Cálculo Exemplos

$- \cos (x) \ln (\sec (x) + \tan (x))$

Etapa 1

Como $- 1$ é constante em relação a $x$ , a derivada de $- \cos (x) \ln (\sec (x) + \tan (x))$ em relação a $x$ é $- \frac{d}{d x} [\cos (x) \ln (\sec (x) + \tan (x))]$ .

$- \frac{d}{d x} [\cos (x) \ln (\sec (x) + \tan (x))]$

Etapa 2

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 $f (x) = \cos (x)$ e $g (x) = \ln (\sec (x) + \tan (x))$ .

$- (\cos (x) \frac{d}{d x} [\ln (\sec (x) + \tan (x))] + \ln (\sec (x) + \tan (x)) \frac{d}{d x} [\cos (x)])$

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 $f (x) = \ln (x)$ e $g (x) = \sec (x) + \tan (x)$ .

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Etapa 3.1

Para aplicar a regra da cadeia, defina $u$ como $\sec (x) + \tan (x)$ .

$- (\cos (x) (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [\sec (x) + \tan (x)]) + \ln (\sec (x) + \tan (x)) \frac{d}{d x} [\cos (x)])$

Etapa 3.2

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

$- (\cos (x) (\frac{1}{u} \frac{d}{d x} [\sec (x) + \tan (x)]) + \ln (\sec (x) + \tan (x)) \frac{d}{d x} [\cos (x)])$

Etapa 3.3

Substitua todas as ocorrências de $u$ por $\sec (x) + \tan (x)$ .

$- (\cos (x) (\frac{1}{\sec (x) + \tan (x)} \frac{d}{d x} [\sec (x) + \tan (x)]) + \ln (\sec (x) + \tan (x)) \frac{d}{d x} [\cos (x)])$

Etapa 4

Diferencie usando a regra da soma.

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Etapa 4.1

Combine $\frac{1}{\sec (x) + \tan (x)}$ e $\cos (x)$ .

$- (\frac{\cos (x)}{\sec (x) + \tan (x)} \frac{d}{d x} [\sec (x) + \tan (x)] + \ln (\sec (x) + \tan (x)) \frac{d}{d x} [\cos (x)])$

Etapa 4.2

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

$- (\frac{\cos (x)}{\sec (x) + \tan (x)} (\frac{d}{d x} [\sec (x)] + \frac{d}{d x} [\tan (x)]) + \ln (\sec (x) + \tan (x)) \frac{d}{d x} [\cos (x)])$

Etapa 5

A derivada de $\sec (x)$ em relação a $x$ é $\sec (x) \tan (x)$ .

$- (\frac{\cos (x)}{\sec (x) + \tan (x)} (\sec (x) \tan (x) + \frac{d}{d x} [\tan (x)]) + \ln (\sec (x) + \tan (x)) \frac{d}{d x} [\cos (x)])$

Etapa 6

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

$- (\frac{\cos (x)}{\sec (x) + \tan (x)} (\sec (x) \tan (x) + \sec^{2} (x)) + \ln (\sec (x) + \tan (x)) \frac{d}{d x} [\cos (x)])$

Etapa 7

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

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

Etapa 8

Simplifique.

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Etapa 8.1

Aplique a propriedade distributiva.

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

Etapa 8.2

Combine os termos.

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Etapa 8.2.1

Multiplique $- 1$ por $- 1$ .

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

Etapa 8.2.2

Multiplique $\ln (\sec (x) + \tan (x))$ por $1$ .

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

Etapa 8.3

Reordene os termos.

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

Etapa 8.4

Simplifique cada termo.

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Etapa 8.4.1

Simplifique cada termo.

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Etapa 8.4.1.1

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

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

Etapa 8.4.1.2

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

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

Etapa 8.4.1.3

Multiplique $\frac{1}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$ .

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Etapa 8.4.1.3.1

Multiplique $\frac{1}{\cos (x)}$ por $\frac{\sin (x)}{\cos (x)}$ .

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

Etapa 8.4.1.3.2

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 8.4.1.3.3

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 8.4.1.3.4

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

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

Etapa 8.4.1.3.5

Some $1$ e $1$ .

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

Etapa 8.4.1.4

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

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

Etapa 8.4.1.5

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

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

Etapa 8.4.1.6

Um elevado a qualquer potência é um.

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

Etapa 8.4.2

Aplique a propriedade distributiva.

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

Etapa 8.4.3

Simplifique o denominador.

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Etapa 8.4.3.1

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

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

Etapa 8.4.3.2

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

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

Etapa 8.4.4

Aplique a propriedade distributiva.

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

Etapa 8.4.5

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

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Etapa 8.4.5.1

Mova o negativo de maior ordem em $- \frac{\sin (x)}{\cos^{2} (x)}$ para o numerador.

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

Etapa 8.4.5.2

Fatore $\cos (x)$ de $\cos^{2} (x)$ .

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

Etapa 8.4.5.3

Cancele o fator comum.

Etapa 8.4.5.4

Reescreva a expressão.

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

Etapa 8.4.6

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

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

Etapa 8.4.7

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

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Etapa 8.4.7.1

Mova o negativo de maior ordem em $- \frac{1}{\cos^{2} (x)}$ para o numerador.

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

Etapa 8.4.7.2

Fatore $\cos (x)$ de $\cos^{2} (x)$ .

