7/27/2019 Calculus Practice Makes Perfect Ch2 http://slidepdf.com/reader/full/calculus-practice-makes-perfect-ch2 1/26 15 DIFFERENTIATION Diferentiation is the process o determining the derivative o a unction. Part II begins 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 moves on to nding derivatives using standard ormulas or diferentiation o certain basic unction types. Properties o derivatives, numerical derivatives, implicit di- erentiation, and higher-order derivatives are also presented. · II ·
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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.
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:
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:
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.
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
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
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.
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.
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 ( ) ,
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.
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
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
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
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.