Cálculo Exemplos

$\ln (\sqrt{\frac{1 + \sin (x)}{1 - \sin (x)}})$

Etapa 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ para reescrever $\sqrt{\frac{1 + \sin (x)}{1 - \sin (x)}}$ como ${(\frac{1 + \sin (x)}{1 - \sin (x)})}^{\frac{1}{2}}$ .

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

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 $f (x) = \ln (x)$ e $g (x) = {(\frac{1 + \sin (x)}{1 - \sin (x)})}^{\frac{1}{2}}$ .

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Etapa 2.1

Para aplicar a regra da cadeia, defina $u_{1}$ como ${(\frac{1 + \sin (x)}{1 - \sin (x)})}^{\frac{1}{2}}$ .

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

Etapa 2.2

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

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

Etapa 2.3

Substitua todas as ocorrências de $u_{1}$ por ${(\frac{1 + \sin (x)}{1 - \sin (x)})}^{\frac{1}{2}}$ .

$\frac{1}{{(\frac{1 + \sin (x)}{1 - \sin (x)})}^{\frac{1}{2}}} \frac{d}{d x} [{(\frac{1 + \sin (x)}{1 - \sin (x)})}^{\frac{1}{2}}]$

Etapa 3

Diferencie usando a regra da cadeia, que determina que $\frac{d}{d x} [f (g (x))]$ é $f' (g (x)) g' (x)$ , em que $f (x) = x^{\frac{1}{2}}$ e $g (x) = \frac{1 + \sin (x)}{1 - \sin (x)}$ .

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Etapa 3.1

Para aplicar a regra da cadeia, defina $u_{2}$ como $\frac{1 + \sin (x)}{1 - \sin (x)}$ .

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

Etapa 3.2

Diferencie usando a regra da multiplicação de potências, que determina que $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ é $n {u_{2}}^{n - 1}$ , em que $n = \frac{1}{2}$ .

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

Etapa 3.3

Substitua todas as ocorrências de $u_{2}$ por $\frac{1 + \sin (x)}{1 - \sin (x)}$ .

$\frac{1}{{(\frac{1 + \sin (x)}{1 - \sin (x)})}^{\frac{1}{2}}} (\frac{1}{2} {(\frac{1 + \sin (x)}{1 - \sin (x)})}^{\frac{1}{2} - 1} \frac{d}{d x} [\frac{1 + \sin (x)}{1 - \sin (x)}])$

Etapa 4

Para escrever $- 1$ como fração com um denominador comum, multiplique por $\frac{2}{2}$ .

$\frac{1}{{(\frac{1 + \sin (x)}{1 - \sin (x)})}^{\frac{1}{2}}} (\frac{1}{2} {(\frac{1 + \sin (x)}{1 - \sin (x)})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [\frac{1 + \sin (x)}{1 - \sin (x)}])$

Etapa 5

Combine $- 1$ e $\frac{2}{2}$ .

$\frac{1}{{(\frac{1 + \sin (x)}{1 - \sin (x)})}^{\frac{1}{2}}} (\frac{1}{2} {(\frac{1 + \sin (x)}{1 - \sin (x)})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [\frac{1 + \sin (x)}{1 - \sin (x)}])$

Etapa 6

Combine os numeradores em relação ao denominador comum.

$\frac{1}{{(\frac{1 + \sin (x)}{1 - \sin (x)})}^{\frac{1}{2}}} (\frac{1}{2} {(\frac{1 + \sin (x)}{1 - \sin (x)})}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [\frac{1 + \sin (x)}{1 - \sin (x)}])$

Etapa 7

Simplifique o numerador.

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Etapa 7.1

Multiplique $- 1$ por $2$ .

$\frac{1}{{(\frac{1 + \sin (x)}{1 - \sin (x)})}^{\frac{1}{2}}} (\frac{1}{2} {(\frac{1 + \sin (x)}{1 - \sin (x)})}^{\frac{1 - 2}{2}} \frac{d}{d x} [\frac{1 + \sin (x)}{1 - \sin (x)}])$

Etapa 7.2

Subtraia $2$ de $1$ .

