Cálculo Exemplos

$f (x) = 3 \ln (\sec (x) + \tan (x))$

Etapa 1

Encontre a primeira derivada.

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Etapa 1.1

Como $3$ é constante em relação a $x$ , a derivada de $3 \ln (\sec (x) + \tan (x))$ em relação a $x$ é $3 \frac{d}{d x} [\ln (\sec (x) + \tan (x))]$ .

$3 \frac{d}{d x} [\ln (\sec (x) + \tan (x))]$

Etapa 1.2

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 $g (x) = \sec (x) + \tan (x)$ .

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Etapa 1.2.1

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

$3 (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [\sec (x) + \tan (x)])$

Etapa 1.2.2

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

$3 (\frac{1}{u} \frac{d}{d x} [\sec (x) + \tan (x)])$

Etapa 1.2.3

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

$3 (\frac{1}{\sec (x) + \tan (x)} \frac{d}{d x} [\sec (x) + \tan (x)])$

Etapa 1.3

Diferencie usando a regra da soma.

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Etapa 1.3.1

Combine $\frac{1}{\sec (x) + \tan (x)}$ e $3$ .

$\frac{3}{\sec (x) + \tan (x)} \frac{d}{d x} [\sec (x) + \tan (x)]$

Etapa 1.3.2

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

$\frac{3}{\sec (x) + \tan (x)} (\frac{d}{d x} [\sec (x)] + \frac{d}{d x} [\tan (x)])$

Etapa 1.4

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

$\frac{3}{\sec (x) + \tan (x)} (\sec (x) \tan (x) + \frac{d}{d x} [\tan (x)])$

Etapa 1.5

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

$\frac{3}{\sec (x) + \tan (x)} (\sec (x) \tan (x) + \sec^{2} (x))$

Etapa 1.6

Simplifique.

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Etapa 1.6.1

Reordene os fatores de $\frac{3}{\sec (x) + \tan (x)} (\sec (x) \tan (x) + \sec^{2} (x))$ .

$(\sec (x) \tan (x) + \sec^{2} (x)) \frac{3}{\sec (x) + \tan (x)}$

Etapa 1.6.2

Aplique a propriedade distributiva.

$\sec (x) \tan (x) \frac{3}{\sec (x) + \tan (x)} + \sec^{2} (x) \frac{3}{\sec (x) + \tan (x)}$

Etapa 1.6.3

Multiplique $\sec (x) \tan (x) \frac{3}{\sec (x) + \tan (x)}$ .

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Etapa 1.6.3.1

Combine $\frac{3}{\sec (x) + \tan (x)}$ e $\sec (x)$ .

$\frac{3 \sec (x)}{\sec (x) + \tan (x)} \tan (x) + \sec^{2} (x) \frac{3}{\sec (x) + \tan (x)}$

Etapa 1.6.3.2

Combine $\frac{3 \sec (x)}{\sec (x) + \tan (x)}$ e $\tan (x)$ .

$\frac{3 \sec (x) \tan (x)}{\sec (x) + \tan (x)} + \sec^{2} (x) \frac{3}{\sec (x) + \tan (x)}$

Etapa 1.6.4

Combine $\sec^{2} (x)$ e $\frac{3}{\sec (x) + \tan (x)}$ .

$\frac{3 \sec (x) \tan (x)}{\sec (x) + \tan (x)} + \frac{\sec^{2} (x) \cdot 3}{\sec (x) + \tan (x)}$

Etapa 1.6.5

Mova $3$ para a esquerda de $\sec^{2} (x)$ .

$\frac{3 \sec (x) \tan (x)}{\sec (x) + \tan (x)} + \frac{3 \sec^{2} (x)}{\sec (x) + \tan (x)}$

Etapa 1.6.6

Combine os numeradores em relação ao denominador comum.

$\frac{3 \sec (x) \tan (x) + 3 \sec^{2} (x)}{\sec (x) + \tan (x)}$

Etapa 1.6.7

Fatore $3 \sec (x)$ de $3 \sec (x) \tan (x) + 3 \sec^{2} (x)$ .

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Etapa 1.6.7.1

Fatore $3 \sec (x)$ de $3 \sec (x) \tan (x)$ .

$\frac{3 \sec (x) (\tan (x)) + 3 \sec^{2} (x)}{\sec (x) + \tan (x)}$

Etapa 1.6.7.2

Fatore $3 \sec (x)$ de $3 \sec^{2} (x)$ .

$\frac{3 \sec (x) (\tan (x)) + 3 \sec (x) (\sec (x))}{\sec (x) + \tan (x)}$

Etapa 1.6.7.3

Fatore $3 \sec (x)$ de $3 \sec (x) (\tan (x)) + 3 \sec (x) (\sec (x))$ .