$\frac{- \sin (x)}{\cos (x) (\frac{1}{\cos (x)} + \frac{\sin (x)}{\cos (x)})} + \frac{- 1}{\cos (x) \cos (x)} \cdot \frac{\cos (x)}{\frac{1}{\cos (x)} + \frac{\sin (x)}{\cos (x)}} + \sin (x) \ln (\sec (x) + \tan (x))$

Etapa 8.4.7.3

Cancele o fator comum.

Etapa 8.4.7.4

Reescreva a expressão.

$\frac{- \sin (x)}{\cos (x) (\frac{1}{\cos (x)} + \frac{\sin (x)}{\cos (x)})} + \frac{- 1}{\cos (x)} \cdot \frac{1}{\frac{1}{\cos (x)} + \frac{\sin (x)}{\cos (x)}} + \sin (x) \ln (\sec (x) + \tan (x))$

Etapa 8.4.8

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

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

Etapa 8.4.9

Simplifique cada termo.

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Etapa 8.4.9.1

Mova o número negativo para a frente da fração.

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

Etapa 8.4.9.2

Mova o número negativo para a frente da fração.

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

Etapa 8.4.10

Simplifique cada termo.

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Etapa 8.4.10.1

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

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

Etapa 8.4.10.2

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

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

Etapa 8.5

Para escrever $\sin (x) \ln (\frac{1}{\cos (x)} + \frac{\sin (x)}{\cos (x)})$ como fração com um denominador comum, multiplique por $\frac{\cos (x) (\frac{1}{\cos (x)} + \frac{\sin (x)}{\cos (x)})}{\cos (x) (\frac{1}{\cos (x)} + \frac{\sin (x)}{\cos (x)})}$ .

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

Etapa 8.6

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

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

Etapa 8.7

Combine os numeradores em relação ao denominador comum.

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

Etapa 8.8

Simplifique o numerador.

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Etapa 8.8.1

Fatore $\sin (x)$ de $- \sin (x) + \sin (x) \ln (\frac{1}{\cos (x)} + \frac{\sin (x)}{\cos (x)}) \cos (x) (\frac{1}{\cos (x)} + \frac{\sin (x)}{\cos (x)})$ .

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Etapa 8.8.1.1

Fatore $\sin (x)$ de $- \sin (x)$ .

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

Etapa 8.8.1.2

Fatore $\sin (x)$ de $\sin (x) \ln (\frac{1}{\cos (x)} + \frac{\sin (x)}{\cos (x)}) \cos (x) (\frac{1}{\cos (x)} + \frac{\sin (x)}{\cos (x)})$ .

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

Etapa 8.8.1.3

Fatore $\sin (x)$ de $\sin (x) \cdot - 1 + \sin (x) (\ln (\frac{1}{\cos (x)} + \frac{\sin (x)}{\cos (x)}) \cos (x) (\frac{1}{\cos (x)} + \frac{\sin (x)}{\cos (x)}))$ .

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

Etapa 8.8.2

Aplique a propriedade distributiva.

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

Etapa 8.8.3

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

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Etapa 8.8.3.1

Fatore $\cos (x)$ de $\ln (\frac{1}{\cos (x)} + \frac{\sin (x)}{\cos (x)}) \cos (x)$ .

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

Etapa 8.8.3.2

Cancele o fator comum.

Etapa 8.8.3.3

Reescreva a expressão.

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

Etapa 8.8.4

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

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Etapa 8.8.4.1

Fatore $\cos (x)$ de $\ln (\frac{1}{\cos (x)} + \frac{\sin (x)}{\cos (x)}) \cos (x)$ .

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

Etapa 8.8.4.2

Cancele o fator comum.

Etapa 8.8.4.3

Reescreva a expressão.

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

Etapa 8.9

Combine os numeradores em relação ao denominador comum.

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

Etapa 8.10

Simplifique o numerador.

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Etapa 8.10.1

Aplique a propriedade distributiva.

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

Etapa 8.10.2

Simplifique.

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Etapa 8.10.2.1

Mova $- 1$ para a esquerda de $\sin (x)$ .

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

Etapa 8.10.2.2

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

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Etapa 8.10.2.2.1

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 8.10.2.2.2

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 8.10.2.2.3

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

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

Etapa 8.10.2.2.4

Some $1$ e $1$ .

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

Etapa 8.10.3

Reescreva $- 1 \sin (x)$ como $- \sin (x)$ .

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

Etapa 8.10.4

Reescreva $- \sin (x) + \sin (x) \ln (\frac{1}{\cos (x)} + \frac{\sin (x)}{\cos (x)}) + \sin^{2} (x) \ln (\frac{1}{\cos (x)} + \frac{\sin (x)}{\cos (x)}) - 1$ em uma forma fatorada.

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Etapa 8.10.4.1

Reordene os termos.

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

Etapa 8.10.4.2

Fatore o máximo divisor comum de cada grupo.

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Etapa 8.10.4.2.1

Agrupe os dois primeiros termos e os dois últimos termos.

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

Etapa 8.10.4.2.2

Fatore o máximo divisor comum (MDC) de cada grupo.

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

Etapa 8.10.4.3

Fatore o polinômio desmembrando o máximo divisor comum, $1 + \sin (x)$ .

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

Etapa 8.11

Simplifique cada termo.

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Etapa 8.11.1

Converta de $\frac{1}{\cos (x)}$ em $\sec (x)$ .

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

Etapa 8.11.2

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

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

Etapa 8.12

Simplifique cada termo.

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Etapa 8.12.1

Converta de $\frac{1}{\cos (x)}$ em $\sec (x)$ .

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

Etapa 8.12.2

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

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