Cálculo Exemplos

$y = \sin (\tan (\sqrt{1 + x^{3}}))$

Etapa 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ para reescrever $\sqrt{1 + x^{3}}$ como ${(1 + x^{3})}^{\frac{1}{2}}$ .

$\frac{d}{d x} [\sin (\tan ({(1 + x^{3})}^{\frac{1}{2}}))]$

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 $f (x) = \sin (x)$ e $g (x) = \tan ({(1 + x^{3})}^{\frac{1}{2}})$ .

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Etapa 2.1

Para aplicar a regra da cadeia, defina $u_{1}$ como $\tan ({(1 + x^{3})}^{\frac{1}{2}})$ .

$\frac{d}{d u_{1}} [\sin (u_{1})] \frac{d}{d x} [\tan ({(1 + x^{3})}^{\frac{1}{2}})]$

Etapa 2.2

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

$\cos (u_{1}) \frac{d}{d x} [\tan ({(1 + x^{3})}^{\frac{1}{2}})]$

Etapa 2.3

Substitua todas as ocorrências de $u_{1}$ por $\tan ({(1 + x^{3})}^{\frac{1}{2}})$ .

$\cos (\tan ({(1 + x^{3})}^{\frac{1}{2}})) \frac{d}{d x} [\tan ({(1 + x^{3})}^{\frac{1}{2}})]$

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 $f (x) = \tan (x)$ e $g (x) = {(1 + x^{3})}^{\frac{1}{2}}$ .

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Etapa 3.1

Para aplicar a regra da cadeia, defina $u_{2}$ como ${(1 + x^{3})}^{\frac{1}{2}}$ .

$\cos (\tan ({(1 + x^{3})}^{\frac{1}{2}})) (\frac{d}{d u_{2}} [\tan (u_{2})] \frac{d}{d x} [{(1 + x^{3})}^{\frac{1}{2}}])$

Etapa 3.2

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

$\cos (\tan ({(1 + x^{3})}^{\frac{1}{2}})) (\sec^{2} (u_{2}) \frac{d}{d x} [{(1 + x^{3})}^{\frac{1}{2}}])$

Etapa 3.3

Substitua todas as ocorrências de $u_{2}$ por ${(1 + x^{3})}^{\frac{1}{2}}$ .

$\cos (\tan ({(1 + x^{3})}^{\frac{1}{2}})) (\sec^{2} ({(1 + x^{3})}^{\frac{1}{2}}) \frac{d}{d x} [{(1 + x^{3})}^{\frac{1}{2}}])$

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 $f (x) = x^{\frac{1}{2}}$ e $g (x) = 1 + x^{3}$ .

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Etapa 4.1

Para aplicar a regra da cadeia, defina $u_{3}$ como $1 + x^{3}$ .

$\cos (\tan ({(1 + x^{3})}^{\frac{1}{2}})) \sec^{2} ({(1 + x^{3})}^{\frac{1}{2}}) (\frac{d}{d u_{3}} [{u_{3}}^{\frac{1}{2}}] \frac{d}{d x} [1 + x^{3}])$

Etapa 4.2

Diferencie usando a regra da multiplicação de potências, que determina que $\frac{d}{d u_{3}} [{u_{3}}^{n}]$ é $n {u_{3}}^{n - 1}$ , em que $n = \frac{1}{2}$ .

$\cos (\tan ({(1 + x^{3})}^{\frac{1}{2}})) \sec^{2} ({(1 + x^{3})}^{\frac{1}{2}}) (\frac{1}{2} {u_{3}}^{\frac{1}{2} - 1} \frac{d}{d x} [1 + x^{3}])$

Etapa 4.3

Substitua todas as ocorrências de $u_{3}$ por $1 + x^{3}$ .

