Derivadas

Tabla de derivadas
Funciones elementales

Funciones compuestas

Función
f(x)
f ( x) = k

Derivada
f '(x)
f '(x) = 0

Función
f(u) con u = u(x)

Derivada
f '(x) = f'(u).u'(x)

f ( x) = x

f '(x) = 1

f ( x) = x p p∈R

f ' ( x) = px p −1

f (u ) = u p p∈R

f ' (u ) = pu p −1u '

f ( x) = ln x

f ' ( x) =

1
x

f ( x) = ln u

f '( x) =

u'
u

f ( x) = log a x

f ' ( x) =

1
x ln a

f ( x) = log a u

f ' ( x) =

u'
u ln a

f ( x) = e x

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) =

f ( x) = arcsen x

f ' ( x) =

f ( x) = arcsen u

f ' ( x) =

f ( x) = arccos x

f ' ( x) =

f ( x) = arccos u

f ' ( x) =

f ( x) = arctg xf ( x) = arctg u

f(x) = sh x

1
1+ x2
f '(x) = ch x

f(x) = sh u

u'
1+ u2
f '(x) = ch u u'

f(x) = ch x

f '(x) = sh x

f(x) = ch u

f '(x) = sh u u'

f ( x) =th x

f ' ( x) =

f ( x) = th u

f ' ( x) =

f ( x) = arg sh x

f ' ( x) =

f ( x) = arg sh u

f ' ( x) =

f ( x) = arg ch x

f ' ( x) =

f ( x) = arg ch u

f ' (x) =

f ( x) = arg th x

1
=
cos 2 x
= 1 + tg 2 x
1
1− x2
−1
1− x

2

f ' ( x) =

1
=
ch 2 x
= 1 − th 2 x

f ' ( x) =

1
1+ x2
1
x2 −1

1
1− x2

f ( x) =arg th u

u'
=
cos 2 u
= (1 + tg 2 u ) u '
u'
1− u2
− u'
1− u2

f ' ( x) =

u'
=
ch 2u
= (1 − th 2 u ) u '

f ' ( x) =

u'
1+ u2
u'
u2 −1

u'
1− u2