$f' (x) = \frac{3 \sec (x) (\tan (x) + \sec (x))}{\sec (x) + \tan (x)}$

Etapa 2

Encontre a segunda derivada.

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Etapa 2.1

Como $3$ é constante em relação a $x$ , a derivada de $\frac{3 \sec (x) (\tan (x) + \sec (x))}{\sec (x) + \tan (x)}$ em relação a $x$ é $3 \frac{d}{d x} [\frac{\sec (x) (\tan (x) + \sec (x))}{\sec (x) + \tan (x)}]$ .

$3 \frac{d}{d x} [\frac{\sec (x) (\tan (x) + \sec (x))}{\sec (x) + \tan (x)}]$

Etapa 2.2

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

$3 \frac{(\sec (x) + \tan (x)) \frac{d}{d x} [\sec (x) (\tan (x) + \sec (x))] - \sec (x) (\tan (x) + \sec (x)) \frac{d}{d x} [\sec (x) + \tan (x)]}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.3

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

$3 \frac{(\sec (x) + \tan (x)) (\sec (x) \frac{d}{d x} [\tan (x) + \sec (x)] + (\tan (x) + \sec (x)) \frac{d}{d x} [\sec (x)]) - \sec (x) (\tan (x) + \sec (x)) \frac{d}{d x} [\sec (x) + \tan (x)]}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.4

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

$3 \frac{(\sec (x) + \tan (x)) (\sec (x) (\frac{d}{d x} [\tan (x)] + \frac{d}{d x} [\sec (x)]) + (\tan (x) + \sec (x)) \frac{d}{d x} [\sec (x)]) - \sec (x) (\tan (x) + \sec (x)) \frac{d}{d x} [\sec (x) + \tan (x)]}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.5

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

$3 \frac{(\sec (x) + \tan (x)) (\sec (x) (\sec^{2} (x) + \frac{d}{d x} [\sec (x)]) + (\tan (x) + \sec (x)) \frac{d}{d x} [\sec (x)]) - \sec (x) (\tan (x) + \sec (x)) \frac{d}{d x} [\sec (x) + \tan (x)]}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.6

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

$3 \frac{(\sec (x) + \tan (x)) (\sec (x) (\sec^{2} (x) + \sec (x) \tan (x)) + (\tan (x) + \sec (x)) \frac{d}{d x} [\sec (x)]) - \sec (x) (\tan (x) + \sec (x)) \frac{d}{d x} [\sec (x) + \tan (x)]}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.7

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

$3 \frac{(\sec (x) + \tan (x)) (\sec (x) (\sec^{2} (x) + \sec (x) \tan (x)) + (\tan (x) + \sec (x)) (\sec (x) \tan (x))) - \sec (x) (\tan (x) + \sec (x)) \frac{d}{d x} [\sec (x) + \tan (x)]}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.8

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

$3 \frac{(\sec (x) + \tan (x)) (\sec (x) (\sec^{2} (x) + \sec (x) \tan (x)) + (\tan (x) + \sec (x)) \sec (x) \tan (x)) - \sec (x) (\tan (x) + \sec (x)) (\frac{d}{d x} [\sec (x)] + \frac{d}{d x} [\tan (x)])}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.9

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

$3 \frac{(\sec (x) + \tan (x)) (\sec (x) (\sec^{2} (x) + \sec (x) \tan (x)) + (\tan (x) + \sec (x)) \sec (x) \tan (x)) - \sec (x) (\tan (x) + \sec (x)) (\sec (x) \tan (x) + \frac{d}{d x} [\tan (x)])}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.10

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

$3 \frac{(\sec (x) + \tan (x)) (\sec (x) (\sec^{2} (x) + \sec (x) \tan (x)) + (\tan (x) + \sec (x)) \sec (x) \tan (x)) - \sec (x) (\tan (x) + \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.11

Combine $3$ e $\frac{(\sec (x) + \tan (x)) (\sec (x) (\sec^{2} (x) + \sec (x) \tan (x)) + (\tan (x) + \sec (x)) \sec (x) \tan (x)) - \sec (x) (\tan (x) + \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$ .

$\frac{3 ((\sec (x) + \tan (x)) (\sec (x) (\sec^{2} (x) + \sec (x) \tan (x)) + (\tan (x) + \sec (x)) \sec (x) \tan (x)) - \sec (x) (\tan (x) + \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12

Simplifique.

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Etapa 2.12.1

Aplique a propriedade distributiva.

$\frac{3 ((\sec (x) + \tan (x)) (\sec (x) \sec^{2} (x) + \sec (x) (\sec (x) \tan (x)) + (\tan (x) + \sec (x)) \sec (x) \tan (x)) - \sec (x) (\tan (x) + \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.2

Aplique a propriedade distributiva.

