Cálculo Exemplos

$y = \sqrt{2 x^{3} + 3 x + 10}$

Etapa 1

Encontre a primeira derivada.

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Etapa 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ para reescrever $\sqrt{2 x^{3} + 3 x + 10}$ como ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}]$

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 $f (x) = x^{\frac{1}{2}}$ e $g (x) = 2 x^{3} + 3 x + 10$ .

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Etapa 1.2.1

Para aplicar a regra da cadeia, defina $u$ como $2 x^{3} + 3 x + 10$ .

$\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [2 x^{3} + 3 x + 10]$

Etapa 1.2.2

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

$\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [2 x^{3} + 3 x + 10]$

Etapa 1.2.3

Substitua todas as ocorrências de $u$ por $2 x^{3} + 3 x + 10$ .

$\frac{1}{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2} - 1} \frac{d}{d x} [2 x^{3} + 3 x + 10]$

Etapa 1.3

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

$\frac{1}{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [2 x^{3} + 3 x + 10]$

Etapa 1.4

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

$\frac{1}{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [2 x^{3} + 3 x + 10]$

Etapa 1.5

Combine os numeradores em relação ao denominador comum.

$\frac{1}{2} {(2 x^{3} + 3 x + 10)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [2 x^{3} + 3 x + 10]$

Etapa 1.6

Simplifique o numerador.

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Etapa 1.6.1

Multiplique $- 1$ por $2$ .

$\frac{1}{2} {(2 x^{3} + 3 x + 10)}^{\frac{1 - 2}{2}} \frac{d}{d x} [2 x^{3} + 3 x + 10]$

Etapa 1.6.2

Subtraia $2$ de $1$ .

$\frac{1}{2} {(2 x^{3} + 3 x + 10)}^{\frac{- 1}{2}} \frac{d}{d x} [2 x^{3} + 3 x + 10]$

Etapa 1.7

Combine frações.

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Etapa 1.7.1

Mova o número negativo para a frente da fração.

$\frac{1}{2} {(2 x^{3} + 3 x + 10)}^{- \frac{1}{2}} \frac{d}{d x} [2 x^{3} + 3 x + 10]$

Etapa 1.7.2

Combine $\frac{1}{2}$ e ${(2 x^{3} + 3 x + 10)}^{- \frac{1}{2}}$ .

$\frac{{(2 x^{3} + 3 x + 10)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [2 x^{3} + 3 x + 10]$

Etapa 1.7.3

Mova ${(2 x^{3} + 3 x + 10)}^{- \frac{1}{2}}$ para o denominador usando a regra do expoente negativo $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} \frac{d}{d x} [2 x^{3} + 3 x + 10]$

Etapa 1.8

De acordo com a regra da soma, a derivada de $2 x^{3} + 3 x + 10$ com relação a $x$ é $\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [10]$ .

$\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [10])$

Etapa 1.9

Como $2$ é constante em relação a $x$ , a derivada de $2 x^{3}$ em relação a $x$ é $2 \frac{d}{d x} [x^{3}]$ .

$\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (2 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [10])$

Etapa 1.10

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

$\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (2 (3 x^{2}) + \frac{d}{d x} [3 x] + \frac{d}{d x} [10])$

Etapa 1.11

Multiplique $3$ por $2$ .

$\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + \frac{d}{d x} [3 x] + \frac{d}{d x} [10])$

Etapa 1.12

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

$\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3 \frac{d}{d x} [x] + \frac{d}{d x} [10])$

Etapa 1.13

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

$\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3 \cdot 1 + \frac{d}{d x} [10])$

Etapa 1.14

Multiplique $3$ por $1$ .

$\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3 + \frac{d}{d x} [10])$

Etapa 1.15

Como $10$ é constante em relação a $x$ , a derivada de $10$ em relação a $x$ é $0$ .

$\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3 + 0)$

Etapa 1.16

Some $6 x^{2} + 3$ e $0$ .

$\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3)$

Etapa 1.17

Simplifique.

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Etapa 1.17.1

Reordene os fatores de $\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3)$ .

$(6 x^{2} + 3) \frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}$

Etapa 1.17.2

Multiplique $6 x^{2} + 3$ por $\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}$ .

$\frac{6 x^{2} + 3}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}$

Etapa 1.17.3

Fatore $3$ de $6 x^{2} + 3$ .

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Etapa 1.17.3.1

Fatore $3$ de $6 x^{2}$ .

$\frac{3 (2 x^{2}) + 3}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}$

Etapa 1.17.3.2

Fatore $3$ de $3$ .

$\frac{3 (2 x^{2}) + 3 (1)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}$

Etapa 1.17.3.3

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

$f' (x) = \frac{3 (2 x^{2} + 1)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}$

Etapa 2

Encontre a segunda derivada.

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Etapa 2.1

Como $\frac{3}{2}$ é constante em relação a $x$ , a derivada de $\frac{3 (2 x^{2} + 1)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}$ em relação a $x$ é $\frac{3}{2} \frac{d}{d x} [\frac{2 x^{2} + 1}{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}]$ .

$\frac{3}{2} \frac{d}{d x} [\frac{2 x^{2} + 1}{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}]$

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 $f (x) = 2 x^{2} + 1$ e $g (x) = {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

$\frac{3}{2} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \frac{d}{d x} [2 x^{2} + 1] - (2 x^{2} + 1) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}]}{{({(2 x^{3} + 3 x + 10)}^{\frac{1}{2}})}^{2}}$

Etapa 2.3

Multiplique os expoentes em ${({(2 x^{3} + 3 x + 10)}^{\frac{1}{2}})}^{2}$ .

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Etapa 2.3.1

Aplique a regra da multiplicação de potências e multiplique os expoentes, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{3}{2} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \frac{d}{d x} [2 x^{2} + 1] - (2 x^{2} + 1) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}]}{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2} \cdot 2}}$

Etapa 2.3.2

Cancele o fator comum de $2$ .

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Etapa 2.3.2.1

Cancele o fator comum.

Etapa 2.3.2.2

Reescreva a expressão.

$\frac{3}{2} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \frac{d}{d x} [2 x^{2} + 1] - (2 x^{2} + 1) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}]}{{(2 x^{3} + 3 x + 10)}^{1}}$

Etapa 2.4

Simplifique.

$\frac{3}{2} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \frac{d}{d x} [2 x^{2} + 1] - (2 x^{2} + 1) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}]}{2 x^{3} + 3 x + 10}$

Etapa 2.5

Diferencie.

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Etapa 2.5.1

De acordo com a regra da soma, a derivada de $2 x^{2} + 1$ com relação a $x$ é $\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [1]$ .

$\frac{3}{2} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [1]) - (2 x^{2} + 1) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}]}{2 x^{3} + 3 x + 10}$

Etapa 2.5.2

Como $2$ é constante em relação a $x$ , a derivada de $2 x^{2}$ em relação a $x$ é $2 \frac{d}{d x} [x^{2}]$ .

$\frac{3}{2} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]) - (2 x^{2} + 1) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}]}{2 x^{3} + 3 x + 10}$

Etapa 2.5.3

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

$\frac{3}{2} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 (2 x) + \frac{d}{d x} [1]) - (2 x^{2} + 1) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}]}{2 x^{3} + 3 x + 10}$

Etapa 2.5.4

Multiplique $2$ por $2$ .

$\frac{3}{2} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (4 x + \frac{d}{d x} [1]) - (2 x^{2} + 1) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}]}{2 x^{3} + 3 x + 10}$

Etapa 2.5.5

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

$\frac{3}{2} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (4 x + 0) - (2 x^{2} + 1) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}]}{2 x^{3} + 3 x + 10}$

Etapa 2.5.6

Simplifique a expressão.

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Etapa 2.5.6.1

Some $4 x$ e $0$ .

$\frac{3}{2} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (4 x) - (2 x^{2} + 1) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}]}{2 x^{3} + 3 x + 10}$

Etapa 2.5.6.2

Mova $4$ para a esquerda de ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

$\frac{3}{2} \cdot \frac{4 \cdot {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}]}{2 x^{3} + 3 x + 10}$

Etapa 2.6

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 $g (x) = 2 x^{3} + 3 x + 10$ .

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Etapa 2.6.1

Para aplicar a regra da cadeia, defina $u$ como $2 x^{3} + 3 x + 10$ .

$\frac{3}{2} \cdot \frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [2 x^{3} + 3 x + 10])}{2 x^{3} + 3 x + 10}$

Etapa 2.6.2

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

$\frac{3}{2} \cdot \frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [2 x^{3} + 3 x + 10])}{2 x^{3} + 3 x + 10}$

Etapa 2.6.3

Substitua todas as ocorrências de $u$ por $2 x^{3} + 3 x + 10$ .