$\frac{1}{{(\frac{1 + \sin (x)}{1 - \sin (x)})}^{\frac{1}{2}}} (\frac{1}{2} {(\frac{1 + \sin (x)}{1 - \sin (x)})}^{\frac{- 1}{2}} \frac{d}{d x} [\frac{1 + \sin (x)}{1 - \sin (x)}])$

Etapa 8

Mova o número negativo para a frente da fração.

$\frac{1}{{(\frac{1 + \sin (x)}{1 - \sin (x)})}^{\frac{1}{2}}} (\frac{1}{2} {(\frac{1 + \sin (x)}{1 - \sin (x)})}^{- \frac{1}{2}} \frac{d}{d x} [\frac{1 + \sin (x)}{1 - \sin (x)}])$

Etapa 9

Diferencie usando a regra do quociente, que determina que $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ é $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ , em que $f (x) = 1 + \sin (x)$ e $g (x) = 1 - \sin (x)$ .

$\frac{1}{{(\frac{1 + \sin (x)}{1 - \sin (x)})}^{\frac{1}{2}}} \cdot \frac{1}{2} {(\frac{1 + \sin (x)}{1 - \sin (x)})}^{- \frac{1}{2}} \frac{(1 - \sin (x)) \frac{d}{d x} [1 + \sin (x)] - (1 + \sin (x)) \frac{d}{d x} [1 - \sin (x)]}{{(1 - \sin (x))}^{2}}$

Etapa 10

Diferencie.

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Etapa 10.1

De acordo com a regra da soma, a derivada de $1 + \sin (x)$ com relação a $x$ é $\frac{d}{d x} [1] + \frac{d}{d x} [\sin (x)]$ .

$\frac{1}{{(\frac{1 + \sin (x)}{1 - \sin (x)})}^{\frac{1}{2}}} \cdot \frac{1}{2} {(\frac{1 + \sin (x)}{1 - \sin (x)})}^{- \frac{1}{2}} \frac{(1 - \sin (x)) (\frac{d}{d x} [1] + \frac{d}{d x} [\sin (x)]) - (1 + \sin (x)) \frac{d}{d x} [1 - \sin (x)]}{{(1 - \sin (x))}^{2}}$

Etapa 10.2

Como $1$ é constante em relação a $x$ , a derivada de $1$ em relação a $x$ é $0$ .

$\frac{1}{{(\frac{1 + \sin (x)}{1 - \sin (x)})}^{\frac{1}{2}}} \cdot \frac{1}{2} {(\frac{1 + \sin (x)}{1 - \sin (x)})}^{- \frac{1}{2}} \frac{(1 - \sin (x)) (0 + \frac{d}{d x} [\sin (x)]) - (1 + \sin (x)) \frac{d}{d x} [1 - \sin (x)]}{{(1 - \sin (x))}^{2}}$

Etapa 10.3

Some $0$ e $\frac{d}{d x} [\sin (x)]$ .

$\frac{1}{{(\frac{1 + \sin (x)}{1 - \sin (x)})}^{\frac{1}{2}}} \cdot \frac{1}{2} {(\frac{1 + \sin (x)}{1 - \sin (x)})}^{- \frac{1}{2}} \frac{(1 - \sin (x)) \frac{d}{d x} [\sin (x)] - (1 + \sin (x)) \frac{d}{d x} [1 - \sin (x)]}{{(1 - \sin (x))}^{2}}$

Etapa 11

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

$\frac{1}{{(\frac{1 + \sin (x)}{1 - \sin (x)})}^{\frac{1}{2}}} \cdot \frac{1}{2} {(\frac{1 + \sin (x)}{1 - \sin (x)})}^{- \frac{1}{2}} \frac{(1 - \sin (x)) \cos (x) - (1 + \sin (x)) \frac{d}{d x} [1 - \sin (x)]}{{(1 - \sin (x))}^{2}}$

Etapa 12

Diferencie.

