Machinery's Handbook, 31st Edition
Calculus 131 Note: The power rules when n = –1 and when n = ½ are often stated as their own rules: and ´ = ´ =− = = = ´ = = = − − − − y x y x y x x y x x x x 1 2 1 2 1 1 2 1 2 2 1 2 1 2 1 2 , ,
Logarithmic: For any base a , Exponential: If Trigonometric:
x 1 .
1
= y ´
, then ´ = y
y
then log , x
x
=
if
;
ln
for natural base e , y =
a
ln x a
x
x
x
x
´ =
y e ´ =
y a =
If y a a y e = ln .
,
then
,
then
.
If 2 For a complete list of the differentiation formulas, see Table of Derivatives and Integrals . Derivative Rules.—Just like other functions, derivatives have certain properties: Rule Example f g x f x g x y x x Sum or difference ( ) ( ) ( ) ( ) ± ´ = ´ ± ´ = + − 5 1 9, ( ) ( ) ( ) ( ) ( ) ( ) ln , ´ = ´ = ´ + ´ = ´ − y x fg x f xgx gxfx y x x y x 5 2 4 1 2 Product = + = + ´ = ´ − ´ ( ) 2 2 2 2 1 ln ( ) ln then If x y then If x y y x sin , y x cos , y = ´ = = ´ = − = cos . sin . tan , x sec . x y then ´ = ( ) ( ) x x x x f x g x x g x f x ( ) ( ) y x − − 4 4 9 2 4 8 18 x )( ) ( x x − = − = 2 2 4 9 x 2 Rule Example f g x f x g x ) ( ) ( ) ( ) ± ´ = ´ ± ´ y x x = + − 5 1 9, ( ) ( ) fg x f xgx gxfx ( ) ( ) ( ) ( ) ln , y x x ´ = ´ − y x ´ = ´ + ´ = y x 5 2 4 1 2 = + ´ = ´ ( ) 2 1 ln ( ) ln x x ( ) ( ) f x g x x g x f x ( ) ( ) y x − − x ( = − = + 2 4 9 2
Sum or difference (
Product
Quotient ( ) ( ) ( ) f x g x
( ) ( ) ( ) f x g x − [ ]
2
+
´ = x )( )
x 4 4 9 2 4 8 18 x x − x
,
,
y x x ´ =
y x x
Quotient
x ( )
x ( )
2
2
[
]
2
2
2 2 x 4
4
x
g x
( )
2 2
(
)
(
)
1
1
/
2
/
2
´
´
[
]
[
]
= ´
´
= ´
´
2
2
2
2
Chain rule f u x ( ( ))
f u x
f u ( ( )) ( ) x ( ( )) u x
f u x
u x ( ( )) ( )
y x x = + − = + − 5 1
y x x = + − = + − x x 5 1 5 1 x x 5 1
Chain rule
1 2 5 1 2 5 − x / 2 5 + x
2 5 + x
(
)
(
)
y 1 2 5 1 2 5 − /
2 ´ = + − x x + = x ( )
(
)
2 ´ = + − x x
1 2
1 2
y
+ =
2 ( x 2 + 5 x – 1) 1/2
2 ( x 2 + 5 x – 1) 1/2
Integrals (Antiderivatives) The other fundamental calculus operation, antidifferentiation , also called integration , is the reverse process of differentiation. Whereas derivatives are rate-of-change func- tions, antiderivatives are accumulation functions. We define F to be an antiderivative of f if F ´( x ) = f ( x ). Each function in a family of antiderivatives F ( x ) + C , where c is a constant, has the same derivative f , since ( F + C )´ = F ´ + C ´ = F ´ + 0 = F ´. For this reason, an antiderivative function F ( x ) also is called an indefinite integral. The operation of antidifferentiation is indicated by the notation F x f x dx ( ) ( ) . = ∫ A definite integral indicates the area under f on a closed interval [ a , b ] of its domain (Fig. 2). It is denoted by F b F a f x dx a b ( ) ( ) ( ) , − = ∫ where a and b are the bounds of integra- tion , and F ( a ) and F ( b ) are the antiderivative values of f at these bounds. Just as velocity is the derivative of displacement, so displacement is the antiderivative of velocity. On a time interval [ t 1 , t 2 ], displacement is found by s t s t v t dt t t ( ) ( ) ( ) . 2 1 1 2 − = ∫
Fig. 2. The definite integral on [ a, b ] is the area under the curve from x = a to x = b .
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