Cálculo Ejemplos

$y = {\sin (x)}^{\tan (x)}$

Paso 1

Usa las propiedades de los logaritmos para simplificar la diferenciación.

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Paso 1.1

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

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

Paso 1.2

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

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

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

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Paso 2.1

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

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

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 $a$ = $e$ .

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

Paso 2.3

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

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

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

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

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

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Paso 4.1

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

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

Paso 4.2

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

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

Paso 4.3

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

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

Paso 5

Convierte de $\frac{1}{\sin (x)}$ a $\csc (x)$ .

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

Paso 6

La derivada de $\sin (x)$ con respecto a $x$ es $\cos (x)$ .

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

Paso 7

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

$e^{\tan (x) \ln (\sin (x))} (\tan (x) \csc (x) \cos (x) + \ln (\sin (x)) \sec^{2} (x))$

Paso 8

Simplifica.

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Paso 8.1

Aplica la propiedad distributiva.

$e^{\tan (x) \ln (\sin (x))} (\tan (x) \csc (x) \cos (x)) + e^{\tan (x) \ln (\sin (x))} (\ln (\sin (x)) \sec^{2} (x))$

Paso 8.2

Elimina los paréntesis.

$e^{\tan (x) \ln (\sin (x))} \tan (x) \csc (x) \cos (x) + e^{\tan (x) \ln (\sin (x))} \ln (\sin (x)) \sec^{2} (x)$

Paso 8.3

Reordena los términos.

$e^{\tan (x) \ln (\sin (x))} \cos (x) \csc (x) \tan (x) + e^{\tan (x) \ln (\sin (x))} \sec^{2} (x) \ln (\sin (x))$

Paso 8.4

Simplifica cada término.

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Paso 8.4.1

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

$e^{\frac{\sin (x)}{\cos (x)} \ln (\sin (x))} \cos (x) \csc (x) \tan (x) + e^{\tan (x) \ln (\sin (x))} \sec^{2} (x) \ln (\sin (x))$

Paso 8.4.2

Combina $\frac{\sin (x)}{\cos (x)}$ y $\ln (\sin (x))$ .

$e^{\frac{\sin (x) \ln (\sin (x))}{\cos (x)}} \cos (x) \csc (x) \tan (x) + e^{\tan (x) \ln (\sin (x))} \sec^{2} (x) \ln (\sin (x))$

Paso 8.4.3

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

$e^{\frac{\sin (x) \ln (\sin (x))}{\cos (x)}} \cos (x) \frac{1}{\sin (x)} \tan (x) + e^{\tan (x) \ln (\sin (x))} \sec^{2} (x) \ln (\sin (x))$

Paso 8.4.4

Multiplica $e^{\frac{\sin (x) \ln (\sin (x))}{\cos (x)}} \cos (x) \frac{1}{\sin (x)}$ .

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Paso 8.4.4.1

Combina $\frac{1}{\sin (x)}$ y $e^{\frac{\sin (x) \ln (\sin (x))}{\cos (x)}}$ .

$\frac{e^{\frac{\sin (x) \ln (\sin (x))}{\cos (x)}}}{\sin (x)} \cos (x) \tan (x) + e^{\tan (x) \ln (\sin (x))} \sec^{2} (x) \ln (\sin (x))$

Paso 8.4.4.2

Combina $\frac{e^{\frac{\sin (x) \ln (\sin (x))}{\cos (x)}}}{\sin (x)}$ y $\cos (x)$ .

$\frac{e^{\frac{\sin (x) \ln (\sin (x))}{\cos (x)}} \cos (x)}{\sin (x)} \tan (x) + e^{\tan (x) \ln (\sin (x))} \sec^{2} (x) \ln (\sin (x))$

Paso 8.4.5

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

$\frac{e^{\frac{\sin (x) \ln (\sin (x))}{\cos (x)}} \cos (x)}{\sin (x)} \cdot \frac{\sin (x)}{\cos (x)} + e^{\tan (x) \ln (\sin (x))} \sec^{2} (x) \ln (\sin (x))$

Paso 8.4.6

Combinar.

$\frac{e^{\frac{\sin (x) \ln (\sin (x))}{\cos (x)}} \cos (x) \sin (x)}{\sin (x) \cos (x)} + e^{\tan (x) \ln (\sin (x))} \sec^{2} (x) \ln (\sin (x))$

Paso 8.4.7

Cancela el factor común de $\cos (x)$ .

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Paso 8.4.7.1

Cancela el factor común.

$\frac{e^{\frac{\sin (x) \ln (\sin (x))}{\cos (x)}} \cos (x) \sin (x)}{\sin (x) \cos (x)} + e^{\tan (x) \ln (\sin (x))} \sec^{2} (x) \ln (\sin (x))$

Paso 8.4.7.2

Reescribe la expresión.

$\frac{e^{\frac{\sin (x) \ln (\sin (x))}{\cos (x)}} \sin (x)}{\sin (x)} + e^{\tan (x) \ln (\sin (x))} \sec^{2} (x) \ln (\sin (x))$

Paso 8.4.8

Cancela el factor común de $\sin (x)$ .