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Etapa 12.1

De acordo com a regra da soma, a derivada de $1 - \sin (x)$ com relação a $x$ é $\frac{d}{d x} [1] + \frac{d}{d x} [- \sin (x)]$ .

$\frac{1}{{(\frac{1 + \sin (x)}{1 - \sin (x)})}^{\frac{1}{2}}} \cdot \frac{1}{2} {(\frac{1 + \sin (x)}{1 - \sin (x)})}^{- \frac{1}{2}} \frac{(1 - \sin (x)) \cos (x) - (1 + \sin (x)) (\frac{d}{d x} [1] + \frac{d}{d x} [- \sin (x)])}{{(1 - \sin (x))}^{2}}$

Etapa 12.2

Como $1$ é constante em relação a $x$ , a derivada de $1$ em relação a $x$ é $0$ .

$\frac{1}{{(\frac{1 + \sin (x)}{1 - \sin (x)})}^{\frac{1}{2}}} \cdot \frac{1}{2} {(\frac{1 + \sin (x)}{1 - \sin (x)})}^{- \frac{1}{2}} \frac{(1 - \sin (x)) \cos (x) - (1 + \sin (x)) (0 + \frac{d}{d x} [- \sin (x)])}{{(1 - \sin (x))}^{2}}$

Etapa 12.3

Some $0$ e $\frac{d}{d x} [- \sin (x)]$ .

Etapa 12.4

Como $- 1$ é constante em relação a $x$ , a derivada de $- \sin (x)$ em relação a $x$ é $- \frac{d}{d x} [\sin (x)]$ .

$\frac{1}{{(\frac{1 + \sin (x)}{1 - \sin (x)})}^{\frac{1}{2}}} \cdot \frac{1}{2} {(\frac{1 + \sin (x)}{1 - \sin (x)})}^{- \frac{1}{2}} \frac{(1 - \sin (x)) \cos (x) - (1 + \sin (x)) (- \frac{d}{d x} [\sin (x)])}{{(1 - \sin (x))}^{2}}$

Etapa 12.5

Multiplique.

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Etapa 12.5.1

Multiplique $- 1$ por $- 1$ .

$\frac{1}{{(\frac{1 + \sin (x)}{1 - \sin (x)})}^{\frac{1}{2}}} \cdot \frac{1}{2} {(\frac{1 + \sin (x)}{1 - \sin (x)})}^{- \frac{1}{2}} \frac{(1 - \sin (x)) \cos (x) + 1 (1 + \sin (x)) \frac{d}{d x} [\sin (x)]}{{(1 - \sin (x))}^{2}}$

Etapa 12.5.2

Multiplique $1 + \sin (x)$ por $1$ .

$\frac{1}{{(\frac{1 + \sin (x)}{1 - \sin (x)})}^{\frac{1}{2}}} \cdot \frac{1}{2} {(\frac{1 + \sin (x)}{1 - \sin (x)})}^{- \frac{1}{2}} \frac{(1 - \sin (x)) \cos (x) + (1 + \sin (x)) \frac{d}{d x} [\sin (x)]}{{(1 - \sin (x))}^{2}}$

Etapa 13

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

Etapa 14

Combine frações.

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Etapa 14.1

Multiplique $\frac{(1 - \sin (x)) \cos (x) + (1 + \sin (x)) \cos (x)}{{(1 - \sin (x))}^{2}}$ por $\frac{1}{2}$ .

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

Etapa 14.2

Mova $2$ para a esquerda de ${(1 - \sin (x))}^{2}$ .

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

Etapa 15

Simplifique.

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Etapa 15.1

Altere o sinal do expoente reescrevendo a base como seu inverso.

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

Etapa 15.2

Aplique a regra do produto a $\frac{1 + \sin (x)}{1 - \sin (x)}$ .

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

Etapa 15.3

Aplique a regra do produto a $\frac{1 - \sin (x)}{1 + \sin (x)}$ .

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

Etapa 15.4

Aplique a propriedade distributiva.

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

Etapa 15.5

Aplique a propriedade distributiva.

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

Etapa 15.6

Combine os termos.