$\frac{3 ((\sec (x) + \tan (x)) (\sec (x) \sec^{2} (x) + \sec (x) (\sec (x) \tan (x)) + (\tan (x) \sec (x) + \sec (x) \sec (x)) \tan (x)) - \sec (x) (\tan (x) + \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.3

Aplique a propriedade distributiva.

$\frac{3 ((\sec (x) + \tan (x)) (\sec (x) \sec^{2} (x) + \sec (x) (\sec (x) \tan (x)) + \tan (x) \sec (x) \tan (x) + \sec (x) \sec (x) \tan (x)) - \sec (x) (\tan (x) + \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.4

Aplique a propriedade distributiva.

$\frac{3 ((\sec (x) + \tan (x)) (\sec (x) \sec^{2} (x) + \sec (x) (\sec (x) \tan (x)) + \tan (x) \sec (x) \tan (x) + \sec (x) \sec (x) \tan (x)) + (- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.5

Aplique a propriedade distributiva.

$\frac{3 ((\sec (x) + \tan (x)) (\sec (x) \sec^{2} (x) + \sec (x) (\sec (x) \tan (x)) + \tan (x) \sec (x) \tan (x) + \sec (x) \sec (x) \tan (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6

Simplifique o numerador.

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Etapa 2.12.6.1

Simplifique cada termo.

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Etapa 2.12.6.1.1

Simplifique cada termo.

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Etapa 2.12.6.1.1.1

Multiplique $\sec (x)$ por $\sec^{2} (x)$ somando os expoentes.

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Etapa 2.12.6.1.1.1.1

Multiplique $\sec (x)$ por $\sec^{2} (x)$ .

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Etapa 2.12.6.1.1.1.1.1

Eleve $\sec (x)$ à potência de $1$ .

$\frac{3 ((\sec (x) + \tan (x)) (\sec^{1} (x) \sec^{2} (x) + \sec (x) (\sec (x) \tan (x)) + \tan (x) \sec (x) \tan (x) + \sec (x) \sec (x) \tan (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.1.1.1.2

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

$\frac{3 ((\sec (x) + \tan (x)) ({\sec (x)}^{1 + 2} + \sec (x) (\sec (x) \tan (x)) + \tan (x) \sec (x) \tan (x) + \sec (x) \sec (x) \tan (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.1.1.2

Some $1$ e $2$ .

$\frac{3 ((\sec (x) + \tan (x)) (\sec^{3} (x) + \sec (x) (\sec (x) \tan (x)) + \tan (x) \sec (x) \tan (x) + \sec (x) \sec (x) \tan (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.1.2

Multiplique $\sec (x) (\sec (x) \tan (x))$ .

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Etapa 2.12.6.1.1.2.1

Eleve $\sec (x)$ à potência de $1$ .

$\frac{3 ((\sec (x) + \tan (x)) (\sec^{3} (x) + \sec^{1} (x) \sec (x) \tan (x) + \tan (x) \sec (x) \tan (x) + \sec (x) \sec (x) \tan (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.1.2.2

Eleve $\sec (x)$ à potência de $1$ .

$\frac{3 ((\sec (x) + \tan (x)) (\sec^{3} (x) + \sec^{1} (x) \sec^{1} (x) \tan (x) + \tan (x) \sec (x) \tan (x) + \sec (x) \sec (x) \tan (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.1.2.3

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

$\frac{3 ((\sec (x) + \tan (x)) (\sec^{3} (x) + {\sec (x)}^{1 + 1} \tan (x) + \tan (x) \sec (x) \tan (x) + \sec (x) \sec (x) \tan (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.1.2.4

Some $1$ e $1$ .

$\frac{3 ((\sec (x) + \tan (x)) (\sec^{3} (x) + \sec^{2} (x) \tan (x) + \tan (x) \sec (x) \tan (x) + \sec (x) \sec (x) \tan (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.1.3

Multiplique $\tan (x) \sec (x) \tan (x)$ .

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Etapa 2.12.6.1.1.3.1

Eleve $\tan (x)$ à potência de $1$ .

$\frac{3 ((\sec (x) + \tan (x)) (\sec^{3} (x) + \sec^{2} (x) \tan (x) + \tan^{1} (x) \tan (x) \sec (x) + \sec (x) \sec (x) \tan (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.1.3.2

Eleve $\tan (x)$ à potência de $1$ .

