7/18/2019 Derivative Rules

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DERIVATIVE RULES

( ) 1nd    n x nx

dx

−=   ( )sin cosd 

 x xdx

=   ( )cos sind 

 x xdx

= −  

( ) ln x xd a a

dx

= ⋅ a   ( ) 2tan secd 

 x x

dx

=   ( ) 2cot cscd 

 x x

dx

= −  

( )( ) ( ) ( ) ( ) ( ) ( )d 

 f x g x f x g x g x f xdx

′ ′⋅ = ⋅ + ⋅   ( )sec sec tand 

 x xdx

=   x   ( )csc csc cotd 

 x x xdx

= −  

( )2

( ) ( ) ( ) ( ) ( )

( ) ( )

d f x g x f x f x g x

dx g x   g x

′ ′⎛ ⎞   ⋅ − ⋅=⎜ ⎟

⎝ ⎠  ( )

2

1arcsin

1

d  x

dx   x=

−  ( ) 2

1arctan

1

d  x

dx x=

+ 

( )( ( )) ( ( )) ( )d 

 f g x f g x g xdx

′= ⋅   ′   ( )2

1arcsec

1

d  x

dx   x x=

− 

( )1

lnd 

 xdx x

=   ( )sinh coshd 

 x xdx

=   ( )cosh sinhd 

 x xdx

=  

INTEGRAL RULES

11, 1

1

n n x dx x c n

n

+= ++∫   ≠ −   sin cos xdx x c= − +∫   2csc cot xdx x c= − +∫  

1

ln

 xa dx a c

a=∫

  x +   cos sin xdx x c= +∫   sec tan sec x xdx x c= +∫  

1lndx x c

 x= +∫   2sec tan xdx x c= +∫   csc cot csc x xdx x c= − +∫  

2

arcsin1

dx x c

 x=

−∫  +   sinh cosh xdx x c= +

∫  cosh sinh xdx x c= +

∫ 

2arctan

1

dx x c

 x= +

+∫  

2arcsec

1

dx x c

 x x=

−∫   +