$\frac{3}{2} \cdot \frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{1}{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2} - 1} \frac{d}{d x} [2 x^{3} + 3 x + 10])}{2 x^{3} + 3 x + 10}$

Etapa 2.7

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

$\frac{3}{2} \cdot \frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{1}{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [2 x^{3} + 3 x + 10])}{2 x^{3} + 3 x + 10}$

Etapa 2.8

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

$\frac{3}{2} \cdot \frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{1}{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [2 x^{3} + 3 x + 10])}{2 x^{3} + 3 x + 10}$

Etapa 2.9

Combine os numeradores em relação ao denominador comum.

$\frac{3}{2} \cdot \frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{1}{2} {(2 x^{3} + 3 x + 10)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [2 x^{3} + 3 x + 10])}{2 x^{3} + 3 x + 10}$

Etapa 2.10

Simplifique o numerador.

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Etapa 2.10.1

Multiplique $- 1$ por $2$ .

$\frac{3}{2} \cdot \frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{1}{2} {(2 x^{3} + 3 x + 10)}^{\frac{1 - 2}{2}} \frac{d}{d x} [2 x^{3} + 3 x + 10])}{2 x^{3} + 3 x + 10}$

Etapa 2.10.2

Subtraia $2$ de $1$ .

$\frac{3}{2} \cdot \frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{1}{2} {(2 x^{3} + 3 x + 10)}^{\frac{- 1}{2}} \frac{d}{d x} [2 x^{3} + 3 x + 10])}{2 x^{3} + 3 x + 10}$

Etapa 2.11

Combine frações.

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Etapa 2.11.1

Mova o número negativo para a frente da fração.

$\frac{3}{2} \cdot \frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{1}{2} {(2 x^{3} + 3 x + 10)}^{- \frac{1}{2}} \frac{d}{d x} [2 x^{3} + 3 x + 10])}{2 x^{3} + 3 x + 10}$

Etapa 2.11.2

Combine $\frac{1}{2}$ e ${(2 x^{3} + 3 x + 10)}^{- \frac{1}{2}}$ .

$\frac{3}{2} \cdot \frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{{(2 x^{3} + 3 x + 10)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [2 x^{3} + 3 x + 10])}{2 x^{3} + 3 x + 10}$

Etapa 2.11.3

Mova ${(2 x^{3} + 3 x + 10)}^{- \frac{1}{2}}$ para o denominador usando a regra do expoente negativo $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{3}{2} \cdot \frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} \frac{d}{d x} [2 x^{3} + 3 x + 10])}{2 x^{3} + 3 x + 10}$

Etapa 2.12

De acordo com a regra da soma, a derivada de $2 x^{3} + 3 x + 10$ com relação a $x$ é $\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [10]$ .

$\frac{3}{2} \cdot \frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [10]))}{2 x^{3} + 3 x + 10}$

Etapa 2.13

Como $2$ é constante em relação a $x$ , a derivada de $2 x^{3}$ em relação a $x$ é $2 \frac{d}{d x} [x^{3}]$ .

$\frac{3}{2} \cdot \frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (2 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [10]))}{2 x^{3} + 3 x + 10}$

Etapa 2.14

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

$\frac{3}{2} \cdot \frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (2 (3 x^{2}) + \frac{d}{d x} [3 x] + \frac{d}{d x} [10]))}{2 x^{3} + 3 x + 10}$

Etapa 2.15

Multiplique $3$ por $2$ .

$\frac{3}{2} \cdot \frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + \frac{d}{d x} [3 x] + \frac{d}{d x} [10]))}{2 x^{3} + 3 x + 10}$

Etapa 2.16

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

$\frac{3}{2} \cdot \frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3 \frac{d}{d x} [x] + \frac{d}{d x} [10]))}{2 x^{3} + 3 x + 10}$

Etapa 2.17

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

$\frac{3}{2} \cdot \frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3 \cdot 1 + \frac{d}{d x} [10]))}{2 x^{3} + 3 x + 10}$

Etapa 2.18

Multiplique $3$ por $1$ .

$\frac{3}{2} \cdot \frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3 + \frac{d}{d x} [10]))}{2 x^{3} + 3 x + 10}$

Etapa 2.19

Como $10$ é constante em relação a $x$ , a derivada de $10$ em relação a $x$ é $0$ .

$\frac{3}{2} \cdot \frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3 + 0))}{2 x^{3} + 3 x + 10}$

Etapa 2.20

Combine frações.

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Etapa 2.20.1

Some $6 x^{2} + 3$ e $0$ .

$\frac{3}{2} \cdot \frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3))}{2 x^{3} + 3 x + 10}$

Etapa 2.20.2

Multiplique $\frac{3}{2}$ por $\frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3))}{2 x^{3} + 3 x + 10}$ .

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3)))}{2 (2 x^{3} + 3 x + 10)}$

Etapa 2.21

Simplifique.

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Etapa 2.21.1

Aplique a propriedade distributiva.

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + (- (2 x^{2}) - 1 \cdot 1) (\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3)))}{2 (2 x^{3} + 3 x + 10)}$

Etapa 2.21.2

Aplique a propriedade distributiva.

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x) + 3 ((- (2 x^{2}) - 1 \cdot 1) (\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3)))}{2 (2 x^{3} + 3 x + 10)}$

Etapa 2.21.3

Aplique a propriedade distributiva.

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x) + 3 ((- (2 x^{2}) - 1 \cdot 1) (\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3)))}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4

Simplifique o numerador.

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Etapa 2.21.4.1

Fatore $3$ de $3 \cdot 4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + 3 (- 1 \cdot 2 x^{2} - 1 \cdot 1) \frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3)$ .

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Etapa 2.21.4.1.1

Fatore $3$ de $3 \cdot 4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x$ .

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x) + 3 (- 1 \cdot 2 x^{2} - 1 \cdot 1) \frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3)}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.1.2

Fatore $3$ de $3 (- 1 \cdot 2 x^{2} - 1 \cdot 1) \frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3)$ .

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x) + 3 ((- 1 \cdot 2 x^{2} - 1 \cdot 1) \frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3))}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.1.3

Fatore $3$ de $3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x) + 3 ((- 1 \cdot 2 x^{2} - 1 \cdot 1) \frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3))$ .

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + (- 1 \cdot 2 x^{2} - 1 \cdot 1) \frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3))}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.2

Simplifique cada termo.

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Etapa 2.21.4.2.1

Multiplique $- 1$ por $2$ .

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + (- 2 x^{2} - 1 \cdot 1) \frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3))}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.2.2

Multiplique $- 1$ por $1$ .

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + (- 2 x^{2} - 1) \frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3))}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.3

Multiplique $- 2 x^{2} - 1$ por $\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}$ .

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + \frac{- 2 x^{2} - 1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3))}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.4

Multiplique $\frac{- 2 x^{2} - 1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}$ por $6 x^{2} + 3$ .

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + \frac{(- 2 x^{2} - 1) (6 x^{2} + 3)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}})}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.5

Simplifique o numerador.

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Etapa 2.21.4.5.1

Fatore $3$ de $6 x^{2} + 3$ .

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Etapa 2.21.4.5.1.1

Fatore $3$ de $6 x^{2}$ .

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + \frac{(- 2 x^{2} - 1) (3 (2 x^{2}) + 3)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}})}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.5.1.2

Fatore $3$ de $3$ .

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + \frac{(- 2 x^{2} - 1) (3 (2 x^{2}) + 3 (1))}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}})}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.5.1.3

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

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + \frac{(- 2 x^{2} - 1) (3 (2 x^{2} + 1))}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}})}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + \frac{(- 2 x^{2} - 1) \cdot 3 (2 x^{2} + 1)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}})}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.5.2

Combine expoentes.

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Etapa 2.21.4.5.2.1

Fatore $- 1$ de $- 2 x^{2}$ .

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + \frac{(- (2 x^{2}) - 1) \cdot 3 (2 x^{2} + 1)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}})}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.5.2.2

Reescreva $- 1$ como $- 1 (1)$ .

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + \frac{(- (2 x^{2}) - 1 (1)) \cdot 3 (2 x^{2} + 1)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}})}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.5.2.3

Fatore $- 1$ de $- (2 x^{2}) - 1 (1)$ .

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + \frac{- (2 x^{2} + 1) \cdot 3 (2 x^{2} + 1)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}})}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.5.2.4

Reescreva $- (2 x^{2} + 1)$ como $- 1 (2 x^{2} + 1)$ .

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + \frac{- 1 (2 x^{2} + 1) \cdot 3 (2 x^{2} + 1)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}})}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.5.2.5

Eleve $2 x^{2} + 1$ à potência de $1$ .

