TABLA DE DERIVADAS

TABLA DE DERIVADAS
Funciones elementales
Función
f(x)
Funciones compuestas
Derivada
f '(x)
f(x) = k
f '(x) = 0
f(x) = x
f '(x) = 1
f ( x) = x p p∈R
f ' ( x) = px p −1
f ( x) = ln x
f ' ( x) =
f ( x) = log a x
f ' ( x) =
f ( x) = e x
Función
f(u) con u = u(x)
Derivada
f '(x) = f '(u).u'(x)
f (u ) = u p p∈R
f ' (u ) = pu p−1u '
1
x
f ( x) = ln u
f ' ( x) =
u'
u
1
x ln a
f ( x) = log a u
f ' ( x) =
u'
u ln a
f ( x) = e x
f (u ) = e u
f ' (u ) = e u u '
f ( x) = a x
f ( x) = a x ln a
f (u ) = a u
f ' (u ) = a u ln a u '
f ( x) = g ( x) h ( x )
f ( x) = h( x) g ( x) h ( x ) −1 g ' ( x) + g ( x) h ( x ) ln g ( x) h' ( x)
f(x) = sen x
f '(x) = cos x
f(x) = sen u
f '(x) = cos u u'
f(x) = cos x
f '(x) = − sen x
f(x) = cos u
f '(x) = − sen u u'
f ( x) = tg x
f ' ( x) =
f ( x) = tg u
f ' ( x) =
1
=
cos 2 x
= (1 + tg 2 u ) u '
= 1 + tg 2 x
f ( x) = arcsen x
f ' ( x) =
f ( x) = arccos x
f ' ( x) =
1
1− x2
−1
1− x2
f(x) = sh x
f(x) = ch x
1
1+ x2
f '(x) = ch x
f '(x) = sh x
f ( x) = th x
f ' ( x) =
f ( x) = arctg x
f ' ( x) =
1
=
ch 2 x
f ( x) = arcsen u
f ' ( x) =
f ( x) = arccos u
f ' ( x) =
f ' ( x) =
f ( x) = arg ch x
f ' ( x) =
f ( x) = arg th x
f ' ( x) =
1
1+ x2
1
x −1
2
1
1− x2
u'
1− u2
− u'
1− u2
f(x) = sh u
f(x) = ch u
u'
1+ u2
f '(x) = ch u u'
f '(x) = sh u u'
f ( x) = th u
f ' ( x) =
f ( x) = arctg u
= 1 − th 2 x
f ( x) = arg sh x
u'
=
cos 2 u
f ' ( x) =
u'
=
ch 2u
= (1 − th 2 u ) u '
f ( x) = arg sh u
f ' ( x) =
f ( x) = arg ch u
f ' ( x) =
f ( x) = arg th u
f ' ( x) =
u'
1+ u2
u'
u2 −1
u'
1− u2