Limits And Derivatives Formulas 1g4q3x
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Limits and Derivatives Formulas 1. Limits
Power rule
Properties if lim f ( x) = l and lim g ( x) = m , then x→a
x→a
lim [ f ( x ) ± g ( x)] = l ± m x→a
lim [ f ( x) ⋅ g ( x) ] = l ⋅ m x→a
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
n
x −a = na n −1 x→a 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 − 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
( ) ( )
)
www.mathportal.org 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)
n
Leibniz’s 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 m−n ( 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
x
mx
x
n
n
x
n
n
mx
n
a
( sin x )( )
nπ = sin x + 2
( cos x )( )
nπ = cos x + 2
n
n
n ( n − 1) 1⋅ 2
f(
n − 2)
g ′′ + ... + fg (
n)
Limits and Derivatives Formulas 1. Limits
Power rule
Properties if lim f ( x) = l and lim g ( x) = m , then x→a
x→a
lim [ f ( x ) ± g ( x)] = l ± m x→a
lim [ f ( x) ⋅ g ( x) ] = l ⋅ m x→a
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
n
x −a = na n −1 x→a 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 − 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
( ) ( )
)
www.mathportal.org 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)
n
Leibniz’s 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 m−n ( 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
x
mx
x
n
n
x
n
n
mx
n
a
( sin x )( )
nπ = sin x + 2
( cos x )( )
nπ = cos x + 2
n
n
n ( n − 1) 1⋅ 2
f(
n − 2)
g ′′ + ... + fg (
n)
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