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + \frac{- 1 \cdot 3 ({(2 x^{2} + 1)}^{1} (2 x^{2} + 1))}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}})}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.5.2.6

Eleve $2 x^{2} + 1$ à potência de $1$ .

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + \frac{- 1 \cdot 3 ({(2 x^{2} + 1)}^{1} {(2 x^{2} + 1)}^{1})}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}})}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.5.2.7

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

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + \frac{- 1 \cdot 3 {(2 x^{2} + 1)}^{1 + 1}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}})}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.5.2.8

Some $1$ e $1$ .

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + \frac{- 1 \cdot 3 {(2 x^{2} + 1)}^{2}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}})}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.5.2.9

Multiplique $- 1$ por $3$ .

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + \frac{- 3 {(2 x^{2} + 1)}^{2}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}})}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.6

Mova o número negativo para a frente da fração.

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - \frac{3 {(2 x^{2} + 1)}^{2}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}})}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.7

Para escrever $4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x$ como fração com um denominador comum, multiplique por $\frac{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}$ .

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x \cdot \frac{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} - \frac{3 {(2 x^{2} + 1)}^{2}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}})}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.8

Combine $4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x$ e $\frac{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}$ .

$\frac{3 (\frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x (2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}})}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} - \frac{3 {(2 x^{2} + 1)}^{2}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}})}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.9

Combine os numeradores em relação ao denominador comum.

$\frac{3 \frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x (2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}) - 3 {(2 x^{2} + 1)}^{2}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.10

Reordene os termos.

$\frac{3 \frac{- 3 {(2 x^{2} + 1)}^{2} + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.11

Reescreva $\frac{- 3 {(2 x^{2} + 1)}^{2} + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}$ em uma forma fatorada.

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Etapa 2.21.4.11.1

Reescreva ${(2 x^{2} + 1)}^{2}$ como $(2 x^{2} + 1) (2 x^{2} + 1)$ .

$\frac{3 \frac{- 3 ((2 x^{2} + 1) (2 x^{2} + 1)) + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.11.2

Expanda $(2 x^{2} + 1) (2 x^{2} + 1)$ usando o método FOIL.

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Etapa 2.21.4.11.2.1

Aplique a propriedade distributiva.

$\frac{3 \frac{- 3 (2 x^{2} (2 x^{2} + 1) + 1 (2 x^{2} + 1)) + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.11.2.2

Aplique a propriedade distributiva.

$\frac{3 \frac{- 3 (2 x^{2} (2 x^{2}) + 2 x^{2} \cdot 1 + 1 (2 x^{2} + 1)) + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.11.2.3

Aplique a propriedade distributiva.

$\frac{3 \frac{- 3 (2 x^{2} (2 x^{2}) + 2 x^{2} \cdot 1 + 1 (2 x^{2}) + 1 \cdot 1) + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.11.3

Simplifique e combine termos semelhantes.

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Etapa 2.21.4.11.3.1

Simplifique cada termo.

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Etapa 2.21.4.11.3.1.1

Reescreva usando a propriedade comutativa da multiplicação.

$\frac{3 \frac{- 3 (2 \cdot 2 x^{2} x^{2} + 2 x^{2} \cdot 1 + 1 (2 x^{2}) + 1 \cdot 1) + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.11.3.1.2

Multiplique $x^{2}$ por $x^{2}$ somando os expoentes.

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Etapa 2.21.4.11.3.1.2.1

Mova $x^{2}$ .

$\frac{3 \frac{- 3 (2 \cdot 2 (x^{2} x^{2}) + 2 x^{2} \cdot 1 + 1 (2 x^{2}) + 1 \cdot 1) + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.11.3.1.2.2

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

$\frac{3 \frac{- 3 (2 \cdot 2 x^{2 + 2} + 2 x^{2} \cdot 1 + 1 (2 x^{2}) + 1 \cdot 1) + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.11.3.1.2.3

Some $2$ e $2$ .

$\frac{3 \frac{- 3 (2 \cdot 2 x^{4} + 2 x^{2} \cdot 1 + 1 (2 x^{2}) + 1 \cdot 1) + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.11.3.1.3

Multiplique $2$ por $2$ .

$\frac{3 \frac{- 3 (4 x^{4} + 2 x^{2} \cdot 1 + 1 (2 x^{2}) + 1 \cdot 1) + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.11.3.1.4

Multiplique $2$ por $1$ .

$\frac{3 \frac{- 3 (4 x^{4} + 2 x^{2} + 1 (2 x^{2}) + 1 \cdot 1) + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.11.3.1.5

Multiplique $2 x^{2}$ por $1$ .

$\frac{3 \frac{- 3 (4 x^{4} + 2 x^{2} + 2 x^{2} + 1 \cdot 1) + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.11.3.1.6

Multiplique $1$ por $1$ .

$\frac{3 \frac{- 3 (4 x^{4} + 2 x^{2} + 2 x^{2} + 1) + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.11.3.2

Some $2 x^{2}$ e $2 x^{2}$ .

$\frac{3 \frac{- 3 (4 x^{4} + 4 x^{2} + 1) + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.11.4

Aplique a propriedade distributiva.

$\frac{3 \frac{- 3 (4 x^{4}) - 3 (4 x^{2}) - 3 \cdot 1 + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.11.5

Simplifique.

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Etapa 2.21.4.11.5.1

Multiplique $4$ por $- 3$ .

$\frac{3 \frac{- 12 x^{4} - 3 (4 x^{2}) - 3 \cdot 1 + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.11.5.2

Multiplique $4$ por $- 3$ .

$\frac{3 \frac{- 12 x^{4} - 12 x^{2} - 3 \cdot 1 + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.11.5.3

Multiplique $- 3$ por $1$ .

$\frac{3 \frac{- 12 x^{4} - 12 x^{2} - 3 + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.11.6

Multiplique ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ por ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ somando os expoentes.

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Etapa 2.21.4.11.6.1

Mova ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

$\frac{3 \frac{- 12 x^{4} - 12 x^{2} - 3 + 2 \cdot 4 x ({(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}})}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.11.6.2

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

$\frac{3 \frac{- 12 x^{4} - 12 x^{2} - 3 + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2} + \frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.11.6.3

Combine os numeradores em relação ao denominador comum.

$\frac{3 \frac{- 12 x^{4} - 12 x^{2} - 3 + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{1 + 1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.11.6.4

Some $1$ e $1$ .

$\frac{3 \frac{- 12 x^{4} - 12 x^{2} - 3 + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.11.6.5

Divida $2$ por $2$ .

$\frac{3 \frac{- 12 x^{4} - 12 x^{2} - 3 + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{1}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.11.7

Simplifique $2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{1}$ .

$\frac{3 \frac{- 12 x^{4} - 12 x^{2} - 3 + 2 \cdot 4 x (2 x^{3} + 3 x + 10)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.11.8

Multiplique $2$ por $4$ .

$\frac{3 \frac{- 12 x^{4} - 12 x^{2} - 3 + 8 x (2 x^{3} + 3 x + 10)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.11.9

Aplique a propriedade distributiva.

$\frac{3 \frac{- 12 x^{4} - 12 x^{2} - 3 + 8 x (2 x^{3}) + 8 x (3 x) + 8 x \cdot 10}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.11.10

Simplifique.

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Etapa 2.21.4.11.10.1

Reescreva usando a propriedade comutativa da multiplicação.

$\frac{3 \frac{- 12 x^{4} - 12 x^{2} - 3 + 8 \cdot 2 x \cdot x^{3} + 8 x (3 x) + 8 x \cdot 10}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.11.10.2

Reescreva usando a propriedade comutativa da multiplicação.

$\frac{3 \frac{- 12 x^{4} - 12 x^{2} - 3 + 8 \cdot 2 x \cdot x^{3} + 8 \cdot 3 x \cdot x + 8 x \cdot 10}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.11.10.3

Multiplique $10$ por $8$ .

$\frac{3 \frac{- 12 x^{4} - 12 x^{2} - 3 + 8 \cdot 2 x \cdot x^{3} + 8 \cdot 3 x \cdot x + 80 x}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.11.11

Simplifique cada termo.

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Etapa 2.21.4.11.11.1

Multiplique $x$ por $x^{3}$ somando os expoentes.

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Etapa 2.21.4.11.11.1.1

Mova $x^{3}$ .

$\frac{3 \frac{- 12 x^{4} - 12 x^{2} - 3 + 8 \cdot 2 (x^{3} x) + 8 \cdot 3 x \cdot x + 80 x}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.11.11.1.2

Multiplique $x^{3}$ por $x$ .

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Etapa 2.21.4.11.11.1.2.1

Eleve $x$ à potência de $1$ .