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Paso 8.4.8.1

Cancela el factor común.

$\frac{e^{\frac{\sin (x) \ln (\sin (x))}{\cos (x)}} \sin (x)}{\sin (x)} + e^{\tan (x) \ln (\sin (x))} \sec^{2} (x) \ln (\sin (x))$

Paso 8.4.8.2

Divide $e^{\frac{\sin (x) \ln (\sin (x))}{\cos (x)}}$ por $1$ .

$e^{\frac{\sin (x) \ln (\sin (x))}{\cos (x)}} + e^{\tan (x) \ln (\sin (x))} \sec^{2} (x) \ln (\sin (x))$

Paso 8.4.9

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

$e^{\frac{\sin (x) \ln (\sin (x))}{\cos (x)}} + e^{\frac{\sin (x)}{\cos (x)} \ln (\sin (x))} \sec^{2} (x) \ln (\sin (x))$

Paso 8.4.10

Combina $\frac{\sin (x)}{\cos (x)}$ y $\ln (\sin (x))$ .

$e^{\frac{\sin (x) \ln (\sin (x))}{\cos (x)}} + e^{\frac{\sin (x) \ln (\sin (x))}{\cos (x)}} \sec^{2} (x) \ln (\sin (x))$

Paso 8.4.11

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

$e^{\frac{\sin (x) \ln (\sin (x))}{\cos (x)}} + e^{\frac{\sin (x) \ln (\sin (x))}{\cos (x)}} {(\frac{1}{\cos (x)})}^{2} \ln (\sin (x))$

Paso 8.4.12

Aplica la regla del producto a $\frac{1}{\cos (x)}$ .

$e^{\frac{\sin (x) \ln (\sin (x))}{\cos (x)}} + e^{\frac{\sin (x) \ln (\sin (x))}{\cos (x)}} \frac{1^{2}}{\cos^{2} (x)} \ln (\sin (x))$

Paso 8.4.13

Uno elevado a cualquier potencia es uno.

$e^{\frac{\sin (x) \ln (\sin (x))}{\cos (x)}} + e^{\frac{\sin (x) \ln (\sin (x))}{\cos (x)}} \frac{1}{\cos^{2} (x)} \ln (\sin (x))$

Paso 8.4.14

Combina $e^{\frac{\sin (x) \ln (\sin (x))}{\cos (x)}}$ y $\frac{1}{\cos^{2} (x)}$ .

$e^{\frac{\sin (x) \ln (\sin (x))}{\cos (x)}} + \frac{e^{\frac{\sin (x) \ln (\sin (x))}{\cos (x)}}}{\cos^{2} (x)} \ln (\sin (x))$

Paso 8.4.15

Combina $\frac{e^{\frac{\sin (x) \ln (\sin (x))}{\cos (x)}}}{\cos^{2} (x)}$ y $\ln (\sin (x))$ .

$e^{\frac{\sin (x) \ln (\sin (x))}{\cos (x)}} + \frac{e^{\frac{\sin (x) \ln (\sin (x))}{\cos (x)}} \ln (\sin (x))}{\cos^{2} (x)}$

Paso 8.5

Simplifica cada término.

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Paso 8.5.1

Separa las fracciones.

$e^{\frac{\ln (\sin (x))}{1} \cdot \frac{\sin (x)}{\cos (x)}} + \frac{e^{\frac{\sin (x) \ln (\sin (x))}{\cos (x)}} \ln (\sin (x))}{\cos^{2} (x)}$

Paso 8.5.2

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

$e^{\frac{\ln (\sin (x))}{1} \tan (x)} + \frac{e^{\frac{\sin (x) \ln (\sin (x))}{\cos (x)}} \ln (\sin (x))}{\cos^{2} (x)}$

Paso 8.5.3

Divide $\ln (\sin (x))$ por $1$ .

$e^{\ln (\sin (x)) \tan (x)} + \frac{e^{\frac{\sin (x) \ln (\sin (x))}{\cos (x)}} \ln (\sin (x))}{\cos^{2} (x)}$

Paso 8.5.4

Simplifica el numerador.

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Paso 8.5.4.1

Separa las fracciones.

$e^{\ln (\sin (x)) \tan (x)} + \frac{e^{\frac{\ln (\sin (x))}{1} \cdot \frac{\sin (x)}{\cos (x)}} \ln (\sin (x))}{\cos^{2} (x)}$

Paso 8.5.4.2

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

$e^{\ln (\sin (x)) \tan (x)} + \frac{e^{\frac{\ln (\sin (x))}{1} \tan (x)} \ln (\sin (x))}{\cos^{2} (x)}$

Paso 8.5.4.3

Divide $\ln (\sin (x))$ por $1$ .

$e^{\ln (\sin (x)) \tan (x)} + \frac{e^{\ln (\sin (x)) \tan (x)} \ln (\sin (x))}{\cos^{2} (x)}$