$\frac{3 ((\sec (x) + \tan (x)) (\sec^{3} (x) + \sec^{2} (x) \tan (x) + \tan^{1} (x) \tan^{1} (x) \sec (x) + \sec (x) \sec (x) \tan (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.1.3.3

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

$\frac{3 ((\sec (x) + \tan (x)) (\sec^{3} (x) + \sec^{2} (x) \tan (x) + {\tan (x)}^{1 + 1} \sec (x) + \sec (x) \sec (x) \tan (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.1.3.4

Some $1$ e $1$ .

$\frac{3 ((\sec (x) + \tan (x)) (\sec^{3} (x) + \sec^{2} (x) \tan (x) + \tan^{2} (x) \sec (x) + \sec (x) \sec (x) \tan (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.1.4

Multiplique $\sec (x) \sec (x)$ .

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Etapa 2.12.6.1.1.4.1

Eleve $\sec (x)$ à potência de $1$ .

$\frac{3 ((\sec (x) + \tan (x)) (\sec^{3} (x) + \sec^{2} (x) \tan (x) + \tan^{2} (x) \sec (x) + \sec^{1} (x) \sec (x) \tan (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.1.4.2

Eleve $\sec (x)$ à potência de $1$ .

$\frac{3 ((\sec (x) + \tan (x)) (\sec^{3} (x) + \sec^{2} (x) \tan (x) + \tan^{2} (x) \sec (x) + \sec^{1} (x) \sec^{1} (x) \tan (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.1.4.3

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

$\frac{3 ((\sec (x) + \tan (x)) (\sec^{3} (x) + \sec^{2} (x) \tan (x) + \tan^{2} (x) \sec (x) + {\sec (x)}^{1 + 1} \tan (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.1.4.4

Some $1$ e $1$ .

$\frac{3 ((\sec (x) + \tan (x)) (\sec^{3} (x) + \sec^{2} (x) \tan (x) + \tan^{2} (x) \sec (x) + \sec^{2} (x) \tan (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.2

Some $\sec^{2} (x) \tan (x)$ e $\sec^{2} (x) \tan (x)$ .

$\frac{3 ((\sec (x) + \tan (x)) (\sec^{3} (x) + 2 \sec^{2} (x) \tan (x) + \tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.3

Expanda $(\sec (x) + \tan (x)) (\sec^{3} (x) + 2 \sec^{2} (x) \tan (x) + \tan^{2} (x) \sec (x))$ multiplicando cada termo na primeira expressão por cada um dos termos na segunda expressão.

$\frac{3 (\sec (x) \sec^{3} (x) + \sec (x) (2 \sec^{2} (x) \tan (x)) + \sec (x) (\tan^{2} (x) \sec (x)) + \tan (x) \sec^{3} (x) + \tan (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.4

Simplifique cada termo.

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Etapa 2.12.6.1.4.1

Multiplique $\sec (x)$ por $\sec^{3} (x)$ somando os expoentes.

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Etapa 2.12.6.1.4.1.1

Multiplique $\sec (x)$ por $\sec^{3} (x)$ .

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Etapa 2.12.6.1.4.1.1.1

Eleve $\sec (x)$ à potência de $1$ .

$\frac{3 (\sec^{1} (x) \sec^{3} (x) + \sec (x) (2 \sec^{2} (x) \tan (x)) + \sec (x) (\tan^{2} (x) \sec (x)) + \tan (x) \sec^{3} (x) + \tan (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.4.1.1.2

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

$\frac{3 ({\sec (x)}^{1 + 3} + \sec (x) (2 \sec^{2} (x) \tan (x)) + \sec (x) (\tan^{2} (x) \sec (x)) + \tan (x) \sec^{3} (x) + \tan (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.4.1.2

Some $1$ e $3$ .

$\frac{3 (\sec^{4} (x) + \sec (x) (2 \sec^{2} (x) \tan (x)) + \sec (x) (\tan^{2} (x) \sec (x)) + \tan (x) \sec^{3} (x) + \tan (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.4.2

Reescreva usando a propriedade comutativa da multiplicação.

$\frac{3 (\sec^{4} (x) + 2 \sec (x) \sec^{2} (x) \tan (x) + \sec (x) (\tan^{2} (x) \sec (x)) + \tan (x) \sec^{3} (x) + \tan (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.4.3

Multiplique $\sec (x)$ por $\sec^{2} (x)$ somando os expoentes.

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Etapa 2.12.6.1.4.3.1

Mova $\sec^{2} (x)$ .

$\frac{3 (\sec^{4} (x) + 2 (\sec^{2} (x) \sec (x)) \tan (x) + \sec (x) (\tan^{2} (x) \sec (x)) + \tan (x) \sec^{3} (x) + \tan (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.4.3.2

Multiplique $\sec^{2} (x)$ por $\sec (x)$ .