$\frac{3 \frac{- 12 x^{4} - 12 x^{2} - 3 + 8 \cdot 2 (x^{3} x^{1}) + 8 \cdot 3 x \cdot x + 80 x}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.11.11.1.2.2

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

$\frac{3 \frac{- 12 x^{4} - 12 x^{2} - 3 + 8 \cdot 2 x^{3 + 1} + 8 \cdot 3 x \cdot x + 80 x}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.11.11.1.3

Some $3$ e $1$ .

$\frac{3 \frac{- 12 x^{4} - 12 x^{2} - 3 + 8 \cdot 2 x^{4} + 8 \cdot 3 x \cdot x + 80 x}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.11.11.2

Multiplique $8$ por $2$ .

$\frac{3 \frac{- 12 x^{4} - 12 x^{2} - 3 + 16 x^{4} + 8 \cdot 3 x \cdot x + 80 x}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.11.11.3

Multiplique $x$ por $x$ somando os expoentes.

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Etapa 2.21.4.11.11.3.1

Mova $x$ .

$\frac{3 \frac{- 12 x^{4} - 12 x^{2} - 3 + 16 x^{4} + 8 \cdot 3 (x \cdot x) + 80 x}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.11.11.3.2

Multiplique $x$ por $x$ .

$\frac{3 \frac{- 12 x^{4} - 12 x^{2} - 3 + 16 x^{4} + 8 \cdot 3 x^{2} + 80 x}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.11.11.4

Multiplique $8$ por $3$ .

$\frac{3 \frac{- 12 x^{4} - 12 x^{2} - 3 + 16 x^{4} + 24 x^{2} + 80 x}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.11.12

Some $- 12 x^{4}$ e $16 x^{4}$ .

$\frac{3 \frac{4 x^{4} - 12 x^{2} - 3 + 24 x^{2} + 80 x}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.11.13

Some $- 12 x^{2}$ e $24 x^{2}$ .

$\frac{3 \frac{4 x^{4} + 12 x^{2} - 3 + 80 x}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.4.11.14

Reordene os termos.

$\frac{3 \frac{4 x^{4} + 12 x^{2} + 80 x - 3}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.5

Combine os termos.

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Etapa 2.21.5.1

Combine $3$ e $\frac{4 x^{4} + 12 x^{2} + 80 x - 3}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}$ .

$\frac{\frac{3 (4 x^{4} + 12 x^{2} + 80 x - 3)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.5.2

Multiplique $2$ por $2$ .

$\frac{\frac{3 (4 x^{4} + 12 x^{2} + 80 x - 3)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{4 x^{3} + 2 (3 x) + 2 \cdot 10}$

Etapa 2.21.5.3

Multiplique $3$ por $2$ .

$\frac{\frac{3 (4 x^{4} + 12 x^{2} + 80 x - 3)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{4 x^{3} + 6 x + 2 \cdot 10}$

Etapa 2.21.5.4

Multiplique $2$ por $10$ .

$\frac{\frac{3 (4 x^{4} + 12 x^{2} + 80 x - 3)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{4 x^{3} + 6 x + 20}$

Etapa 2.21.5.5

Reescreva $\frac{\frac{3 (4 x^{4} + 12 x^{2} + 80 x - 3)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{4 x^{3} + 6 x + 20}$ como um produto.

$\frac{3 (4 x^{4} + 12 x^{2} + 80 x - 3)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} \cdot \frac{1}{4 x^{3} + 6 x + 20}$

Etapa 2.21.5.6

Multiplique $\frac{3 (4 x^{4} + 12 x^{2} + 80 x - 3)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}$ por $\frac{1}{4 x^{3} + 6 x + 20}$ .

$\frac{3 (4 x^{4} + 12 x^{2} + 80 x - 3)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (4 x^{3} + 6 x + 20)}$

Etapa 2.21.6

Simplifique o denominador.

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Etapa 2.21.6.1

Fatore $2$ de $4 x^{3} + 6 x + 20$ .

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Etapa 2.21.6.1.1

Fatore $2$ de $4 x^{3}$ .

$\frac{3 (4 x^{4} + 12 x^{2} + 80 x - 3)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 (2 x^{3}) + 6 x + 20)}$

Etapa 2.21.6.1.2

Fatore $2$ de $6 x$ .

$\frac{3 (4 x^{4} + 12 x^{2} + 80 x - 3)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 (2 x^{3}) + 2 (3 x) + 20)}$

Etapa 2.21.6.1.3

Fatore $2$ de $20$ .

$\frac{3 (4 x^{4} + 12 x^{2} + 80 x - 3)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 (2 x^{3}) + 2 (3 x) + 2 (10))}$

Etapa 2.21.6.1.4

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

$\frac{3 (4 x^{4} + 12 x^{2} + 80 x - 3)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 (2 x^{3} + 3 x) + 2 (10))}$

Etapa 2.21.6.1.5

Fatore $2$ de $2 (2 x^{3} + 3 x) + 2 (10)$ .

$\frac{3 (4 x^{4} + 12 x^{2} + 80 x - 3)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 (2 x^{3} + 3 x + 10))}$

$\frac{3 (4 x^{4} + 12 x^{2} + 80 x - 3)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \cdot 2 (2 x^{3} + 3 x + 10)}$

Etapa 2.21.6.2

Combine expoentes.

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Etapa 2.21.6.2.1

Multiplique $2$ por $2$ .

$\frac{3 (4 x^{4} + 12 x^{2} + 80 x - 3)}{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 x^{3} + 3 x + 10)}$

Etapa 2.21.6.2.2

Eleve $2 x^{3} + 3 x + 10$ à potência de $1$ .

$\frac{3 (4 x^{4} + 12 x^{2} + 80 x - 3)}{4 ({(2 x^{3} + 3 x + 10)}^{1} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}})}$

Etapa 2.21.6.2.3

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

$\frac{3 (4 x^{4} + 12 x^{2} + 80 x - 3)}{4 {(2 x^{3} + 3 x + 10)}^{1 + \frac{1}{2}}}$

Etapa 2.21.6.2.4

Escreva $1$ como uma fração com um denominador comum.

$\frac{3 (4 x^{4} + 12 x^{2} + 80 x - 3)}{4 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2} + \frac{1}{2}}}$

Etapa 2.21.6.2.5

Combine os numeradores em relação ao denominador comum.

$\frac{3 (4 x^{4} + 12 x^{2} + 80 x - 3)}{4 {(2 x^{3} + 3 x + 10)}^{\frac{2 + 1}{2}}}$

Etapa 2.21.6.2.6

Some $2$ e $1$ .

$f'' (x) = \frac{3 (4 x^{4} + 12 x^{2} + 80 x - 3)}{4 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}}$

Etapa 3

Encontre a terceira derivada.

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Etapa 3.1

Como $\frac{3}{4}$ é constante em relação a $x$ , a derivada de $\frac{3 (4 x^{4} + 12 x^{2} + 80 x - 3)}{4 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}}$ em relação a $x$ é $\frac{3}{4} \frac{d}{d x} [\frac{4 x^{4} + 12 x^{2} + 80 x - 3}{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}}]$ .

$\frac{3}{4} \frac{d}{d x} [\frac{4 x^{4} + 12 x^{2} + 80 x - 3}{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}}]$

Etapa 3.2

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 $f (x) = 4 x^{4} + 12 x^{2} + 80 x - 3$ e $g (x) = {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} \frac{d}{d x} [4 x^{4} + 12 x^{2} + 80 x - 3] - (4 x^{4} + 12 x^{2} + 80 x - 3) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}]}{{({(2 x^{3} + 3 x + 10)}^{\frac{3}{2}})}^{2}}$

Etapa 3.3

Diferencie.

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Etapa 3.3.1

Multiplique os expoentes em ${({(2 x^{3} + 3 x + 10)}^{\frac{3}{2}})}^{2}$ .

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Etapa 3.3.1.1

Aplique a regra da multiplicação de potências e multiplique os expoentes, ${(a^{m})}^{n} = a^{m n}$ .

Etapa 3.3.1.2

Cancele o fator comum de $2$ .

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Etapa 3.3.1.2.1

Cancele o fator comum.

Etapa 3.3.1.2.2

Reescreva a expressão.