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

Etapa 5

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

$\cos (\tan ({(1 + x^{3})}^{\frac{1}{2}})) \sec^{2} ({(1 + x^{3})}^{\frac{1}{2}}) (\frac{1}{2} {(1 + x^{3})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [1 + x^{3}])$

Etapa 6

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

$\cos (\tan ({(1 + x^{3})}^{\frac{1}{2}})) \sec^{2} ({(1 + x^{3})}^{\frac{1}{2}}) (\frac{1}{2} {(1 + x^{3})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [1 + x^{3}])$

Etapa 7

Combine os numeradores em relação ao denominador comum.

$\cos (\tan ({(1 + x^{3})}^{\frac{1}{2}})) \sec^{2} ({(1 + x^{3})}^{\frac{1}{2}}) (\frac{1}{2} {(1 + x^{3})}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [1 + x^{3}])$

Etapa 8

Simplifique o numerador.

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Etapa 8.1

Multiplique $- 1$ por $2$ .

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

Etapa 8.2

Subtraia $2$ de $1$ .

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

Etapa 9

Combine frações.

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Etapa 9.1

Mova o número negativo para a frente da fração.

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

Etapa 9.2

Combine $\frac{1}{2}$ e ${(1 + x^{3})}^{- \frac{1}{2}}$ .

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

Etapa 9.3

Mova ${(1 + x^{3})}^{- \frac{1}{2}}$ para o denominador usando a regra do expoente negativo $b^{- n} = \frac{1}{b^{n}}$ .

$\cos (\tan ({(1 + x^{3})}^{\frac{1}{2}})) \sec^{2} ({(1 + x^{3})}^{\frac{1}{2}}) (\frac{1}{2 {(1 + x^{3})}^{\frac{1}{2}}} \frac{d}{d x} [1 + x^{3}])$

Etapa 9.4

Combine $\frac{1}{2 {(1 + x^{3})}^{\frac{1}{2}}}$ e $\cos (\tan ({(1 + x^{3})}^{\frac{1}{2}}))$ .

$\frac{\cos (\tan ({(1 + x^{3})}^{\frac{1}{2}}))}{2 {(1 + x^{3})}^{\frac{1}{2}}} \sec^{2} ({(1 + x^{3})}^{\frac{1}{2}}) \frac{d}{d x} [1 + x^{3}]$

Etapa 9.5

Combine $\frac{\cos (\tan ({(1 + x^{3})}^{\frac{1}{2}}))}{2 {(1 + x^{3})}^{\frac{1}{2}}}$ e $\sec^{2} ({(1 + x^{3})}^{\frac{1}{2}})$ .

$\frac{\cos (\tan ({(1 + x^{3})}^{\frac{1}{2}})) \sec^{2} ({(1 + x^{3})}^{\frac{1}{2}})}{2 {(1 + x^{3})}^{\frac{1}{2}}} \frac{d}{d x} [1 + x^{3}]$

Etapa 10

De acordo com a regra da soma, a derivada de $1 + x^{3}$ com relação a $x$ é $\frac{d}{d x} [1] + \frac{d}{d x} [x^{3}]$ .

$\frac{\cos (\tan ({(1 + x^{3})}^{\frac{1}{2}})) \sec^{2} ({(1 + x^{3})}^{\frac{1}{2}})}{2 {(1 + x^{3})}^{\frac{1}{2}}} (\frac{d}{d x} [1] + \frac{d}{d x} [x^{3}])$

Etapa 11

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

$\frac{\cos (\tan ({(1 + x^{3})}^{\frac{1}{2}})) \sec^{2} ({(1 + x^{3})}^{\frac{1}{2}})}{2 {(1 + x^{3})}^{\frac{1}{2}}} (0 + \frac{d}{d x} [x^{3}])$

Etapa 12

Some $0$ e $\frac{d}{d x} [x^{3}]$ .

$\frac{\cos (\tan ({(1 + x^{3})}^{\frac{1}{2}})) \sec^{2} ({(1 + x^{3})}^{\frac{1}{2}})}{2 {(1 + x^{3})}^{\frac{1}{2}}} \frac{d}{d x} [x^{3}]$

Etapa 13

Diferencie usando a regra da multiplicação de potências, que determina que $\frac{d}{d x} [x^{n}]$ é $n x^{n - 1}$ , em que $n = 3$ .

