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 ·
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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
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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( ) .
Te ollowing examples illustrate the use o this ormula:
U I f x x ( ) , then ` f x ( )
U I y 2x + 5, then ` y
Ud
dx x
¤ ¦ ¥
³ µ ´
4 ·3
EXERCISE
Find the derivative o the given unction.
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( ) .
Te ollowing examples illustrate the use o this ormula:
U I f x x ( ) , then ` f x x ( )
U I y x , then `
y x
Ud
dx x x ( )
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20 Differentiation
4 ·4
EXERCISE
Find the derivative o the given unction.
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.
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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
.
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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·
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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
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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
For problems 11–15, fnd the indicated numerical derivative.
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.
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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
For problems 11–15, fnd the indicated numerical derivative.
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
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Rules of differentiation 2
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
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28 Differentiation
For problems 11–15, fnd the indicated numerical derivative.
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
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Rules of differentiation 2
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
For problems 11–15, fnd the indicated numerical derivative.
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
.
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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
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Rules of differentiation 3
For problems 6–10, fnd the indicated numerical derivative.
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
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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
Find the derivative o the given unction.
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·
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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 )
Furthermore, by the chain rule, i u is a diferentiable unction o x , then
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
Find the derivative o the given unction.
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 )
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Additional derivatives 3
Furthermore, by the chain rule, i u is a diferentiable unction o x , then
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
Find the derivative o the given unction.
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 )
Furthermore, by the chain rule, i u is a diferentiable unction o x , then
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
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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
Find the derivative o the given unction.
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
Furthermore, by the chain rule, i u is a diferentiable unction o x , then
Ud
dx u u
du
dx (sin ) cos
Ud
dx u u
du
dx (cos ) sin
Ud
dx u u
du
dx (tan ) sec
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Additional derivatives 3
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
Find the derivative o the given unction.
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 )
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38 Differentiation
U
d
dx x
x (cot )
Ud
dx x
x x (sec )
| |
Ud
dx x
x x (csc )
| |
Furthermore, by the chain rule, i u is a diferentiable unction o x , then
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
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Additional derivatives 3
6·6
EXERCISE
Find the derivative o the given unction.
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
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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.