Etapa 3.3.2

De acordo com a regra da soma, a derivada de $4 x^{4} + 12 x^{2} + 80 x - 3$ com relação a $x$ é $\frac{d}{d x} [4 x^{4}] + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [80 x] + \frac{d}{d x} [- 3]$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (\frac{d}{d x} [4 x^{4}] + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [80 x] + \frac{d}{d x} [- 3]) - (4 x^{4} + 12 x^{2} + 80 x - 3) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}]}{{(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.3.3

Como $4$ é constante em relação a $x$ , a derivada de $4 x^{4}$ em relação a $x$ é $4 \frac{d}{d x} [x^{4}]$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (4 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [80 x] + \frac{d}{d x} [- 3]) - (4 x^{4} + 12 x^{2} + 80 x - 3) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}]}{{(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.3.4

Diferencie usando a regra da multiplicação de potências, que determina que $\frac{d}{d x} [x^{n}]$ é $n x^{n - 1}$ , em que $n = 4$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (4 (4 x^{3}) + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [80 x] + \frac{d}{d x} [- 3]) - (4 x^{4} + 12 x^{2} + 80 x - 3) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}]}{{(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.3.5

Multiplique $4$ por $4$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [80 x] + \frac{d}{d x} [- 3]) - (4 x^{4} + 12 x^{2} + 80 x - 3) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}]}{{(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.3.6

Como $12$ é constante em relação a $x$ , a derivada de $12 x^{2}$ em relação a $x$ é $12 \frac{d}{d x} [x^{2}]$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 12 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [80 x] + \frac{d}{d x} [- 3]) - (4 x^{4} + 12 x^{2} + 80 x - 3) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}]}{{(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.3.7

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

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 12 (2 x) + \frac{d}{d x} [80 x] + \frac{d}{d x} [- 3]) - (4 x^{4} + 12 x^{2} + 80 x - 3) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}]}{{(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.3.8

Multiplique $2$ por $12$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + \frac{d}{d x} [80 x] + \frac{d}{d x} [- 3]) - (4 x^{4} + 12 x^{2} + 80 x - 3) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}]}{{(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.3.9

Como $80$ é constante em relação a $x$ , a derivada de $80 x$ em relação a $x$ é $80 \frac{d}{d x} [x]$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80 \frac{d}{d x} [x] + \frac{d}{d x} [- 3]) - (4 x^{4} + 12 x^{2} + 80 x - 3) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}]}{{(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.3.10

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

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80 \cdot 1 + \frac{d}{d x} [- 3]) - (4 x^{4} + 12 x^{2} + 80 x - 3) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}]}{{(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.3.11

Multiplique $80$ por $1$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80 + \frac{d}{d x} [- 3]) - (4 x^{4} + 12 x^{2} + 80 x - 3) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}]}{{(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.3.12

Como $- 3$ é constante em relação a $x$ , a derivada de $- 3$ em relação a $x$ é $0$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80 + 0) - (4 x^{4} + 12 x^{2} + 80 x - 3) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}]}{{(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.3.13

Some $16 x^{3} + 24 x + 80$ e $0$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) - (4 x^{4} + 12 x^{2} + 80 x - 3) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}]}{{(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.4

Diferencie usando a regra da cadeia, que determina que $\frac{d}{d x} [f (g (x))]$ é $f' (g (x)) g' (x)$ , em que $f (x) = x^{\frac{3}{2}}$ e $g (x) = 2 x^{3} + 3 x + 10$ .

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Etapa 3.4.1

Para aplicar a regra da cadeia, defina $u$ como $2 x^{3} + 3 x + 10$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) - (4 x^{4} + 12 x^{2} + 80 x - 3) (\frac{d}{d u} [u^{\frac{3}{2}}] \frac{d}{d x} [2 x^{3} + 3 x + 10])}{{(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.4.2

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

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) - (4 x^{4} + 12 x^{2} + 80 x - 3) (\frac{3}{2} u^{\frac{3}{2} - 1} \frac{d}{d x} [2 x^{3} + 3 x + 10])}{{(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.4.3

Substitua todas as ocorrências de $u$ por $2 x^{3} + 3 x + 10$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) - (4 x^{4} + 12 x^{2} + 80 x - 3) (\frac{3}{2} {(2 x^{3} + 3 x + 10)}^{\frac{3}{2} - 1} \frac{d}{d x} [2 x^{3} + 3 x + 10])}{{(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.5

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

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) - (4 x^{4} + 12 x^{2} + 80 x - 3) (\frac{3}{2} {(2 x^{3} + 3 x + 10)}^{\frac{3}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [2 x^{3} + 3 x + 10])}{{(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.6

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

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) - (4 x^{4} + 12 x^{2} + 80 x - 3) (\frac{3}{2} {(2 x^{3} + 3 x + 10)}^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [2 x^{3} + 3 x + 10])}{{(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.7

Combine os numeradores em relação ao denominador comum.

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) - (4 x^{4} + 12 x^{2} + 80 x - 3) (\frac{3}{2} {(2 x^{3} + 3 x + 10)}^{\frac{3 - 1 \cdot 2}{2}} \frac{d}{d x} [2 x^{3} + 3 x + 10])}{{(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.8

Simplifique o numerador.

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Etapa 3.8.1

Multiplique $- 1$ por $2$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) - (4 x^{4} + 12 x^{2} + 80 x - 3) (\frac{3}{2} {(2 x^{3} + 3 x + 10)}^{\frac{3 - 2}{2}} \frac{d}{d x} [2 x^{3} + 3 x + 10])}{{(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.8.2

Subtraia $2$ de $3$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) - (4 x^{4} + 12 x^{2} + 80 x - 3) (\frac{3}{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \frac{d}{d x} [2 x^{3} + 3 x + 10])}{{(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.9

Combine $\frac{3}{2}$ e ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) - (4 x^{4} + 12 x^{2} + 80 x - 3) (\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} \frac{d}{d x} [2 x^{3} + 3 x + 10])}{{(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.10

De acordo com a regra da soma, a derivada de $2 x^{3} + 3 x + 10$ com relação a $x$ é $\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [10]$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) - (4 x^{4} + 12 x^{2} + 80 x - 3) (\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [10]))}{{(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.11

Como $2$ é constante em relação a $x$ , a derivada de $2 x^{3}$ em relação a $x$ é $2 \frac{d}{d x} [x^{3}]$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) - (4 x^{4} + 12 x^{2} + 80 x - 3) (\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (2 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [10]))}{{(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.12

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

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) - (4 x^{4} + 12 x^{2} + 80 x - 3) (\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (2 (3 x^{2}) + \frac{d}{d x} [3 x] + \frac{d}{d x} [10]))}{{(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.13

Multiplique $3$ por $2$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) - (4 x^{4} + 12 x^{2} + 80 x - 3) (\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + \frac{d}{d x} [3 x] + \frac{d}{d x} [10]))}{{(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.14

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

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) - (4 x^{4} + 12 x^{2} + 80 x - 3) (\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3 \frac{d}{d x} [x] + \frac{d}{d x} [10]))}{{(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.15

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

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) - (4 x^{4} + 12 x^{2} + 80 x - 3) (\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3 \cdot 1 + \frac{d}{d x} [10]))}{{(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.16

Multiplique $3$ por $1$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) - (4 x^{4} + 12 x^{2} + 80 x - 3) (\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3 + \frac{d}{d x} [10]))}{{(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.17

Como $10$ é constante em relação a $x$ , a derivada de $10$ em relação a $x$ é $0$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) - (4 x^{4} + 12 x^{2} + 80 x - 3) (\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3 + 0))}{{(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.18

Combine frações.

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Etapa 3.18.1

Some $6 x^{2} + 3$ e $0$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) - (4 x^{4} + 12 x^{2} + 80 x - 3) (\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3))}{{(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.18.2

Multiplique $\frac{3}{4}$ por $\frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) - (4 x^{4} + 12 x^{2} + 80 x - 3) (\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3))}{{(2 x^{3} + 3 x + 10)}^{3}}$ .

$\frac{3 ({(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) - (4 x^{4} + 12 x^{2} + 80 x - 3) (\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3)))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19

Simplifique.

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Etapa 3.19.1

Aplique a propriedade distributiva.

$\frac{3 ({(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) + (- (4 x^{4}) - (12 x^{2}) - (80 x) - - 3) (\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3)))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.2

Aplique a propriedade distributiva.

$\frac{3 ({(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80)) + 3 ((- (4 x^{4}) - (12 x^{2}) - (80 x) - - 3) (\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3)))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3

Simplifique o numerador.

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Etapa 3.19.3.1

Fatore $3$ de $3 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) + 3 (- 1 \cdot 4 x^{4} - 1 \cdot 12 x^{2} - 1 \cdot 80 x - - 3) \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3)$ .

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Etapa 3.19.3.1.1

Fatore $3$ de $3 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80)$ .

$\frac{3 ({(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80)) + 3 (- 1 \cdot 4 x^{4} - 1 \cdot 12 x^{2} - 1 \cdot 80 x - - 3) \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3)}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.1.2

Fatore $3$ de $3 (- 1 \cdot 4 x^{4} - 1 \cdot 12 x^{2} - 1 \cdot 80 x - - 3) \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3)$ .