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Etapa 2.12.6.1.4.3.2.1

Eleve $\sec (x)$ à potência de $1$ .

$\frac{3 (\sec^{4} (x) + 2 (\sec^{2} (x) \sec^{1} (x)) \tan (x) + \sec (x) (\tan^{2} (x) \sec (x)) + \tan (x) \sec^{3} (x) + \tan (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.4.3.2.2

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

$\frac{3 (\sec^{4} (x) + 2 {\sec (x)}^{2 + 1} \tan (x) + \sec (x) (\tan^{2} (x) \sec (x)) + \tan (x) \sec^{3} (x) + \tan (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.4.3.3

Some $2$ e $1$ .

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + \sec (x) (\tan^{2} (x) \sec (x)) + \tan (x) \sec^{3} (x) + \tan (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.4.4

Multiplique $\sec (x) (\tan^{2} (x) \sec (x))$ .

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Etapa 2.12.6.1.4.4.1

Eleve $\sec (x)$ à potência de $1$ .

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + \sec^{1} (x) \sec (x) \tan^{2} (x) + \tan (x) \sec^{3} (x) + \tan (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.4.4.2

Eleve $\sec (x)$ à potência de $1$ .

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + \sec^{1} (x) \sec^{1} (x) \tan^{2} (x) + \tan (x) \sec^{3} (x) + \tan (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.4.4.3

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

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + {\sec (x)}^{1 + 1} \tan^{2} (x) + \tan (x) \sec^{3} (x) + \tan (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.4.4.4

Some $1$ e $1$ .

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + \sec^{2} (x) \tan^{2} (x) + \tan (x) \sec^{3} (x) + \tan (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.4.5

Reescreva usando a propriedade comutativa da multiplicação.

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + \sec^{2} (x) \tan^{2} (x) + \tan (x) \sec^{3} (x) + 2 \tan (x) \sec^{2} (x) \tan (x) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.4.6

Multiplique $2 \tan (x) \sec^{2} (x) \tan (x)$ .

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Etapa 2.12.6.1.4.6.1

Eleve $\tan (x)$ à potência de $1$ .

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + \sec^{2} (x) \tan^{2} (x) + \tan (x) \sec^{3} (x) + 2 (\tan^{1} (x) \tan (x)) \sec^{2} (x) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.4.6.2

Eleve $\tan (x)$ à potência de $1$ .

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + \sec^{2} (x) \tan^{2} (x) + \tan (x) \sec^{3} (x) + 2 (\tan^{1} (x) \tan^{1} (x)) \sec^{2} (x) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.4.6.3

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

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + \sec^{2} (x) \tan^{2} (x) + \tan (x) \sec^{3} (x) + 2 {\tan (x)}^{1 + 1} \sec^{2} (x) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.4.6.4

Some $1$ e $1$ .

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + \sec^{2} (x) \tan^{2} (x) + \tan (x) \sec^{3} (x) + 2 \tan^{2} (x) \sec^{2} (x) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.4.7

Multiplique $\tan (x)$ por $\tan^{2} (x)$ somando os expoentes.

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Etapa 2.12.6.1.4.7.1

Mova $\tan^{2} (x)$ .

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + \sec^{2} (x) \tan^{2} (x) + \tan (x) \sec^{3} (x) + 2 \tan^{2} (x) \sec^{2} (x) + \tan^{2} (x) \tan (x) \sec (x)) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.4.7.2

Multiplique $\tan^{2} (x)$ por $\tan (x)$ .

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Etapa 2.12.6.1.4.7.2.1

Eleve $\tan (x)$ à potência de $1$ .

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + \sec^{2} (x) \tan^{2} (x) + \tan (x) \sec^{3} (x) + 2 \tan^{2} (x) \sec^{2} (x) + \tan^{2} (x) \tan^{1} (x) \sec (x)) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.4.7.2.2

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

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + \sec^{2} (x) \tan^{2} (x) + \tan (x) \sec^{3} (x) + 2 \tan^{2} (x) \sec^{2} (x) + {\tan (x)}^{2 + 1} \sec (x)) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.4.7.3

Some $2$ e $1$ .

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + \sec^{2} (x) \tan^{2} (x) + \tan (x) \sec^{3} (x) + 2 \tan^{2} (x) \sec^{2} (x) + \tan^{3} (x) \sec (x)) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.5

Reordene os fatores de $\tan (x) \sec^{3} (x)$ .

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + \sec^{2} (x) \tan^{2} (x) + \sec^{3} (x) \tan (x) + 2 \tan^{2} (x) \sec^{2} (x) + \tan^{3} (x) \sec (x)) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.6

Some $2 \sec^{3} (x) \tan (x)$ e $\sec^{3} (x) \tan (x)$ .