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

Etapa 14

Combine frações.

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Etapa 14.1

Combine $3$ e $\frac{\cos (\tan ({(1 + x^{3})}^{\frac{1}{2}})) \sec^{2} ({(1 + x^{3})}^{\frac{1}{2}})}{2 {(1 + x^{3})}^{\frac{1}{2}}}$ .

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

Etapa 14.2

Combine $\frac{3 (\cos (\tan ({(1 + x^{3})}^{\frac{1}{2}})) \sec^{2} ({(1 + x^{3})}^{\frac{1}{2}}))}{2 {(1 + x^{3})}^{\frac{1}{2}}}$ e $x^{2}$ .

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

Etapa 15

Simplifique.

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Etapa 15.1

Reordene os termos.

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

Etapa 15.2

Simplifique o numerador.

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Etapa 15.2.1

Reescreva $\sec ({(1 + x^{3})}^{\frac{1}{2}})$ em termos de senos e cossenos.

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

Etapa 15.2.2

Aplique a regra do produto a $\frac{1}{\cos ({(1 + x^{3})}^{\frac{1}{2}})}$ .

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

Etapa 15.2.3

Um elevado a qualquer potência é um.

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

Etapa 15.2.4

Combine expoentes.

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Etapa 15.2.4.1

Combine $3$ e $\frac{1}{\cos^{2} ({(1 + x^{3})}^{\frac{1}{2}})}$ .

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

Etapa 15.2.4.2

Combine $x^{2}$ e $\frac{3}{\cos^{2} ({(1 + x^{3})}^{\frac{1}{2}})}$ .

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

Etapa 15.2.4.3

Combine $\frac{x^{2} \cdot 3}{\cos^{2} ({(1 + x^{3})}^{\frac{1}{2}})}$ e $\cos (\tan ({(1 + x^{3})}^{\frac{1}{2}}))$ .

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

Etapa 15.2.5

Mova $3$ para a esquerda de $x^{2}$ .

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

Etapa 15.3

Multiplique o numerador pelo inverso do denominador.

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

Etapa 15.4

Combine.

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

Etapa 15.5

Multiplique $3$ por $1$ .

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

Etapa 15.6

Mova $2$ para a esquerda de $\cos^{2} ({(1 + x^{3})}^{\frac{1}{2}})$ .

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

Etapa 15.7

Fatore $\cos ({(1 + x^{3})}^{\frac{1}{2}})$ de $\cos^{2} ({(1 + x^{3})}^{\frac{1}{2}})$ .

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

Etapa 15.8

Separe as frações.

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

Etapa 15.9

Reescreva $\frac{\cos (\tan ({(1 + x^{3})}^{\frac{1}{2}}))}{\cos ({(1 + x^{3})}^{\frac{1}{2}})}$ como um produto.

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

Etapa 15.10

Escreva $\cos (\tan ({(1 + x^{3})}^{\frac{1}{2}}))$ como uma fração com denominador $1$ .

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

Etapa 15.11

Simplifique.

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Etapa 15.11.1

Divida $\cos (\tan ({(1 + x^{3})}^{\frac{1}{2}}))$ por $1$ .

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

Etapa 15.11.2

Converta de $\frac{1}{\cos ({(1 + x^{3})}^{\frac{1}{2}})}$ em $\sec ({(1 + x^{3})}^{\frac{1}{2}})$ .

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

Etapa 15.12

Multiplique $\frac{3 x^{2}}{2 \cos ({(1 + x^{3})}^{\frac{1}{2}}) {(1 + x^{3})}^{\frac{1}{2}}} (\cos (\tan ({(1 + x^{3})}^{\frac{1}{2}})) \sec ({(1 + x^{3})}^{\frac{1}{2}}))$ .