$\frac{3 ({(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80)) + 3 ((- 1 \cdot 4 x^{4} - 1 \cdot 12 x^{2} - 1 \cdot 80 x - - 3) \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.1.3

Fatore $3$ de $3 ({(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80)) + 3 ((- 1 \cdot 4 x^{4} - 1 \cdot 12 x^{2} - 1 \cdot 80 x - - 3) \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3))$ .

$\frac{3 ({(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) + (- 1 \cdot 4 x^{4} - 1 \cdot 12 x^{2} - 1 \cdot 80 x - - 3) \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.2

Aplique a propriedade distributiva.

$\frac{3 ({(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3}) + {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (24 x) + {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} \cdot 80 + (- 1 \cdot 4 x^{4} - 1 \cdot 12 x^{2} - 1 \cdot 80 x - - 3) \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.3

Simplifique.

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Etapa 3.19.3.3.1

Reescreva usando a propriedade comutativa da multiplicação.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (24 x) + {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} \cdot 80 + (- 1 \cdot 4 x^{4} - 1 \cdot 12 x^{2} - 1 \cdot 80 x - - 3) \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.3.2

Reescreva usando a propriedade comutativa da multiplicação.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} \cdot 80 + (- 1 \cdot 4 x^{4} - 1 \cdot 12 x^{2} - 1 \cdot 80 x - - 3) \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.3.3

Mova $80$ para a esquerda de ${(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 \cdot {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 1 \cdot 4 x^{4} - 1 \cdot 12 x^{2} - 1 \cdot 80 x - - 3) \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.4

Simplifique cada termo.

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Etapa 3.19.3.4.1

Multiplique $- 1$ por $4$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 4 x^{4} - 1 \cdot 12 x^{2} - 1 \cdot 80 x - - 3) \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.4.2

Multiplique $- 1$ por $12$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 4 x^{4} - 12 x^{2} - 1 \cdot 80 x - - 3) \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.4.3

Multiplique $- 1$ por $80$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 4 x^{4} - 12 x^{2} - 80 x - - 3) \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.4.4

Multiplique $- 1$ por $- 3$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 4 x^{4} - 12 x^{2} - 80 x + 3) \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.5

Aplique a propriedade distributiva.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 4 x^{4} \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} - 12 x^{2} \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} - 80 x \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} + 3 \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.6

Simplifique.

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Etapa 3.19.3.6.1

Cancele o fator comum de $2$ .

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Etapa 3.19.3.6.1.1

Fatore $2$ de $- 4 x^{4}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (2 (- 2 x^{4}) \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} - 12 x^{2} \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} - 80 x \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} + 3 \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.6.1.2

Cancele o fator comum.

Etapa 3.19.3.6.1.3

Reescreva a expressão.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 2 x^{4} (3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}) - 12 x^{2} \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} - 80 x \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} + 3 \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.6.2

Multiplique $3$ por $- 2$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 6 x^{4} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 12 x^{2} \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} - 80 x \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} + 3 \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.6.3

Cancele o fator comum de $2$ .

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Etapa 3.19.3.6.3.1

Fatore $2$ de $- 12 x^{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 6 x^{4} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + 2 (- 6 x^{2}) \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} - 80 x \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} + 3 \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.6.3.2

Cancele o fator comum.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 6 x^{4} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + 2 (- 6 x^{2}) \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} - 80 x \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} + 3 \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.6.3.3

Reescreva a expressão.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 6 x^{4} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 6 x^{2} (3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}) - 80 x \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} + 3 \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.6.4

Multiplique $3$ por $- 6$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 6 x^{4} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 80 x \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} + 3 \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.6.5

Cancele o fator comum de $2$ .

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Etapa 3.19.3.6.5.1

Fatore $2$ de $- 80 x$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 6 x^{4} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + 2 (- 40 x) \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} + 3 \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.6.5.2

Cancele o fator comum.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 6 x^{4} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + 2 (- 40 x) \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} + 3 \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.6.5.3

Reescreva a expressão.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 6 x^{4} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 40 x (3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}) + 3 \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.6.6

Multiplique $3$ por $- 40$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 6 x^{4} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + 3 \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.6.7

Multiplique $3 \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2}$ .

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Etapa 3.19.3.6.7.1

Combine $3$ e $\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 6 x^{4} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + \frac{3 (3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}})}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.6.7.2

Multiplique $3$ por $3$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 6 x^{4} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.7

Para escrever $- 6 x^{4} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ como fração com um denominador comum, multiplique por $\frac{2}{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 6 x^{4} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \cdot \frac{2}{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.8

Combine $- 6 x^{4} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ e $\frac{2}{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + \frac{- 6 x^{4} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \cdot 2}{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.9

Combine os numeradores em relação ao denominador comum.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + \frac{- 6 x^{4} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \cdot 2 + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.10

Simplifique o numerador.

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Etapa 3.19.3.10.1

Fatore $3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ de $- 6 x^{4} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \cdot 2 + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

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Etapa 3.19.3.10.1.1

Reordene a expressão.

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Etapa 3.19.3.10.1.1.1

Mova ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + \frac{- 6 x^{4} \cdot 2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.10.1.1.2

Mova $x^{4}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + \frac{- 6 \cdot 2 x^{4} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.10.1.2

Fatore $3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ de $- 6 \cdot 2 x^{4} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 2 \cdot 2 x^{4}) + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.10.1.3

Fatore $3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ de $9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 2 \cdot 2 x^{4}) + 3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (3)}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.10.1.4

Fatore $3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ de $3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 2 \cdot 2 x^{4}) + 3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (3)$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 2 \cdot 2 x^{4} + 3)}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.10.2

Multiplique $- 2$ por $2$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} + 3)}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.11

Para escrever $- 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ como fração com um denominador comum, multiplique por $\frac{2}{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \cdot \frac{2}{2} + \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} + 3)}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.12

Combine $- 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ e $\frac{2}{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + \frac{- 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \cdot 2}{2} + \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} + 3)}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.13

Combine os numeradores em relação ao denominador comum.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + \frac{- 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \cdot 2 + 3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} + 3)}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.14

Simplifique o numerador.

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Etapa 3.19.3.14.1

Fatore $3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ de $- 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \cdot 2 + 3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} + 3)$ .

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Etapa 3.19.3.14.1.1

Reordene a expressão.

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Etapa 3.19.3.14.1.1.1

Mova ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + \frac{- 18 x^{2} \cdot 2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + 3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} + 3)}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.14.1.1.2

Mova $x^{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + \frac{- 18 \cdot 2 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + 3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} + 3)}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.14.1.2

Fatore $3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ de $- 18 \cdot 2 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 6 \cdot 2 x^{2}) + 3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} + 3)}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.14.1.3

Fatore $3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ de $3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 6 \cdot 2 x^{2}) + 3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} + 3)$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 6 \cdot 2 x^{2} - 4 x^{4} + 3)}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.14.2

Multiplique $- 6$ por $2$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 12 x^{2} - 4 x^{4} + 3)}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.14.3

Reordene os termos.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} + 3)}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.15

Para escrever $- 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ como fração com um denominador comum, multiplique por $\frac{2}{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \cdot \frac{2}{2} + \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} + 3)}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.16

Combine $- 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ e $\frac{2}{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (\frac{- 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \cdot 2}{2} + \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} + 3)}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.17

Combine os numeradores em relação ao denominador comum.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + \frac{- 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \cdot 2 + 3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} + 3)}{2} (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.18

Simplifique o numerador.

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Etapa 3.19.3.18.1

Fatore $3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ de $- 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \cdot 2 + 3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} + 3)$ .

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Etapa 3.19.3.18.1.1

Reordene a expressão.

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Etapa 3.19.3.18.1.1.1

Mova ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + \frac{- 120 x \cdot 2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + 3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} + 3)}{2} (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.18.1.1.2

Mova $x$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + \frac{- 120 \cdot 2 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + 3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} + 3)}{2} (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.18.1.2

Fatore $3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ de $- 120 \cdot 2 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 40 \cdot 2 x) + 3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} + 3)}{2} (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.18.1.3

Fatore $3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ de $3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 40 \cdot 2 x) + 3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} + 3)$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 40 \cdot 2 x - 4 x^{4} - 12 x^{2} + 3)}{2} (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.18.2

Multiplique $- 40$ por $2$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 80 x - 4 x^{4} - 12 x^{2} + 3)}{2} (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.18.3

Reordene os termos.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2} (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.19

Aplique a propriedade distributiva.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2} (6 x^{2}) + \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2} \cdot 3)}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.20

Reescreva usando a propriedade comutativa da multiplicação.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 6 \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2} x^{2} + \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2} \cdot 3)}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.21

Multiplique $\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2} \cdot 3$ .

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Etapa 3.19.3.21.1

Combine $\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2}$ e $3$ .

