Táboa de derivadas e integrais inmediatas.

TABLA DE DERIVADAS E INTEGRAIS
DERIVADAS
f(x)=g(x)+h(x)
f(x)=k.g(x)
f’(x)=g’(x)+h’(x)
f’(x)=k.g’(x)
f(x)=g(x).h(x) f’(x)=g’(x).h(x)+g(x).h’(x)
TIPO
f ( x) =
INTEGRAIS
g ( x)
h( x )
f '( x) =
f ( x ) = g[h( x )]
SIMPLE
g '( x ). h( x) − g ( x ). h'( x )
∫ kf ( x)dx = k ∫ f ( x)dx
∫ [ f ( x) + g( x)]dx = ∫ f ( x)dx + ∫ g( x)dx
[h ( x ) ]
2
f '( x ) = g ' [h( x )]. h'( x)
COMPOSTA
SIMPLE
COMPOSTA
CONSTANTE
f(x)=k
f’(x)=0
∫ kdx = kx + C
IDENTIDADE
f(x)=x
f’(x)=1
∫ dx = x + C
POTENCIAL
EXPONENCIAL
TANXENTE
f '( x ) = e x
f ( x) = e g ( x )
f '( x ) = e g ( x ) . g '( x )
∫ e dx = e
f ( x) = a x
f '( x ) = a x .ln a
f ( x) = a g ( x )
f '( x ) = a g ( x ) .ln a. g '( x )
∫ a x dx =
f ( x ) = ln x
f ' ( x) =
n −1
n
1
x
f ( x ) = ln g ( x )
1
x.ln a
f ( x ) = log a g ( x )
1
g '( x )
. g '( x ) =
g( x)
g( x)
1
g '( x )
f '( x ) =
. g '( x ) =
g ( x ).ln a
g ( x ).ln a
f '( x ) =
f '( x ) = cos x
f ( x ) = seng ( x )
f '( x ) = cos g ( x ). g '( x )
f ( x ) = cos x
f '( x ) = − senx
f ( x ) = cos g ( x )
f ' ( x ) = − seng ( x ). g ' ( x )
f ( x) = tgx
f '( x) =
1
= 1 + tg 2 x
cos2 x
f ( x ) = arc tgx
f '( x ) =
ARCO
SENO
f ( x ) = arc senx
f '( x ) =
f ( x ) = arc cos
f '( x ) =
1
1+ x2
1
1− x2
−1
1− x2
f ( x ) = tg[ g ( x )]
f ( x ) = arc tg[ g ( x )]
f '( x ) =
f '( x ) =
f ( x ) = arc sen[ g ( x )] f '( x ) =
f ( x ) = arc cos[ g ( x )]
Si n ≠ − 1
f '( x ) =
1
g '( x )
. g '( x ) =
cos g ( x )
cos2 g ( x )
2
1
1 + [ g ( x )]
2
1
1 − [ g ( x )]
2
−1
1 − [ g ( x )]
2
. g '( x ) =
. g' ( x) =
. g' ( x) =
x n dx =
x
f ( x ) = senx
ARCO
TANXENTE
ARCO
COSENO
∫
f ( x) = e x
f ( x ) = log a x f ' ( x ) =
COSENO
f '( x ) = n.[ g ( x )] . g '( x )
f '( x ) = n. x n −1
LOGARÍTMICA
SENO
f ( x ) = [ g ( x )]
f ( x) = x n
1
∫ cos
2
1 + [ g ( x )]
1 − [ g ( x )]
− g' ( x)
1 − [ g ( x )]
2
ax
+C
ln a
∫ senxdx = − cos x + C
∫ cos xdx = senx + C
∫
∫
x
dx =
n
n+ 1
∫e
+C
f ( x)
∫
+C
. f '( x )dx = e f ( x ) + C
∫ a f ( x ) . f ' ( x)dx =
1
2
2
∫
∫ x dx = ln x + C
g' ( x)
g '( x )
x
f ( x)
[ f ( x)] . f '( x)dx = [ ]
n +1
x n+1
+C
n+ 1
a f ( x)
+C
ln a
f '( x )
dx = ln f ( x ) + C
f ( x)
∫ senf ( x). f '( x)dx = − cos f ( x) + C
∫ cos f ( x). f '( x)dx = senf ( x) + C
2
∫ (1 + tg x)dx = tgx + C ∫ cos2 f ( x) dx = ∫ (1 + tg [ f ( x)]). f '( x)dx = tg[ f ( x)] + C
2
1
∫ 1 + x 2 dx = arc tgx + C
1
dx = arc senx + C
1− x2
−1
dx = arc cos x + C
1− x2
f '( x)
f ' ( x)
∫ 1 + [ f ( x)] dx = arc tgf ( x) + C
2
∫
∫
f ' ( x)
1 − [ f ( x)]
2
− f '( x )
1 − [ f ( x )]
2
dx = arc senf ( x ) + C
dx = arc cos f ( x ) + C