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Etapa 15.12.1

Combine $\cos (\tan ({(1 + x^{3})}^{\frac{1}{2}}))$ e $\frac{3 x^{2}}{2 \cos ({(1 + x^{3})}^{\frac{1}{2}}) {(1 + x^{3})}^{\frac{1}{2}}}$ .

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

Etapa 15.12.2

Combine $\frac{\cos (\tan ({(1 + x^{3})}^{\frac{1}{2}})) (3 x^{2})}{2 \cos ({(1 + x^{3})}^{\frac{1}{2}}) {(1 + x^{3})}^{\frac{1}{2}}}$ e $\sec ({(1 + x^{3})}^{\frac{1}{2}})$ .

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

Etapa 15.13

Separe as frações.

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

Etapa 15.14

Reescreva $\frac{\cos (\tan ({(1 + x^{3})}^{\frac{1}{2}}))}{\cos ({(1 + x^{3})}^{\frac{1}{2}})}$ como um produto.

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

Etapa 15.15

Escreva $\cos (\tan ({(1 + x^{3})}^{\frac{1}{2}}))$ como uma fração com denominador $1$ .

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

Etapa 15.16

Simplifique.

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Etapa 15.16.1

Divida $\cos (\tan ({(1 + x^{3})}^{\frac{1}{2}}))$ por $1$ .

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

Etapa 15.16.2

Converta de $\frac{1}{\cos ({(1 + x^{3})}^{\frac{1}{2}})}$ em $\sec ({(1 + x^{3})}^{\frac{1}{2}})$ .

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

Etapa 15.17

Multiplique $\frac{(3 x^{2}) \sec ({(1 + x^{3})}^{\frac{1}{2}})}{2 {(1 + x^{3})}^{\frac{1}{2}}} (\cos (\tan ({(1 + x^{3})}^{\frac{1}{2}})) \sec ({(1 + x^{3})}^{\frac{1}{2}}))$ .

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Etapa 15.17.1

Combine $\cos (\tan ({(1 + x^{3})}^{\frac{1}{2}}))$ e $\frac{(3 x^{2}) \sec ({(1 + x^{3})}^{\frac{1}{2}})}{2 {(1 + x^{3})}^{\frac{1}{2}}}$ .

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

Etapa 15.17.2

Combine $\frac{\cos (\tan ({(1 + x^{3})}^{\frac{1}{2}})) ((3 x^{2}) \sec ({(1 + x^{3})}^{\frac{1}{2}}))}{2 {(1 + x^{3})}^{\frac{1}{2}}}$ e $\sec ({(1 + x^{3})}^{\frac{1}{2}})$ .

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

Etapa 15.17.3

Eleve $\sec ({(1 + x^{3})}^{\frac{1}{2}})$ à potência de $1$ .

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

Etapa 15.17.4

Eleve $\sec ({(1 + x^{3})}^{\frac{1}{2}})$ à potência de $1$ .

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

Etapa 15.17.5

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

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

Etapa 15.17.6

Some $1$ e $1$ .

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

Etapa 15.18

Simplifique o numerador.

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Etapa 15.18.1

Reescreva.

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

Etapa 15.18.2

Eleve $\sec ({(1 + x^{3})}^{\frac{1}{2}})$ à potência de $1$ .

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

Etapa 15.18.3

Eleve $\sec ({(1 + x^{3})}^{\frac{1}{2}})$ à potência de $1$ .

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

Etapa 15.18.4

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

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

Etapa 15.18.5

Some $1$ e $1$ .

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

Etapa 15.18.6

Remova os parênteses desnecessários.

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

Etapa 15.19

Mova $3$ para a esquerda de $\cos (\tan ({(1 + x^{3})}^{\frac{1}{2}}))$ .

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

Etapa 15.20

Reordene os fatores em $\frac{3 \cos (\tan ({(1 + x^{3})}^{\frac{1}{2}})) x^{2} \sec^{2} ({(1 + x^{3})}^{\frac{1}{2}})}{2 {(1 + x^{3})}^{\frac{1}{2}}}$ .

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