Etapa 3.19.3.21.2

Multiplique $3$ por $3$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 6 \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.22

Simplifique cada termo.

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Etapa 3.19.3.22.1

Cancele o fator comum de $2$ .

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Etapa 3.19.3.22.1.1

Fatore $2$ de $6$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 2 (3) \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.22.1.2

Cancele o fator comum.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 2 \cdot 3 \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.22.1.3

Reescreva a expressão.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 3 (3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)) x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.22.2

Multiplique $3$ por $3$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 9 ({(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)) x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.22.3

Aplique a propriedade distributiva.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 9 ({(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4}) + {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 12 x^{2}) + {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 80 x) + {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \cdot 3) x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.22.4

Simplifique.

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Etapa 3.19.3.22.4.1

Reescreva usando a propriedade comutativa da multiplicação.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 9 (- 4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} + {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 12 x^{2}) + {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 80 x) + {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \cdot 3) x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.22.4.2

Reescreva usando a propriedade comutativa da multiplicação.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 9 (- 4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} - 12 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 80 x) + {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \cdot 3) x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.22.4.3

Reescreva usando a propriedade comutativa da multiplicação.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 9 (- 4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} - 12 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} - 80 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \cdot 3) x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.22.4.4

Mova $3$ para a esquerda de ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 9 (- 4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} - 12 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} - 80 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + 3 \cdot {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}) x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.22.5

Aplique a propriedade distributiva.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (9 (- 4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4}) + 9 (- 12 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2}) + 9 (- 80 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x) + 9 (3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}})) x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.22.6

Simplifique.

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Etapa 3.19.3.22.6.1

Multiplique $- 4$ por $9$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 36 ({(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4}) + 9 (- 12 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2}) + 9 (- 80 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x) + 9 (3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}})) x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.22.6.2

Multiplique $- 12$ por $9$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 36 ({(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4}) - 108 ({(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2}) + 9 (- 80 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x) + 9 (3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}})) x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.22.6.3

Multiplique $- 80$ por $9$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 36 ({(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4}) - 108 ({(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2}) - 720 ({(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x) + 9 (3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}})) x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.22.6.4

Multiplique $3$ por $9$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 36 ({(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4}) - 108 ({(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2}) - 720 ({(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x) + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}) x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.22.7

Remova os parênteses.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 36 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} - 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}) x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.22.8

Aplique a propriedade distributiva.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 36 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} x^{2} - 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} x^{2} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x \cdot x^{2} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.22.9

Simplifique.

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Etapa 3.19.3.22.9.1

Multiplique $x^{4}$ por $x^{2}$ somando os expoentes.

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Etapa 3.19.3.22.9.1.1

Mova $x^{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 36 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (x^{2} x^{4}) - 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} x^{2} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x \cdot x^{2} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.22.9.1.2

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

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 36 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2 + 4} - 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} x^{2} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x \cdot x^{2} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.22.9.1.3

Some $2$ e $4$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 36 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{6} - 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} x^{2} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x \cdot x^{2} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.22.9.2

Multiplique $x^{2}$ por $x^{2}$ somando os expoentes.

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Etapa 3.19.3.22.9.2.1

Mova $x^{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 36 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{6} - 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (x^{2} x^{2}) - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x \cdot x^{2} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.22.9.2.2

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

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 36 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{6} - 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2 + 2} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x \cdot x^{2} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.22.9.2.3

Some $2$ e $2$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 36 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{6} - 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x \cdot x^{2} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.22.9.3

Multiplique $x$ por $x^{2}$ somando os expoentes.

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Etapa 3.19.3.22.9.3.1

Mova $x^{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 36 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{6} - 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (x^{2} x) + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.22.9.3.2

Multiplique $x^{2}$ por $x$ .

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Etapa 3.19.3.22.9.3.2.1

Eleve $x$ à potência de $1$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 36 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{6} - 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (x^{2} x^{1}) + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.22.9.3.2.2

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

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 36 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{6} - 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2 + 1} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.22.9.3.3

Some $2$ e $1$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 36 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{6} - 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.23

Para escrever $- 36 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{6}$ como fração com um denominador comum, multiplique por $\frac{2}{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} - 36 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{6} \cdot \frac{2}{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.24

Combine $- 36 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{6}$ e $\frac{2}{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{- 36 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{6} \cdot 2}{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.25

Combine os numeradores em relação ao denominador comum.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{- 36 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{6} \cdot 2 + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.26

Simplifique o numerador.

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Etapa 3.19.3.26.1

Fatore $9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ de $- 36 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{6} \cdot 2 + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)$ .

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Etapa 3.19.3.26.1.1

Reordene a expressão.

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Etapa 3.19.3.26.1.1.1

Mova ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{- 36 x^{6} \cdot 2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.26.1.1.2

Mova $x^{6}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{- 36 \cdot 2 x^{6} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.26.1.2

Fatore $9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ de $- 36 \cdot 2 x^{6} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 \cdot 2 x^{6}) + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.26.1.3

Fatore $9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ de $9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 \cdot 2 x^{6}) + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 \cdot 2 x^{6} - 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.26.2

Multiplique $- 4$ por $2$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.27

Para escrever $- 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4}$ como fração com um denominador comum, multiplique por $\frac{2}{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} - 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} \cdot \frac{2}{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.28

Combine $- 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4}$ e $\frac{2}{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{- 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} \cdot 2}{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.29

Combine os numeradores em relação ao denominador comum.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{- 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} \cdot 2 + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.30

Simplifique o numerador.

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Etapa 3.19.3.30.1

Fatore $9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ de $- 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} \cdot 2 + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 4 x^{4} - 12 x^{2} - 80 x + 3)$ .

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Etapa 3.19.3.30.1.1

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Etapa 3.19.3.30.1.1.1

Mova ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{- 108 x^{4} \cdot 2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.30.1.1.2

Mova $x^{4}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{- 108 \cdot 2 x^{4} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.30.1.2

Fatore $9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ de $- 108 \cdot 2 x^{4} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 12 \cdot 2 x^{4}) + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.30.1.3

Fatore $9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ de $9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 12 \cdot 2 x^{4}) + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 4 x^{4} - 12 x^{2} - 80 x + 3)$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 12 \cdot 2 x^{4} - 8 x^{6} - 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.30.2

Multiplique $- 12$ por $2$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 24 x^{4} - 8 x^{6} - 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.30.3

Subtraia $4 x^{4}$ de $- 24 x^{4}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.31

Para escrever $- 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3}$ como fração com um denominador comum, multiplique por $\frac{2}{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} \cdot \frac{2}{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.32

Combine $- 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3}$ e $\frac{2}{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{- 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} \cdot 2}{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.33

Combine os numeradores em relação ao denominador comum.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{- 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} \cdot 2 + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.34

Simplifique o numerador.

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Etapa 3.19.3.34.1

Fatore $9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ de $- 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} \cdot 2 + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 12 x^{2} - 80 x + 3)$ .

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Etapa 3.19.3.34.1.1

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Etapa 3.19.3.34.1.1.1

Mova ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{- 720 x^{3} \cdot 2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.34.1.1.2

Mova $x^{3}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{- 720 \cdot 2 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.34.1.2

Fatore $9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ de $- 720 \cdot 2 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 80 \cdot 2 x^{3}) + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.34.1.3

Fatore $9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ de $9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 80 \cdot 2 x^{3}) + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 12 x^{2} - 80 x + 3)$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 80 \cdot 2 x^{3} - 8 x^{6} - 28 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.34.2

Multiplique $- 80$ por $2$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 160 x^{3} - 8 x^{6} - 28 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.34.3

Reordene os termos.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.35

Para escrever $27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2}$ como fração com um denominador comum, multiplique por $\frac{2}{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} \cdot \frac{2}{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.36

Combine $27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2}$ e $\frac{2}{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + \frac{27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} \cdot 2}{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.37

Combine os numeradores em relação ao denominador comum.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + \frac{27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} \cdot 2 + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.38

Simplifique o numerador.

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Etapa 3.19.3.38.1

Fatore $9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ de $27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} \cdot 2 + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 12 x^{2} - 80 x + 3)$ .

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Etapa 3.19.3.38.1.1

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Etapa 3.19.3.38.1.1.1

Mova ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + \frac{27 x^{2} \cdot 2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.38.1.1.2

Mova $x^{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + \frac{27 \cdot 2 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.38.1.2

Fatore $9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ de $27 \cdot 2 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (3 \cdot 2 x^{2}) + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.38.1.3

Fatore $9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ de $9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (3 \cdot 2 x^{2}) + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 12 x^{2} - 80 x + 3)$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (3 \cdot 2 x^{2} - 8 x^{6} - 28 x^{4} - 160 x^{3} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.38.2

Multiplique $3$ por $2$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (6 x^{2} - 8 x^{6} - 28 x^{4} - 160 x^{3} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.38.3

Subtraia $12 x^{2}$ de $6 x^{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.39

Para escrever $16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3}$ como fração com um denominador comum, multiplique por $\frac{2}{2}$ .

