Cálculo Exemplos

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

Etapa 1

Use as propriedades dos logaritmos para simplificar a diferenciação.

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Etapa 1.1

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

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

Etapa 1.2

Expanda $\ln ({\sin (x)}^{\tan (x)})$ movendo $\tan (x)$ para fora do logaritmo.

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

Etapa 2

Diferencie usando a regra da cadeia, que determina que $\frac{d}{d x} [f (g (x))]$ é $f' (g (x)) g' (x)$ , em que $f (x) = e^{x}$ e $g (x) = \tan (x) \ln (\sin (x))$ .

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Etapa 2.1

Para aplicar a regra da cadeia, defina $u_{1}$ como $\tan (x) \ln (\sin (x))$ .

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

Etapa 2.2

Diferencie usando a regra exponencial, que determina que $\frac{d}{d u_{1}} [a^{u_{1}}]$ é $a^{u_{1}} \ln (a)$ , em que $a$ = $e$ .

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

Etapa 2.3

Substitua todas as ocorrências de $u_{1}$ por $\tan (x) \ln (\sin (x))$ .

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

Etapa 3

Diferencie usando a regra do produto, que determina que $\frac{d}{d x} [f (x) g (x)]$ é $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ , em que $f (x) = \tan (x)$ e $g (x) = \ln (\sin (x))$ .

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

Etapa 4

Diferencie usando a regra da cadeia, que determina que $\frac{d}{d x} [f (g (x))]$ é $f' (g (x)) g' (x)$ , em que $f (x) = \ln (x)$ e $g (x) = \sin (x)$ .

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Etapa 4.1

Para aplicar a regra da cadeia, defina $u_{2}$ como $\sin (x)$ .

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

Etapa 4.2

A derivada de $\ln (u_{2})$ em relação a $u_{2}$ é $\frac{1}{u_{2}}$ .

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

Etapa 4.3

Substitua todas as ocorrências de $u_{2}$ por $\sin (x)$ .

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

Etapa 5

Converta de $\frac{1}{\sin (x)}$ em $\csc (x)$ .

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

Etapa 6

A derivada de $\sin (x)$ em relação a $x$ é $\cos (x)$ .

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

Etapa 7

A derivada de $\tan (x)$ em relação a $x$ é $\sec^{2} (x)$ .

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

Etapa 8

Simplifique.

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Etapa 8.1

Aplique a propriedade distributiva.

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

Etapa 8.2

Remova os parênteses.

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

Etapa 8.3

Reordene os termos.

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

Etapa 8.4

Simplifique cada termo.

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Etapa 8.4.1

Reescreva $\tan (x)$ em termos de senos e cossenos.

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

Etapa 8.4.2

Combine $\frac{\sin (x)}{\cos (x)}$ e $\ln (\sin (x))$ .

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

Etapa 8.4.3

Reescreva $\csc (x)$ em termos de senos e cossenos.

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

Etapa 8.4.4

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

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Etapa 8.4.4.1

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

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

Etapa 8.4.4.2

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

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

Etapa 8.4.5

Reescreva $\tan (x)$ em termos de senos e cossenos.

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

Etapa 8.4.6

Combine.

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

Etapa 8.4.7

Cancele o fator comum de $\cos (x)$ .

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Etapa 8.4.7.1

Cancele o fator comum.

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

Etapa 8.4.7.2

Reescreva a expressão.

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

Etapa 8.4.8

Cancele o fator comum de $\sin (x)$ .

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Etapa 8.4.8.1

Cancele o fator comum.

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

Etapa 8.4.8.2

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

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

Etapa 8.4.9

Reescreva $\tan (x)$ em termos de senos e cossenos.

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

Etapa 8.4.10

Combine $\frac{\sin (x)}{\cos (x)}$ e $\ln (\sin (x))$ .

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

Etapa 8.4.11

Reescreva $\sec (x)$ em termos de senos e cossenos.

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

Etapa 8.4.12

Aplique a regra do produto a $\frac{1}{\cos (x)}$ .

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

Etapa 8.4.13

Um elevado a qualquer potência é um.

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

Etapa 8.4.14

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

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

Etapa 8.4.15

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

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

Etapa 8.5

Simplifique cada termo.

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Etapa 8.5.1

Separe as frações.

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

Etapa 8.5.2

Converta de $\frac{\sin (x)}{\cos (x)}$ em $\tan (x)$ .

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

Etapa 8.5.3

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

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

Etapa 8.5.4

Simplifique o numerador.

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Etapa 8.5.4.1

Separe as frações.

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

Etapa 8.5.4.2

Converta de $\frac{\sin (x)}{\cos (x)}$ em $\tan (x)$ .

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

Etapa 8.5.4.3

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

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