Cálculo Ejemplos

x^{\sec (x)}

$x^{\sec (x)}$

Paso 1

Usa las propiedades de los logaritmos para simplificar la diferenciación.

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Paso 1.1

Reescribe

x^{\sec (x)}

$x^{\sec (x)}$ como

e^{\ln (x^{\sec (x)})}

$e^{\ln (x^{\sec (x)})}$ .

\frac{d}{d x} [e^{\ln (x^{\sec (x)})}]

$\frac{d}{d x} [e^{\ln (x^{\sec (x)})}]$

Paso 1.2

Expande

\ln (x^{\sec (x)})

$\ln (x^{\sec (x)})$ ; para ello, mueve

\sec (x)

$\sec (x)$ fuera del logaritmo.

\frac{d}{d x} [e^{\sec (x) \ln (x)}]

$\frac{d}{d x} [e^{\sec (x) \ln (x)}]$

\frac{d}{d x} [e^{\sec (x) \ln (x)}]

$\frac{d}{d x} [e^{\sec (x) \ln (x)}]$

Paso 2

Diferencia con la regla de la cadena, que establece que

\frac{d}{d x} [f (g (x))]

$\frac{d}{d x} [f (g (x))]$ es

f' (g (x)) g' (x)

$f' (g (x)) g' (x)$ donde

f (x) = e^{x}

$f (x) = e^{x}$ y

g (x) = \sec (x) \ln (x)

$g (x) = \sec (x) \ln (x)$ .

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Paso 2.1

Para aplicar la regla de la cadena, establece

u

$u$ como

\sec (x) \ln (x)

$\sec (x) \ln (x)$ .

\frac{d}{d u} [e^{u}] \frac{d}{d x} [\sec (x) \ln (x)]

$\frac{d}{d u} [e^{u}] \frac{d}{d x} [\sec (x) \ln (x)]$

Paso 2.2

Diferencia con la regla exponencial, que establece que

\frac{d}{d u} [a^{u}]

$\frac{d}{d u} [a^{u}]$ es

a^{u} \ln (a)

$a^{u} \ln (a)$ donde

a

$a$ =

e

$e$ .

e^{u} \frac{d}{d x} [\sec (x) \ln (x)]

$e^{u} \frac{d}{d x} [\sec (x) \ln (x)]$

Paso 2.3

Reemplaza todos los casos de

u

$u$ con

\sec (x) \ln (x)

$\sec (x) \ln (x)$ .

e^{\sec (x) \ln (x)} \frac{d}{d x} [\sec (x) \ln (x)]

$e^{\sec (x) \ln (x)} \frac{d}{d x} [\sec (x) \ln (x)]$

e^{\sec (x) \ln (x)} \frac{d}{d x} [\sec (x) \ln (x)]

$e^{\sec (x) \ln (x)} \frac{d}{d x} [\sec (x) \ln (x)]$

Paso 3

Diferencia con la regla del producto, que establece que

\frac{d}{d x} [f (x) g (x)]

$\frac{d}{d x} [f (x) g (x)]$ es

f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]

$f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ donde

f (x) = \sec (x)

$f (x) = \sec (x)$ y

g (x) = \ln (x)

$g (x) = \ln (x)$ .

e^{\sec (x) \ln (x)} (\sec (x) \frac{d}{d x} [\ln (x)] + \ln (x) \frac{d}{d x} [\sec (x)])

$e^{\sec (x) \ln (x)} (\sec (x) \frac{d}{d x} [\ln (x)] + \ln (x) \frac{d}{d x} [\sec (x)])$

Paso 4

La derivada de

\ln (x)

$\ln (x)$ con respecto a

x

$x$ es

\frac{1}{x}

$\frac{1}{x}$ .

e^{\sec (x) \ln (x)} (\sec (x) \frac{1}{x} + \ln (x) \frac{d}{d x} [\sec (x)])

$e^{\sec (x) \ln (x)} (\sec (x) \frac{1}{x} + \ln (x) \frac{d}{d x} [\sec (x)])$

Paso 5

Combina

\sec (x)

$\sec (x)$ y

\frac{1}{x}

$\frac{1}{x}$ .

e^{\sec (x) \ln (x)} (\frac{\sec (x)}{x} + \ln (x) \frac{d}{d x} [\sec (x)])

$e^{\sec (x) \ln (x)} (\frac{\sec (x)}{x} + \ln (x) \frac{d}{d x} [\sec (x)])$

Paso 6

La derivada de

\sec (x)

$\sec (x)$ con respecto a

x

$x$ es

\sec (x) \tan (x)

$\sec (x) \tan (x)$ .

e^{\sec (x) \ln (x)} (\frac{\sec (x)}{x} + \ln (x) (\sec (x) \tan (x)))

$e^{\sec (x) \ln (x)} (\frac{\sec (x)}{x} + \ln (x) (\sec (x) \tan (x)))$

Paso 7

Simplifica.

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Paso 7.1

Aplica la propiedad distributiva.

e^{\sec (x) \ln (x)} \frac{\sec (x)}{x} + e^{\sec (x) \ln (x)} (\ln (x) (\sec (x) \tan (x)))

$e^{\sec (x) \ln (x)} \frac{\sec (x)}{x} + e^{\sec (x) \ln (x)} (\ln (x) (\sec (x) \tan (x)))$

Paso 7.2

Combina

e^{\sec (x) \ln (x)}

$e^{\sec (x) \ln (x)}$ y

\frac{\sec (x)}{x}

$\frac{\sec (x)}{x}$ .

\frac{e^{\sec (x) \ln (x)} \sec (x)}{x} + e^{\sec (x) \ln (x)} \ln (x) \sec (x) \tan (x)

$\frac{e^{\sec (x) \ln (x)} \sec (x)}{x} + e^{\sec (x) \ln (x)} \ln (x) \sec (x) \tan (x)$

Paso 7.3

Reordena los términos.

e^{\sec (x) \ln (x)} \sec (x) \tan (x) \ln (x) + \frac{e^{\sec (x) \ln (x)} \sec (x)}{x}

$e^{\sec (x) \ln (x)} \sec (x) \tan (x) \ln (x) + \frac{e^{\sec (x) \ln (x)} \sec (x)}{x}$

e^{\sec (x) \ln (x)} \sec (x) \tan (x) \ln (x) + \frac{e^{\sec (x) \ln (x)} \sec (x)}{x}

$e^{\sec (x) \ln (x)} \sec (x) \tan (x) \ln (x) + \frac{e^{\sec (x) \ln (x)} \sec (x)}{x}$