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