Limits and Derivatives Formulas
Limits and Derivatives Formulas
Limits and Derivatives Formulas
org
Power rule
Properties
if lim f ( x) = l and lim g ( x) = m , then
xa
xa
lim [ f ( x ) g ( x)] = l m
xa
lim [ f ( x) g ( x) ] = l m
xa
lim c f ( x) = c l
x a
x a
( )
Chain rule
d
f ( g ( x ) ) = f ( g ( x) ) g ( x)
dx
1
1
= where l 0
f ( x) l
Formulas
n
1
lim 1 + = e
x
n
1
lim (1 + n ) n = e
x
sin x
lim
=1
x 0
x
tan x
lim
=1
x 0
x
cos x 1
lim
=0
x 0
x
n
x a
= na n 1
xa x a
lim
an 1
lim
= ln a
x 0
x
d
(c) = 0
dx
d
( x) = 1
dx
d
( sin x ) = cos x
dx
d
( cos x ) = sin x
dx
d
1
( tan x ) = 2 = sec2 x
dx
cos x
d
( sec x ) = sec x tan x
dx
d
( csc x ) = csc cot x
dx
d
1
( cot x ) = 2 = csc2 x
dx
sin x
d
1
sin 1 x =
dx
1 x2
2. Common Derivatives
Basic Properties and Formulas
( cf ) = cf ( x)
g ) = f ( x) + g ( x)
Product rule
( f g ) =
f g + f g
Quotient rule
f g f g
=
g2
d
1
cos 1 x =
dx
1 x2
d
1
tan 1 x =
dx
1 + x2
d x
a = a x ln a
dx
d x
e = ex
dx
d
1
( ln x ) = , x > 0
dx
x
d
1
ln x ) = , x 0
(
dx
x
d
1
, x>0
( log a x ) =
dx
x ln a
(f
Common Derivatives
f ( x) l
=
where m 0
lim
x a g ( x)
m
lim
d n
x = nx n 1
dx
( )
( )
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3. Higher-order Derivatives
Definitions and properties
Second derivative
d dy d 2 y
dx dx dx 2
f =
Higher-Order derivative
f( ) = f(
n
n 1)
( n)
= f ( ) + g(
n
n)
(n)
= f ( ) g(
n)
(f
+ g)
(f
g)
Leibnizs Formulas
( f g ) =
f g + 2 f g + f .g
( f g ) =
f g + 3 f g + 3 f g + f g
( f g )(
n)
= f ( ) g + nf (
n
n 1)
g+
Important Formulas
(x )
m
(x )
n
(n)
(n)
m!
x mn
( m n )!
= n!
n 1
( log a x )
(n)
( 1) ( n 1)!
x n ln a
n 1
( ln x )
(n)
( n)
( 1) ( n 1)!
xn
( a ) = a ln a
( )
(e ) = e
( )
( a ) = m a ln
x
mx
mx
( sin x )( )
= sin x +
( cos x )( )
= cos x +
n ( n 1)
1 2
f(
n 2)
g + ... + fg (
n)