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Etapa 15.6.1

Multiplique pelo inverso da fração para dividir por $\frac{{(1 + \sin (x))}^{\frac{1}{2}}}{{(1 - \sin (x))}^{\frac{1}{2}}}$ .

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

Etapa 15.6.2

Multiplique $\frac{{(1 - \sin (x))}^{\frac{1}{2}}}{{(1 + \sin (x))}^{\frac{1}{2}}}$ por $1$ .

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

Etapa 15.6.3

Multiplique $\cos (x)$ por $1$ .

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

Etapa 15.6.4

Multiplique $\cos (x)$ por $1$ .

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

Etapa 15.6.5

Some $\cos (x)$ e $\cos (x)$ .

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

Etapa 15.6.6

Some $- \sin (x) \cos (x)$ e $\sin (x) \cos (x)$ .

$\frac{{(1 - \sin (x))}^{\frac{1}{2}}}{{(1 + \sin (x))}^{\frac{1}{2}}} \cdot \frac{0 + 2 \cos (x)}{2 {(1 - \sin (x))}^{2}} \frac{{(1 - \sin (x))}^{\frac{1}{2}}}{{(1 + \sin (x))}^{\frac{1}{2}}}$

Etapa 15.6.7

Some $0$ e $2 \cos (x)$ .

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

Etapa 15.6.8

Cancele o fator comum de $2$ .

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Etapa 15.6.8.1

Cancele o fator comum.

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

Etapa 15.6.8.2

Reescreva a expressão.

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

Etapa 15.6.9

Multiplique $\frac{{(1 - \sin (x))}^{\frac{1}{2}}}{{(1 + \sin (x))}^{\frac{1}{2}}}$ por $\frac{\cos (x)}{{(1 - \sin (x))}^{2}}$ .

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

Etapa 15.6.10

Mova ${(1 - \sin (x))}^{\frac{1}{2}}$ para o denominador usando a regra do expoente negativo $b^{n} = \frac{1}{b^{- n}}$ .

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

Etapa 15.6.11

Multiplique ${(1 - \sin (x))}^{2}$ por ${(1 - \sin (x))}^{- \frac{1}{2}}$ somando os expoentes.

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Etapa 15.6.11.1

Mova ${(1 - \sin (x))}^{- \frac{1}{2}}$ .

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

Etapa 15.6.11.2

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

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

Etapa 15.6.11.3

Para escrever $2$ como fração com um denominador comum, multiplique por $\frac{2}{2}$ .

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

Etapa 15.6.11.4

Combine $2$ e $\frac{2}{2}$ .

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

Etapa 15.6.11.5

Combine os numeradores em relação ao denominador comum.

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

Etapa 15.6.11.6

Simplifique o numerador.

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Etapa 15.6.11.6.1

Multiplique $2$ por $2$ .

$\frac{\cos (x)}{{(1 + \sin (x))}^{\frac{1}{2}} {(1 - \sin (x))}^{\frac{- 1 + 4}{2}}} \cdot \frac{{(1 - \sin (x))}^{\frac{1}{2}}}{{(1 + \sin (x))}^{\frac{1}{2}}}$

Etapa 15.6.11.6.2

Some $- 1$ e $4$ .

$\frac{\cos (x)}{{(1 + \sin (x))}^{\frac{1}{2}} {(1 - \sin (x))}^{\frac{3}{2}}} \cdot \frac{{(1 - \sin (x))}^{\frac{1}{2}}}{{(1 + \sin (x))}^{\frac{1}{2}}}$

Etapa 15.6.12

Multiplique $\frac{\cos (x)}{{(1 + \sin (x))}^{\frac{1}{2}} {(1 - \sin (x))}^{\frac{3}{2}}}$ por $\frac{{(1 - \sin (x))}^{\frac{1}{2}}}{{(1 + \sin (x))}^{\frac{1}{2}}}$ .

