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Common Derivatives and Integrals

Common Derivatives and Integrals

Derivatives

Integrals

Basic Properties/Formulas/Rules
d
( cf ( x ) ) = cf ¢ ( x ) , c is any constant. ( f ( x ) ± g ( x ) )¢ = f ¢ ( x ) ± g ¢ ( x )
dx
dn
d
( c ) = 0 , c is any constant.
( x ) = nxn-1 , n is any number.
dx
dx
æ f ö¢ f ¢ g - f g ¢
– (Quotient Rule)
( f g )¢ = f ¢ g + f g ¢ – (Product Rule)ç ÷ =
g2
ègø
d
f ( g ( x ) ) = f ¢ ( g ( x ) ) g ¢ ( x ) (Chain Rule)
dx
g¢ x
d
d g ( x)
gx
e
= g ¢( x ) e ( )
( ln g ( x )) = g (x )
dx
dx
()

(

)

()

Common Derivatives
Polynomials
d
d
(c) = 0
( x) = 1
dx
dx

d
( cx ) = c
dx

Trig Functions
d
( sin x ) = cos x
dx
d
( sec x ) = sec x tan x
dx

d
( cos x ) = - sin x
dx
d
( csc x ) = - csc x cot xdx

d
( tan x ) = sec2 x
dx
d
( cot x ) = - csc2 x
dx

Inverse Trig Functions
d
( sin -1 x ) = 1 2
dx
1- x
d
( sec-1 x ) = 12
dx
x x -1

d
( cos-1 x ) = - 1 2
dx
1- x
d
( csc-1 x ) = - 12
dx
x x -1

d
( tan -1 x ) = 1 +1x2
dx
d
( cot -1 x ) = - 1 +1x2
dx

dn
( x ) = nxn-1
dx

d
( cxn ) = ncxn -1
dx

Basic Properties/Formulas/Rules
ò cf ( x ) dx = c ò f (x ) dx , c is a constant.

ò f ( x ) ± g ( x ) dx = ò f ( x ) dx ± ò g ( x ) dx
ò a f ( x ) dx = F ( x ) a = F (b ) - F ( a ) where F ( x ) = ò f ( x ) dx
b

b

b

b

b

b

b

ò a cf ( x ) dx = c ò a f ( x ) dx , c is a constant. ò a f ( x ) ± g ( x ) dx = ò a f ( x ) dx ± òa g ( x ) dx
a

b

ò a f ( x ) dx = 0
b

a

ò a f ( x ) dx = -òb f ( x ) dx
c

b

b

ò a f( x ) dx = ò a f ( x ) dx + òc f ( x ) dx
If f ( x ) ³ 0 on a £ x £ b then

ò a c dx = c ( b - a )

b

ò a f ( x ) dx ³ 0

If f ( x ) ³ g ( x ) on a £ x £ b then

b

b

ò a f ( x ) dx ³ ò a g ( x ) dx

Common Integrals
Polynomials
1

ò dx = x + c

ò k dx = k x + c

ò x dx = n + 1 x

ó 1 dx = ln x + c
ô
õx

òx

òx

-1

n

dx = ln x + c

ó 1 dx = 1 ln ax + b+ c
ô
õ ax + b
a

òx

p
q

-n

dx =

p
q

+ c, n ¹ -1

1
x - n +1 + c , n ¹ 1
-n + 1

p

dx =

n+1

1 q +1
q
x +c=
x
p+ q
+1

Trig Functions
ò cos u du = sin u + c

p+q
q

+c

ò sin u du = - cos u + c
ò sec u du = tan u + c
ò sec u tan u du = sec u + c ò csc u cot udu = - csc u + c ò csc u du = - cot u + c
ò tan u du = ln sec u + c
ò cot u du = lnsin u + c
1
ò sec u du = ln sec u + tan u + c
ò sec u du = 2 ( sec u tan u + ln sec u + tan u ) + c
2

2

3

Exponential/Logarithm Functions
dx
dx
( a ) = a x ln ( a )
(e ) = ex
dx
dx
d
d
( ln ( x ) ) = 1 , x > 0
( ln x ) = 1 , x ¹ 0
dx
x
dx
x
Hyperbolic Trig Functions
d
( sinh x ) = cosh x
dx
d
( sech x ) = - sech x tanh x
dx

ò csc u du = ln csc u - cot u + c
d1
( log a ( x ) ) = x ln a , x > 0
dx

Visit http://tutorial.math.lamar.edu for a complete set of Calculus I & II notes.

© 2005 Paul

3

u du =

1
( - csc u cot u + ln csc u - cot u ) + c
2

Exponential/Logarithm Functions
u
u
ò e du = e + c

d
d
( cosh x ) = sinh x
( tanh x ) = sech 2 x
dx
dx
d
d
( csch x ) = - csch x coth x
( coth x ) = - csch 2 x
dx
dx

ò cscu
ò a du =

au
+c
ln a

e au
( a sin ( bu ) - b cos ( bu ) ) + c
a + b2
e au
eau cos ( bu ) du = 2
( a cos ( bu ) + b sin ( bu ) ) + c
ò
a + b2

òe

Dawkins

au

sin ( bu ) du =

2

Visit http://tutorial.math.lamar.edu for a complete set of Calculus I & II notes.

ò ln u du = u ln ( u ) - u + c
ò ue du = ( u - 1) e
u

u

+c

ó 1 du = ln ln u + c
ô
õ u ln u
©2005 Paul

Dawkins

Common Derivatives and Integrals

Inverse Trig Functions
1
ó
æu ö
du = sin -1 ç ÷ + c
ô
2
2
èaø
õ a -u

ò sin

1
ó1
æuö
du = tan -1 ç ÷ + c
ô2
2
a
õ a +u
èaø
1
1
ó
æuö
du = sec-1 ç ÷ + c
ô
a
èaø
õ u u2 - a2
Hyperbolic Trig Functions
ò sinh u du = cosh u + c

ò sech tanh u du = - sech u + c
ò tanh u du = ln ( cosh u ) + c

Common...