BASIC DERIVATIVE RULES
Note: a and k are constants.
Sum Rule:
d
dx
[ f (x) + g(x) ] =
Constant Rule:
d
dx
[ k ! f (x) ] = k ! f "(x)
Power Rule:
d
dx
!" x n #$ = n % x n&1
d
dx
Product Rule:
[ f (x) ! g(x) ] =
d
dx
Chain Rule:
d
dx
Quotient Rule:
f "(x)g(x) + f (x)g"(x)
[ f (g(x)) ] =
! f (x) $ =
"# g(x) %&
f !(x) + g!(x)
f !(g(x)) " g!(x)
f ' (x)g(x)( f (x) g ' (x)
g 2 (x)
Derivatives of Special Functions:
d
dx
(constant) = 0
d
dx
(x) = 1
d
dx
(sin x) = cos x
d
dx
(cos x) = ! sin x
d
dx
(tan x) = sec 2 x
d
dx
(cot x) = ! csc 2 x
d
dx
(sec x) = sec x tan x
d
dx
(csc x) = ! csc x cot x
d
dx
(a x ) = ( ln a ) a x
d
dx
(e x ) = e x
d
dx
(sin !1 x) =
d
dx
(ln x) =
d
dx
(cos!1 x) = !
d
dx
(tan !1 x) =
d
dx
(sec !1 x) =
1
1! x 2
1
1! x 2
1
1+ x 2
1
x
1
x
x 2 !1
BASIC INTEGRAL FORMULAS
Sum Rule:
! [ f (x) + g(x) ] dx = ! f (x)!dx + ! g(x) dx
Constant Rule:
" k ! f (x)dx = k " f (x)dx
Integrals of Special Functions:
!
! (e x ) dx = e x + C
!
(x n )dx =
" ln x dx = x ln x ! x + C
!
( ) dx = ln
! (cos x)dx = sin x + C
" (sin x)dx = ! cos x + C
! (sec2 x)dx = tan x + C
! (sec x tan x)dx = sec x + C
"
(csc x cot x)dx = ! csc x + C
" (csc2 x)dx = ! cot x + C
"
"
dx
1+ x 2
(k)dx = kx + C
dx
1! x 2
= sin !1 x + C
1
x
1
n+1
x n+1 + C
x +C
= tan !1 x + C
x
! (a x )dx = lna a + C
"x
dx
x 2 !1
= sec !1 x + C