$\frac{3 (\sec^{4} (x) + 3 \sec^{3} (x) \tan (x) + \sec^{2} (x) \tan^{2} (x) + 2 \tan^{2} (x) \sec^{2} (x) + \tan^{3} (x) \sec (x)) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.7

Reordene os fatores de $2 \tan^{2} (x) \sec^{2} (x)$ .

$\frac{3 (\sec^{4} (x) + 3 \sec^{3} (x) \tan (x) + \sec^{2} (x) \tan^{2} (x) + 2 \sec^{2} (x) \tan^{2} (x) + \tan^{3} (x) \sec (x)) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.8

Some $\sec^{2} (x) \tan^{2} (x)$ e $2 \sec^{2} (x) \tan^{2} (x)$ .

$\frac{3 (\sec^{4} (x) + 3 \sec^{3} (x) \tan (x) + 3 \sec^{2} (x) \tan^{2} (x) + \tan^{3} (x) \sec (x)) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.9

Aplique a propriedade distributiva.

$\frac{3 \sec^{4} (x) + 3 (3 \sec^{3} (x) \tan (x)) + 3 (3 \sec^{2} (x) \tan^{2} (x)) + 3 (\tan^{3} (x) \sec (x)) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.10

Simplifique.

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Etapa 2.12.6.1.10.1

Multiplique $3$ por $3$ .

$\frac{3 \sec^{4} (x) + 9 (\sec^{3} (x) \tan (x)) + 3 (3 \sec^{2} (x) \tan^{2} (x)) + 3 (\tan^{3} (x) \sec (x)) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.10.2

Multiplique $3$ por $3$ .

$\frac{3 \sec^{4} (x) + 9 (\sec^{3} (x) \tan (x)) + 9 (\sec^{2} (x) \tan^{2} (x)) + 3 \tan^{3} (x) \sec (x) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.11

Remova os parênteses.

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.12

Multiplique $- \sec (x) \sec (x)$ .

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Etapa 2.12.6.1.12.1

Eleve $\sec (x)$ à potência de $1$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 ((- \sec (x) \tan (x) - (\sec^{1} (x) \sec (x))) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.12.2

Eleve $\sec (x)$ à potência de $1$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 ((- \sec (x) \tan (x) - (\sec^{1} (x) \sec^{1} (x))) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.12.3

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

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 ((- \sec (x) \tan (x) - {\sec (x)}^{1 + 1}) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.12.4

Some $1$ e $1$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 ((- \sec (x) \tan (x) - \sec^{2} (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.13

Expanda $(- \sec (x) \tan (x) - \sec^{2} (x)) (\sec (x) \tan (x) + \sec^{2} (x))$ usando o método FOIL.

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Etapa 2.12.6.1.13.1

Aplique a propriedade distributiva.

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec (x) \tan (x) (\sec (x) \tan (x) + \sec^{2} (x)) - \sec^{2} (x) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.13.2

Aplique a propriedade distributiva.

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec (x) \tan (x) (\sec (x) \tan (x)) - \sec (x) \tan (x) \sec^{2} (x) - \sec^{2} (x) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.13.3

Aplique a propriedade distributiva.

Etapa 2.12.6.1.14

Simplifique e combine termos semelhantes.

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Etapa 2.12.6.1.14.1

Simplifique cada termo.

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Etapa 2.12.6.1.14.1.1

Multiplique $- \sec (x) \tan (x) (\sec (x) \tan (x))$ .

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Etapa 2.12.6.1.14.1.1.1

Eleve $\sec (x)$ à potência de $1$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- (\sec^{1} (x) \sec (x)) \tan (x) \tan (x) - \sec (x) \tan (x) \sec^{2} (x) - \sec^{2} (x) (\sec (x) \tan (x)) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.14.1.1.2

Eleve $\sec (x)$ à potência de $1$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- (\sec^{1} (x) \sec^{1} (x)) \tan (x) \tan (x) - \sec (x) \tan (x) \sec^{2} (x) - \sec^{2} (x) (\sec (x) \tan (x)) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.14.1.1.3

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

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- {\sec (x)}^{1 + 1} \tan (x) \tan (x) - \sec (x) \tan (x) \sec^{2} (x) - \sec^{2} (x) (\sec (x) \tan (x)) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.14.1.1.4

Some $1$ e $1$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) \tan (x) \tan (x) - \sec (x) \tan (x) \sec^{2} (x) - \sec^{2} (x) (\sec (x) \tan (x)) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.14.1.1.5

Eleve $\tan (x)$ à potência de $1$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) (\tan^{1} (x) \tan (x)) - \sec (x) \tan (x) \sec^{2} (x) - \sec^{2} (x) (\sec (x) \tan (x)) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.14.1.1.6