$\frac{\cos (x) {(1 - \sin (x))}^{\frac{1}{2}}}{{(1 + \sin (x))}^{\frac{1}{2}} {(1 - \sin (x))}^{\frac{3}{2}} {(1 + \sin (x))}^{\frac{1}{2}}}$

Etapa 15.6.13

Multiplique ${(1 + \sin (x))}^{\frac{1}{2}}$ por ${(1 + \sin (x))}^{\frac{1}{2}}$ somando os expoentes.

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Etapa 15.6.13.1

Mova ${(1 + \sin (x))}^{\frac{1}{2}}$ .

$\frac{\cos (x) {(1 - \sin (x))}^{\frac{1}{2}}}{{(1 + \sin (x))}^{\frac{1}{2}} {(1 + \sin (x))}^{\frac{1}{2}} {(1 - \sin (x))}^{\frac{3}{2}}}$

Etapa 15.6.13.2

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

$\frac{\cos (x) {(1 - \sin (x))}^{\frac{1}{2}}}{{(1 + \sin (x))}^{\frac{1}{2} + \frac{1}{2}} {(1 - \sin (x))}^{\frac{3}{2}}}$

Etapa 15.6.13.3

Combine os numeradores em relação ao denominador comum.

$\frac{\cos (x) {(1 - \sin (x))}^{\frac{1}{2}}}{{(1 + \sin (x))}^{\frac{1 + 1}{2}} {(1 - \sin (x))}^{\frac{3}{2}}}$

Etapa 15.6.13.4

Some $1$ e $1$ .

$\frac{\cos (x) {(1 - \sin (x))}^{\frac{1}{2}}}{{(1 + \sin (x))}^{\frac{2}{2}} {(1 - \sin (x))}^{\frac{3}{2}}}$

Etapa 15.6.13.5

Divida $2$ por $2$ .

$\frac{\cos (x) {(1 - \sin (x))}^{\frac{1}{2}}}{{(1 + \sin (x))}^{1} {(1 - \sin (x))}^{\frac{3}{2}}}$

Etapa 15.6.14

Simplifique ${(1 + \sin (x))}^{1} {(1 - \sin (x))}^{\frac{3}{2}}$ .

$\frac{\cos (x) {(1 - \sin (x))}^{\frac{1}{2}}}{(1 + \sin (x)) {(1 - \sin (x))}^{\frac{3}{2}}}$

Etapa 15.6.15

Mova ${(1 - \sin (x))}^{\frac{1}{2}}$ para o denominador usando a regra do expoente negativo $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{\cos (x)}{(1 + \sin (x)) {(1 - \sin (x))}^{\frac{3}{2}} {(1 - \sin (x))}^{- \frac{1}{2}}}$

Etapa 15.6.16

Simplifique o denominador.

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Etapa 15.6.16.1

Multiplique ${(1 - \sin (x))}^{\frac{3}{2}}$ por ${(1 - \sin (x))}^{- \frac{1}{2}}$ somando os expoentes.

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Etapa 15.6.16.1.1

Mova ${(1 - \sin (x))}^{- \frac{1}{2}}$ .

$\frac{\cos (x)}{(1 + \sin (x)) ({(1 - \sin (x))}^{- \frac{1}{2}} {(1 - \sin (x))}^{\frac{3}{2}})}$

Etapa 15.6.16.1.2

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

$\frac{\cos (x)}{(1 + \sin (x)) {(1 - \sin (x))}^{- \frac{1}{2} + \frac{3}{2}}}$

Etapa 15.6.16.1.3

Combine os numeradores em relação ao denominador comum.

$\frac{\cos (x)}{(1 + \sin (x)) {(1 - \sin (x))}^{\frac{- 1 + 3}{2}}}$

Etapa 15.6.16.1.4

Some $- 1$ e $3$ .

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

Etapa 15.6.16.1.5

Divida $2$ por $2$ .

$\frac{\cos (x)}{(1 + \sin (x)) {(1 - \sin (x))}^{1}}$

Etapa 15.6.16.2

Simplifique $(1 + \sin (x)) {(1 - \sin (x))}^{1}$ .

$\frac{\cos (x)}{(1 + \sin (x)) (1 - \sin (x))}$