, the unction is said to be expressed in implicit orm. For example, the equation x 2 y 1

denes the unction y

x implicitly in terms o x. In this case, the implicit orm o the equa-

tion can be solved or y as a unction o x ; however, or many implicit orms, it is dicult andsometimes impossible to solve or y in terms o x.

Under the assumption thatdy dx

, the derivative o y with respect to x , exists, you can use the

technique o implicit diferentiation to nd dy dx

when a unction is expressed in implicit orm—

regardless o whether you can express the unction in explicit orm. Use the ollowing steps:

1. Diferentiate every term on both sides o the equation with respect to x.

2. Solve the resulting equation ordy dx

.



30 Differentiation

PROBLEM Given the equation x y , use implicit diferentiation to nddy dx

.

SOLUTION Step 1: Diferentiate every term on both sides o the equation with respect to x :

d

dx x y

d

dx ( ) ( )

d

dx x

d

dx y

d

dx ( ) ( ) ( )

6 x y dy dx

Step 2: Solve the resulting equation ordy dx

.

6 y dy

dx x

dy

dx

x

y

6

Note that in this example,dy

dx is expressed in terms o both x and y. o evaluate such a

derivative, you would need to know both x and y at a particular point (x , y ). You can denote the

numerical derivative asdy

dx x y ( , )

.

Te example that ollows illustrates this situation.

dy

dx

x

y

6 at (3, 1) is given by

dy

dx

x

y ( , ) ( , )

( )

( )

6

6

5·6

EXERCISE

For problems 1–10, use implicit dierentiation to fnd dy

dx.

1. x 2 y 1 4.1 1

9 x y

2. xy 3 3 x 2 y + 5 y 5. x 2 + y 2 16

3. x y 25





6.dy

dx ( , )3 1

when x 2 y 1 9.dy

dx ( , )5 10

when1 1

9 x y

7.dy

dx ( , )5 2

when xy 3 3 x 2 y + 5 y 10.dy

dx ( , )2 1

when x 2 + y 2 16

8.

dy

dx ( , )4 9

when x y 25






33

Derivative of the naturalexponential function ex

Exponential unctions are dened by equations o the orm y f x bx ( )( , ),b bw where b is the base o the exponential unction. Te natural expo-nential unction is the exponential unction whose base is the irrational number e.

Te number e is the limit as n approaches innity o

¤

¦ ¥

³

µ ´ n

n

, which is approxi-mately 2.718281828 (to nine decimal places).

Te natural exponential unction is its own derivative; that is, d

dx e ex x ( ) .

Furthermore, by the chain rule, i u is a diferentiable unction o x , then

d

dx e e

du

dx u u( )

U I f x ex ( ) , 6 then ` f x d

dx e ex x ( ) ( )6 6

U I y e2x , then `

y e

d

dx x e ex x x ( ) ( )

U

d

dx e e

d

dx x e x xex x x x ( ) ( ) ( )

6 6

6·1

EXERCISE


1. x e x ( ) 20 6. x x e x ( ) 15 102

2. y e 3 x 7. g x e x x ( ) 7 2 3

3. g x e x ( ) 5 3

8. t e t

( ).

1000 5

4. y e x 4 5 3

9. g t e t ( ) 2500 2 1

5. h x e x ( ) 10 3

10. x e x

( ) 1

2

2

2

P

·6·

Additional derivatives·6·



34 Differentiation

Derivative of the natural logarithmic function lnx

Logarithmic unctions are dened by equations o the orm y f (x ) logbx i and only i

b x x y ( ), where b is the base o the logarithmic unction, (b w 1, b 0). For a given base, thelogarithmic unction is the inverse unction o the corresponding exponential unction, and re-ciprocally. Te logarithmic unction dened by y x

e log , usually denoted ln ,x is the natural

logarithmic unction. It is the inverse unction o the natural exponential unction y ex .Te derivative o the natural logarithmic unction is as ollows:

d

dx x

x (ln )


d

dx u

u

du

dx (ln )

U I f x x ( ) ln , 6 then ` f x d

dx x

x x ( ) (ln )6 6

6

U I y x ln( ), then ` y x

d

dx x

x x

x

6

( ) ( )

U

d

dx x

x

d

dx x

x x (ln ) ( ) ( )

Te above example illustrates that or any nonzero constant k,

d

dx kx

kx

d

dx kx

kx k

x (ln ) ( ) ( )