Eleve $\tan (x)$ à potência de $1$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) (\tan^{1} (x) \tan^{1} (x)) - \sec (x) \tan (x) \sec^{2} (x) - \sec^{2} (x) (\sec (x) \tan (x)) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.14.1.1.7

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

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) {\tan (x)}^{1 + 1} - \sec (x) \tan (x) \sec^{2} (x) - \sec^{2} (x) (\sec (x) \tan (x)) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.14.1.1.8

Some $1$ e $1$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) \tan^{2} (x) - \sec (x) \tan (x) \sec^{2} (x) - \sec^{2} (x) (\sec (x) \tan (x)) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.14.1.2

Multiplique $\sec (x)$ por $\sec^{2} (x)$ somando os expoentes.

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Etapa 2.12.6.1.14.1.2.1

Mova $\sec^{2} (x)$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) \tan^{2} (x) - (\sec^{2} (x) \sec (x)) \tan (x) - \sec^{2} (x) (\sec (x) \tan (x)) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.14.1.2.2

Multiplique $\sec^{2} (x)$ por $\sec (x)$ .

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Etapa 2.12.6.1.14.1.2.2.1

Eleve $\sec (x)$ à potência de $1$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) \tan^{2} (x) - (\sec^{2} (x) \sec^{1} (x)) \tan (x) - \sec^{2} (x) (\sec (x) \tan (x)) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.14.1.2.2.2

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

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) \tan^{2} (x) - {\sec (x)}^{2 + 1} \tan (x) - \sec^{2} (x) (\sec (x) \tan (x)) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.14.1.2.3

Some $2$ e $1$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) \tan^{2} (x) - \sec^{3} (x) \tan (x) - \sec^{2} (x) (\sec (x) \tan (x)) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.14.1.3

Multiplique $\sec^{2} (x)$ por $\sec (x)$ somando os expoentes.

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Etapa 2.12.6.1.14.1.3.1

Mova $\sec (x)$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) \tan^{2} (x) - \sec^{3} (x) \tan (x) - (\sec (x) \sec^{2} (x)) \tan (x) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.14.1.3.2

Multiplique $\sec (x)$ por $\sec^{2} (x)$ .

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Etapa 2.12.6.1.14.1.3.2.1

Eleve $\sec (x)$ à potência de $1$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) \tan^{2} (x) - \sec^{3} (x) \tan (x) - (\sec^{1} (x) \sec^{2} (x)) \tan (x) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.14.1.3.2.2

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

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) \tan^{2} (x) - \sec^{3} (x) \tan (x) - {\sec (x)}^{1 + 2} \tan (x) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.14.1.3.3

Some $1$ e $2$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) \tan^{2} (x) - \sec^{3} (x) \tan (x) - \sec^{3} (x) \tan (x) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.14.1.4

Multiplique $\sec^{2} (x)$ por $\sec^{2} (x)$ somando os expoentes.

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Etapa 2.12.6.1.14.1.4.1

Mova $\sec^{2} (x)$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) \tan^{2} (x) - \sec^{3} (x) \tan (x) - \sec^{3} (x) \tan (x) - (\sec^{2} (x) \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.14.1.4.2

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

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) \tan^{2} (x) - \sec^{3} (x) \tan (x) - \sec^{3} (x) \tan (x) - {\sec (x)}^{2 + 2})}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.14.1.4.3

Some $2$ e $2$ .

Etapa 2.12.6.1.14.2

Subtraia $\sec^{3} (x) \tan (x)$ de $- \sec^{3} (x) \tan (x)$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) \tan^{2} (x) - 2 \sec^{3} (x) \tan (x) - \sec^{4} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.15

Aplique a propriedade distributiva.

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) \tan^{2} (x)) + 3 (- 2 \sec^{3} (x) \tan (x)) + 3 (- \sec^{4} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.16

Simplifique.

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Etapa 2.12.6.1.16.1

Multiplique $- 1$ por $3$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) - 3 (\sec^{2} (x) \tan^{2} (x)) + 3 (- 2 \sec^{3} (x) \tan (x)) + 3 (- \sec^{4} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.16.2

Multiplique $- 2$ por $3$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) - 3 (\sec^{2} (x) \tan^{2} (x)) - 6 (\sec^{3} (x) \tan (x)) + 3 (- \sec^{4} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.16.3

Multiplique $- 1$ por $3$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) - 3 (\sec^{2} (x) \tan^{2} (x)) - 6 (\sec^{3} (x) \tan (x)) - 3 \sec^{4} (x)}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.1.17

Remova os parênteses.

