Cálculo Ejemplos

${\cot (x)}^{\tan (x)}$

Paso 1

Usa las propiedades de los logaritmos para simplificar la diferenciación.

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Paso 1.1

Reescribe ${\cot (x)}^{\tan (x)}$ como $e^{\ln ({\cot (x)}^{\tan (x)})}$ .

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

Paso 1.2

Expande $\ln ({\cot (x)}^{\tan (x)})$ ; para ello, mueve $\tan (x)$ fuera del logaritmo.

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

Paso 2

Diferencia con la regla de la cadena, que establece que $\frac{d}{d x} [f (g (x))]$ es $f' (g (x)) g' (x)$ donde $f (x) = e^{x}$ y $g (x) = \tan (x) \ln (\cot (x))$ .

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Paso 2.1

Para aplicar la regla de la cadena, establece $u_{1}$ como $\tan (x) \ln (\cot (x))$ .

$\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [\tan (x) \ln (\cot (x))]$

Paso 2.2

Diferencia con la regla exponencial, que establece que $\frac{d}{d u_{1}} [a^{u_{1}}]$ es $a^{u_{1}} \ln (a)$ donde $a$ = $e$ .

$e^{u_{1}} \frac{d}{d x} [\tan (x) \ln (\cot (x))]$

Paso 2.3

Reemplaza todos los casos de $u_{1}$ con $\tan (x) \ln (\cot (x))$ .

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

Paso 3

Diferencia con la regla del producto, que establece que $\frac{d}{d x} [f (x) g (x)]$ es $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ donde $f (x) = \tan (x)$ y $g (x) = \ln (\cot (x))$ .

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

Paso 4

Diferencia con la regla de la cadena, que establece que $\frac{d}{d x} [f (g (x))]$ es $f' (g (x)) g' (x)$ donde $f (x) = \ln (x)$ y $g (x) = \cot (x)$ .

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Paso 4.1

Para aplicar la regla de la cadena, establece $u_{2}$ como $\cot (x)$ .

$e^{\tan (x) \ln (\cot (x))} (\tan (x) (\frac{d}{d u_{2}} [\ln (u_{2})] \frac{d}{d x} [\cot (x)]) + \ln (\cot (x)) \frac{d}{d x} [\tan (x)])$

Paso 4.2

La derivada de $\ln (u_{2})$ con respecto a $u_{2}$ es $\frac{1}{u_{2}}$ .

$e^{\tan (x) \ln (\cot (x))} (\tan (x) (\frac{1}{u_{2}} \frac{d}{d x} [\cot (x)]) + \ln (\cot (x)) \frac{d}{d x} [\tan (x)])$

Paso 4.3

Reemplaza todos los casos de $u_{2}$ con $\cot (x)$ .

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

Paso 5

Reescribe $\cot (x)$ en términos de senos y cosenos.

$e^{\tan (x) \ln (\cot (x))} (\tan (x) (\frac{1}{\frac{\cos (x)}{\sin (x)}} \frac{d}{d x} [\cot (x)]) + \ln (\cot (x)) \frac{d}{d x} [\tan (x)])$

Paso 6

Multiplica por la recíproca de la fracción para dividir por $\frac{\cos (x)}{\sin (x)}$ .

$e^{\tan (x) \ln (\cot (x))} (\tan (x) (\frac{\sin (x)}{\cos (x)} \frac{d}{d x} [\cot (x)]) + \ln (\cot (x)) \frac{d}{d x} [\tan (x)])$

Paso 7

Convierte de $\frac{\sin (x)}{\cos (x)}$ a $\tan (x)$ .

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

Paso 8

Eleva $\tan (x)$ a la potencia de $1$ .

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

Paso 9

Eleva $\tan (x)$ a la potencia de $1$ .

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

Paso 10

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

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

Paso 11

Suma $1$ y $1$ .

$e^{\tan (x) \ln (\cot (x))} (\tan^{2} (x) \frac{d}{d x} [\cot (x)] + \ln (\cot (x)) \frac{d}{d x} [\tan (x)])$

Paso 12

La derivada de $\cot (x)$ con respecto a $x$ es $- \csc^{2} (x)$ .

$e^{\tan (x) \ln (\cot (x))} (\tan^{2} (x) (- \csc^{2} (x)) + \ln (\cot (x)) \frac{d}{d x} [\tan (x)])$

Paso 13

La derivada de $\tan (x)$ con respecto a $x$ es $\sec^{2} (x)$ .

$e^{\tan (x) \ln (\cot (x))} (\tan^{2} (x) (- \csc^{2} (x)) + \ln (\cot (x)) \sec^{2} (x))$

Paso 14

Simplifica.

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Paso 14.1

Aplica la propiedad distributiva.

$e^{\tan (x) \ln (\cot (x))} (\tan^{2} (x) (- \csc^{2} (x))) + e^{\tan (x) \ln (\cot (x))} (\ln (\cot (x)) \sec^{2} (x))$

Paso 14.2

Elimina los paréntesis.

$e^{\tan (x) \ln (\cot (x))} \tan^{2} (x) (- \csc^{2} (x)) + e^{\tan (x) \ln (\cot (x))} \ln (\cot (x)) \sec^{2} (x)$

Paso 14.3

Reordena los términos.

$- e^{\tan (x) \ln (\cot (x))} \csc^{2} (x) \tan^{2} (x) + e^{\tan (x) \ln (\cot (x))} \sec^{2} (x) \ln (\cot (x))$