Derivadas

Derivadas
Reglas de derivación
[f (x) + g (x)]0 = f 0 (x) + g0 (x)
f y g funciones
[f (x) − g (x)]0 = f 0 (x) − g0 (x)
f y g funciones
0
0
[k · f (x)] = k · f (x)
f y g funciones, k númerof (x) 0
f 0 (x)
[k]= k
f y g funciones, k número
0
0
0
[f (x) · g(x)] = f (x) · g(x) + f (x) · g (x)
f y g funciones
"
#0
0
0
f (x) · g (x) − f (x) · g (x)
f (x)
Cociente
=
f y gfunciones
g (x)
g(x)2
Composición (regla de la cadena) (g ◦ f )0 (x) = (g [f (x)])0 = g 0 [f (x)] · f 0 (x)
f y g funciones
1
Función inversa
(f −1 )0 [f (x)] = 0
f y g funciones
f (x)
SumaResta
Producto de función y número
Cociente de función y número
Producto

Derivadas de funciones usuales
Constante

(k)0 = 0

Potencia

(xn )0 = nxn−1

[f (x)n ]0 = nf (x)n−1 f 0 (x)Exponencial

(ax )0 = ax · log a

[af (x) ]0 = af (x) · f 0 (x) · log a

Exponencial de base e

(ex )0 = ex

(ef (x) )0 = ef (x) · f 0 (x)
f 0 (x)
1
(loga [f (x)])0 =
·
f (x) log a
f 0(x)
(log[f (x)])0 =
f (x)
0
(sin[f (x)]) = cos[f (x)] · f 0 (x)
(cos[f (x)])0 = − sin[f (x)] · f 0 (x)
f 0 (x)
(tan[f (x)])0 =
cos2 [f (x)]
f 0 (x)
(cot[f (x)])0 = − 2
sin [f (x)]
f 0 (x)(arcsin[f (x)])0 = q
1 − f (x)2
f 0 (x)
(arccos[f (x)])0 = − q
1 − f (x)2
f 0 (x)
0
(arctan[f (x)]) =
1 + f (x)2
0
(sinh[f (x)]) = cosh[f (x)] · f 0 (x)
(cosh[f (x)]0 = sinh[f (x)] · f 0 (x)f 0 (x)
0
q
(arg sinh[f (x)]) =
1 + f (x)2
f 0 (x)
(arg cos[f (x)])0 = q
f (x)2 − 1

Logaritmo
Logaritmo neperiano (base e)

(loga x)0 =

1
x

·

(log x)0 =

1
log a
1
x(sin x)0 = cos x
(cos x)0 = − sin x
1
Tangente
(tan x)0 =
cos2 x
1
Cotangente
(cot x)0 = − 2
sin x
1
Arcoseno
(arcsin x)0 = √
1 − x2
1
Arcocoseno
(arccos x)0 = − √
1 − x2
1Arcotangente
(arctan x)0 =
1 + x2
0
Seno hiperbólico
(sinh x) = cosh x
Coseno hiperbólico
(cosh x)0 = sinh x
1
Argumento seno hiperbólico
(arg sinh x)0 = √
1 + x2
1
Argumento coseno hiperbólico...