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) - 3 \sec^{2} (x) \tan^{2} (x) - 6 \sec^{3} (x) \tan (x) - 3 \sec^{4} (x)}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.2

Combine os termos opostos em $3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) - 3 \sec^{2} (x) \tan^{2} (x) - 6 \sec^{3} (x) \tan (x) - 3 \sec^{4} (x)$ .

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Etapa 2.12.6.2.1

Subtraia $3 \sec^{4} (x)$ de $3 \sec^{4} (x)$ .

$\frac{9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) - 3 \sec^{2} (x) \tan^{2} (x) - 6 \sec^{3} (x) \tan (x) + 0}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.2.2

Some $9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) - 3 \sec^{2} (x) \tan^{2} (x) - 6 \sec^{3} (x) \tan (x)$ e $0$ .

$\frac{9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) - 3 \sec^{2} (x) \tan^{2} (x) - 6 \sec^{3} (x) \tan (x)}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.3

Subtraia $6 \sec^{3} (x) \tan (x)$ de $9 \sec^{3} (x) \tan (x)$ .

$\frac{3 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) - 3 \sec^{2} (x) \tan^{2} (x)}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.6.4

Subtraia $3 \sec^{2} (x) \tan^{2} (x)$ de $9 \sec^{2} (x) \tan^{2} (x)$ .

$\frac{3 \sec^{3} (x) \tan (x) + 6 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x)}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.7

Simplifique o numerador.

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Etapa 2.12.7.1

Fatore $3 \sec (x) \tan (x)$ de $3 \sec^{3} (x) \tan (x) + 6 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x)$ .

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Etapa 2.12.7.1.1

Fatore $3 \sec (x) \tan (x)$ de $3 \sec^{3} (x) \tan (x)$ .

$\frac{3 \sec (x) \tan (x) (\sec^{2} (x)) + 6 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x)}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.7.1.2

Fatore $3 \sec (x) \tan (x)$ de $6 \sec^{2} (x) \tan^{2} (x)$ .

$\frac{3 \sec (x) \tan (x) (\sec^{2} (x)) + 3 \sec (x) \tan (x) (2 \sec (x) \tan (x)) + 3 \tan^{3} (x) \sec (x)}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.7.1.3

Fatore $3 \sec (x) \tan (x)$ de $3 \tan^{3} (x) \sec (x)$ .

$\frac{3 \sec (x) \tan (x) (\sec^{2} (x)) + 3 \sec (x) \tan (x) (2 \sec (x) \tan (x)) + 3 \sec (x) \tan (x) (\tan^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.7.1.4

Fatore $3 \sec (x) \tan (x)$ de $3 \sec (x) \tan (x) (\sec^{2} (x)) + 3 \sec (x) \tan (x) (2 \sec (x) \tan (x))$ .

$\frac{3 \sec (x) \tan (x) (\sec^{2} (x) + 2 \sec (x) \tan (x)) + 3 \sec (x) \tan (x) (\tan^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.7.1.5

Fatore $3 \sec (x) \tan (x)$ de $3 \sec (x) \tan (x) (\sec^{2} (x) + 2 \sec (x) \tan (x)) + 3 \sec (x) \tan (x) (\tan^{2} (x))$ .

$\frac{3 \sec (x) \tan (x) (\sec^{2} (x) + 2 \sec (x) \tan (x) + \tan^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.7.2

Fatore usando a regra do quadrado perfeito.

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Etapa 2.12.7.2.1

Verifique se o termo do meio é duas vezes o produto dos números ao quadrado no primeiro e no terceiro termos.

$2 \sec (x) \tan (x) = 2 \cdot \sec (x) \cdot \tan (x)$

Etapa 2.12.7.2.2

Reescreva o polinômio.

$\frac{3 \sec (x) \tan (x) (\sec^{2} (x) + 2 \cdot \sec (x) \cdot \tan (x) + \tan^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.7.2.3

Fatore usando a regra do trinômio quadrado perfeito $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , em que $a = \sec (x)$ e $b = \tan (x)$ .

$\frac{3 \sec (x) \tan (x) {(\sec (x) + \tan (x))}^{2}}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.8

Cancele o fator comum de ${(\sec (x) + \tan (x))}^{2}$ .

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Etapa 2.12.8.1

Cancele o fator comum.

$\frac{3 \sec (x) \tan (x) {(\sec (x) + \tan (x))}^{2}}{{(\sec (x) + \tan (x))}^{2}}$

Etapa 2.12.8.2

Divida $3 \sec (x) \tan (x)$ por $1$ .

$f'' (x) = 3 \sec (x) \tan (x)$

Etapa 3

A segunda derivada de $f (x)$ com relação a $x$ é $3 \sec (x) \tan (x)$ .

$3 \sec (x) \tan (x)$