6·2

EXERCISE


1. x x ( ) ln 20 6. x x x ( ) ln 15 102

2. y x ln 3 7. g x x x ( ) ln( ) 7 2 3

3. g x x ( ) ln( ) 5 3 8. t t t ( ) ln( ) 3 5 202

4. y x 4 5 3ln( ) 9. g t et ( ) ln( )

5. h x x ( ) ln( ) 10 3 10. x x ( ) ln(ln )

Derivatives of exponential functionsfor bases other than e

Suppose b is a positive real number (b w 1) , then

d

dx b b bx x ( ) (ln )



Additional derivatives 3


d

dx b b b

du

dx u u( ) (ln )

U I f x x ( ) ( ) , 6 then ` f x d

dx x x ( ) ( ) (ln )6 6

U I y x

, then ` y d

dx x x x x

(ln ) ( ) (ln ) ( ) (ln )

U

d

dx

d

dx x x x x ( ) (ln ) ( ) (ln ) (( ) (ln ) 6 6

x x x

6·3

EXERCISE


1. x x

( ) ( ) 20 3 6. x x x

( ) ( ) 15 10 52 3

2. y x 53 7. g x x x ( ) 37 2 3

3. g x x ( ) 25 3

8. t t

( ).

100

10 0 5

4. y x 4 25 3

( ) 9. g t t ( ) ( ) 2500 52 1

5. h x x ( ) 4 10 3

10. x x

( ) 8

2

2

Derivatives of logarithmic functionsfor bases other than e

Suppose b is a positive real number (b w 1) , then

d

dx x

b x b(log )

(ln )


d

dx u

b u

du

dx b(log )

(ln )

U I f x x ( ) log , 6

then ` f x d dx

x x x

( ) (log )(ln ) ln

6 6

6

U I y x log ( ),

then ` y x

d

dx x

x x

x

6

(ln )( )

(ln )( )

ln

U

d

dx x

x

d

dx x

x (log )

(ln )( )

(ln )( )

x x ln



36 Differentiation

Te above example illustrates that or any nonzero constant k,

d

dx kx

b kx

d

dx kx

b kx k

b(log )

(ln )( )

(ln )( )

x x bln

6·4

EXERCISE


1. x x ( ) log 204

6. x x x ( ) log 15 102

2

2. y x log10

3 7. g x x x ( ) log ( ) 6

37 2

3. g x x ( ) log ( )8

35 8. t t t ( ) log ( ) 16

23 5 20

4. y x 4 58

3log ( ) 9. g t et ( ) log ( )2

5. h x x ( ) log ( ) 5

310 10. x x ( ) log (log )10 10

Derivatives of trigonometric functionsTe derivatives o the trigonometric unctions are as ollows:

Ud

dx x x (sin ) cos

Ud

dx x x (cos ) sin

Ud

dx

x x (tan ) sec

U

d

dx x x (cot ) csc

U

d

dx x x x (sec ) sec tan

Ud

dx x x x (csc ) csc cot


Ud

dx u u

du

dx (sin ) cos

Ud

dx u u

du

dx (cos ) sin

Ud

dx u u

du

dx (tan ) sec




Ud

dx u u

du

dx (cot ) csc

Ud

dx u u u

du

dx (sec ) (sec tan )

Ud

dx u u u

du

dx (csc ) ( csc cot )

U I h x x ( ) sin , then ` h x x d dx

x x x ( ) (cos ) ( ) (cos )( ) cos

U I y x

¤ ¦ ¥

³ µ ´

cos , then `

¤ ¦ ¥

³ µ ´

¤ ¦ ¥

³ µ ´

¤ ¦ ¥

³ µ ´

§

© y

x d

dx

x x

sin sin¨̈

¶

¸·¤ ¦ ¥

³ µ ´

¤ ¦ ¥

³ µ ´

sin

x

U

d

dx x x

d

dx x

d

dx x (tan cot ) (tan ) (cot ) sec (( ) ( ) csc ( ) ( ) x

d

dx x x

d

dx x

[sec ( )]( ) [csc ( )]( ) sec ( ) csc x x x ( )x

6·5

EXERCISE


1. x x ( ) sin 5 3 6. s t t ( ) cot 4 5

2. h x x ( ) cos( )1

42 2 7. g x

x x ( ) tan

¤ ¦ ¥

³ µ ´ 6

2

3203

3. g x x

( ) tan¤ ¦ ¥

³ µ ´ 5

3

58. x x x x ( ) sin cos 2 2

4. x x ( ) sec10 2 9. h x x

x

( )sin

sin

3

1 3

5. y x 2

32 3sec( ) 10. x e x x ( ) sin 4 2

Derivatives of inverse trigonometric functionsTe derivatives o the inverse trigonometric unctions are as ollows:

Ud

dx

x

x

(sin )

Ud

dx x

x (cos )

Ud

dx x

x (tan )



38 Differentiation

U

d

dx x

x (cot )

Ud

dx x

x x (sec )

| |

Ud

dx x

x x (csc )

| |


Ud

dx u

u

du

dx (sin )

U

d

dx u

u

du

dx (cos )

U

d

dx u

u

du

dx (tan )

U

d

dx u u

du

dx (cot )

Ud

dx u

u u

du

dx (sec )

| |

Ud

dx u

u u

du

dx (csc )

| |

U I h x x ( ) ( ),sin then `

h x x

d

dx x

x x ( )

( )( ) ( )

U I y x

¤

¦ ¥

³

µ ´

cos ,

then `

¤ ¦ ¥

³ µ ´

¤

¦ ¥

³

µ ´

¤

¦ ¥

³

y

x

d

dx

x

x

9

µ µ ´

9

9

x

¤ ¦ ¥

³ µ ´

9

9

x x

U

d

dx x x

d

dx x

d

dx x (tan cot ) (tan ) (cot )

x x

Note: An alternative notation or an inverse trigonometric unction is to prex the original unc-

tion with “arc,” as in “arcsin x ,” which is read “arcsine o x ” or “an angle whose sine is x .” Anadvantage o this notation is that it helps you avoid the common error o conusing the inverse

unction; or example, sin ,x with its reciprocal (sin )sin

.x x




6·6

EXERCISE


1. x x ( ) sin ( ) 1 3 6. x x ( ) cos ( ) 1 2

2. h x e x ( ) cos ( ) 1 7. h x x ( ) csc ( ) 1 2

3. g x x ( ) tan ( ) 1 2 8. g x x

( ) sec¤

¦ ¥

³

µ ´

42

1

4. x x ( ) cot ( ) 1 7 5 9. x x x ( ) sin ( ) 1 27

5. y x 1

1551 3sin ( ) 10. y x arcsin( )1 2

Higher-order derivativesFor a given unction f , higher-order derivatives o f , i they exist, are obtained by diferentiating f successively multiple times. Te derivative ` f is called the rst derivative o f. Te derivative o ` f is called the second derivative o f and is denoted `` f . Similarly, the derivative o `` f is called thethird derivative o f and is denoted ``̀ f , and so on.

Other common notations or higher-order derivatives are the ollowing:

U 1st derivative: ` ` f x y dy

dx D f x

x ( ), , , [ ( )]

U 2nd derivative: `̀ `̀ f x y d y

d x D f x

x ( ), , , [ ( )]

U 3rd derivative: `̀ ` `̀ ` f x y d y

d x D f x

x ( ), , , [ ( )]

U 4th derivative: f x y d y

d x D f x

x

( ) ( )( ), , , [ ( )]

U nth derivative: f x y d y

d x D f x n n

n

n x

n( ) ( )( ), , , [ ( )]

Note: Te nth derivative is also called the nth-order derivative. Tus, the rst derivative is the rst-order derivative; the second derivative, the second-order derivative; the third derivative, thethird-order derivative; and so on.

PROBLEM Find the rst three derivatives o f i f (x ) x 100 40x 5.

SOLUTION ` f x x x ( ) 99

`` f x x x ( ) 99 9

``̀ f x x x ( ) 9 9



6·7

EXERCISE

Find the indicated derivative o the given unction.

1. I x x x ( ) , 7 102 fnd ``` x ( ). 6. I s t t t

( ) , 162

3102 fnd ``s t ( ).

2. I h x x ( ) , 3fnd ``h x ( ). 7. I g x x ( ) ln , 3 fnd D g x

x

3[ ( )].

3. I g x x ( ) , 2 fnd g x ( )( ).5 8. I x

x

x ( ) ,

10

55

3

fnd x ( )( ).4

4. I x e x ( ) , 5 fnd x ( )( ).4 9. I x x ( ) , 32 fnd ``̀ x ( ).

5. I y x sin ,3 fndd y

d x

3

3. 10. I y x log ,

25 fnd

d y

d x

4

4.

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