$\frac{3 (24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} \cdot \frac{2}{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.40

Combine $16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3}$ e $\frac{2}{2}$ .

$\frac{3 (24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + \frac{16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} \cdot 2}{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.41

Combine os numeradores em relação ao denominador comum.

$\frac{3 (24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + \frac{16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} \cdot 2 + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.42

Mova $x^{3}$ .

$\frac{3 (80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 24 x {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + \frac{16 \cdot 2 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.43

Para escrever $80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}$ como fração com um denominador comum, multiplique por $\frac{2}{2}$ .

$\frac{3 (24 x {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} \cdot \frac{2}{2} + \frac{16 \cdot 2 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.44

Combine $80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}$ e $\frac{2}{2}$ .

$\frac{3 (24 x {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + \frac{80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} \cdot 2}{2} + \frac{16 \cdot 2 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.45

Combine os numeradores em relação ao denominador comum.

$\frac{3 (24 x {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + \frac{80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} \cdot 2 + 16 \cdot 2 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.46

Para escrever $24 x {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}$ como fração com um denominador comum, multiplique por $\frac{2}{2}$ .

$\frac{3 (24 x {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} \cdot \frac{2}{2} + \frac{80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} \cdot 2 + 16 \cdot 2 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.47

Combine $24 x {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}$ e $\frac{2}{2}$ .

$\frac{3 (\frac{24 x {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} \cdot 2}{2} + \frac{80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} \cdot 2 + 16 \cdot 2 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.48

Combine os numeradores em relação ao denominador comum.

$\frac{3 \frac{24 x {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} \cdot 2 + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} \cdot 2 + 16 \cdot 2 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.49

Reordene os termos.

$\frac{3 \frac{2 \cdot 24 x {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50

Reescreva $\frac{2 \cdot 24 x {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3)}{2}$ em uma forma fatorada.

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Etapa 3.19.3.50.1

Fatore ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ de $2 \cdot 24 x {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3)$ .

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Etapa 3.19.3.50.1.1

Fatore ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ de $2 \cdot 24 x {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 \cdot 24 x {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}}) + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.1.2

Fatore ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ de $2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 \cdot 24 x {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}}) + {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}}) + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.1.3

Fatore ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ de $2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 \cdot 24 x {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}}) + {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}}) + {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}}) + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.1.4

Fatore ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ de $9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3)$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 \cdot 24 x {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}}) + {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}}) + {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}}) + {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.1.5

Fatore ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ de ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 \cdot 24 x {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}}) + {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}})$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 \cdot 24 x {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}}) + {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}}) + {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.1.6

Fatore ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ de ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 \cdot 24 x {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}}) + {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}})$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 \cdot 24 x {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}}) + {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.1.7

Fatore ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ de ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 \cdot 24 x {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}}) + {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 \cdot 24 x {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.2

Multiplique $2$ por $24$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (48 x {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.3

Divida $2$ por $2$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (48 x {(2 x^{3} + 3 x + 10)}^{1} + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.4

Simplifique.

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (48 x (2 x^{3} + 3 x + 10) + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.5

Aplique a propriedade distributiva.

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (48 x (2 x^{3}) + 48 x (3 x) + 48 x \cdot 10 + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.6

Simplifique.

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Etapa 3.19.3.50.6.1

Reescreva usando a propriedade comutativa da multiplicação.

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (48 \cdot 2 x \cdot x^{3} + 48 x (3 x) + 48 x \cdot 10 + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.6.2

Reescreva usando a propriedade comutativa da multiplicação.

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (48 \cdot 2 x \cdot x^{3} + 48 \cdot 3 x \cdot x + 48 x \cdot 10 + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.6.3

Multiplique $10$ por $48$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (48 \cdot 2 x \cdot x^{3} + 48 \cdot 3 x \cdot x + 480 x + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.7

Simplifique cada termo.

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Etapa 3.19.3.50.7.1

Multiplique $x$ por $x^{3}$ somando os expoentes.

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Etapa 3.19.3.50.7.1.1

Mova $x^{3}$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (48 \cdot 2 (x^{3} x) + 48 \cdot 3 x \cdot x + 480 x + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.7.1.2

Multiplique $x^{3}$ por $x$ .

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Etapa 3.19.3.50.7.1.2.1

Eleve $x$ à potência de $1$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (48 \cdot 2 (x^{3} x^{1}) + 48 \cdot 3 x \cdot x + 480 x + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.7.1.2.2

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

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (48 \cdot 2 x^{3 + 1} + 48 \cdot 3 x \cdot x + 480 x + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.7.1.3

Some $3$ e $1$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (48 \cdot 2 x^{4} + 48 \cdot 3 x \cdot x + 480 x + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.7.2

Multiplique $48$ por $2$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 48 \cdot 3 x \cdot x + 480 x + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.7.3

Multiplique $x$ por $x$ somando os expoentes.

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Etapa 3.19.3.50.7.3.1

Mova $x$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 48 \cdot 3 (x \cdot x) + 480 x + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.7.3.2

Multiplique $x$ por $x$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 48 \cdot 3 x^{2} + 480 x + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.7.4

Multiplique $48$ por $3$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.8

Multiplique $2$ por $80$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 160 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.9

Divida $2$ por $2$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 160 {(2 x^{3} + 3 x + 10)}^{1} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.10

Simplifique.

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 160 (2 x^{3} + 3 x + 10) + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.11

Aplique a propriedade distributiva.

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 160 (2 x^{3}) + 160 (3 x) + 160 \cdot 10 + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.12

Simplifique.

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Etapa 3.19.3.50.12.1

Multiplique $2$ por $160$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 160 (3 x) + 160 \cdot 10 + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.12.2

Multiplique $3$ por $160$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 160 \cdot 10 + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.12.3

Multiplique $160$ por $10$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.13

Multiplique $2$ por $16$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 32 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.14

Divida $2$ por $2$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 32 x^{3} {(2 x^{3} + 3 x + 10)}^{1} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.15

Simplifique.

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 32 x^{3} (2 x^{3} + 3 x + 10) + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.16

Aplique a propriedade distributiva.

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 32 x^{3} (2 x^{3}) + 32 x^{3} (3 x) + 32 x^{3} \cdot 10 + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.17

Simplifique.

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Etapa 3.19.3.50.17.1

Reescreva usando a propriedade comutativa da multiplicação.

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 32 \cdot 2 x^{3} x^{3} + 32 x^{3} (3 x) + 32 x^{3} \cdot 10 + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.17.2

Reescreva usando a propriedade comutativa da multiplicação.

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 32 \cdot 2 x^{3} x^{3} + 32 \cdot 3 x^{3} x + 32 x^{3} \cdot 10 + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.17.3

Multiplique $10$ por $32$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 32 \cdot 2 x^{3} x^{3} + 32 \cdot 3 x^{3} x + 320 x^{3} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.18

Simplifique cada termo.

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Etapa 3.19.3.50.18.1

Multiplique $x^{3}$ por $x^{3}$ somando os expoentes.

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Etapa 3.19.3.50.18.1.1

Mova $x^{3}$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 32 \cdot 2 (x^{3} x^{3}) + 32 \cdot 3 x^{3} x + 320 x^{3} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.18.1.2

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

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 32 \cdot 2 x^{3 + 3} + 32 \cdot 3 x^{3} x + 320 x^{3} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.18.1.3

Some $3$ e $3$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 32 \cdot 2 x^{6} + 32 \cdot 3 x^{3} x + 320 x^{3} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.18.2

Multiplique $32$ por $2$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 64 x^{6} + 32 \cdot 3 x^{3} x + 320 x^{3} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.18.3

Multiplique $x^{3}$ por $x$ somando os expoentes.

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Etapa 3.19.3.50.18.3.1

Mova $x$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 64 x^{6} + 32 \cdot 3 (x \cdot x^{3}) + 320 x^{3} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.18.3.2

Multiplique $x$ por $x^{3}$ .

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Etapa 3.19.3.50.18.3.2.1

Eleve $x$ à potência de $1$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 64 x^{6} + 32 \cdot 3 (x^{1} x^{3}) + 320 x^{3} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.18.3.2.2

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

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 64 x^{6} + 32 \cdot 3 x^{1 + 3} + 320 x^{3} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.18.3.3

Some $1$ e $3$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 64 x^{6} + 32 \cdot 3 x^{4} + 320 x^{3} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.18.4

Multiplique $32$ por $3$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 64 x^{6} + 96 x^{4} + 320 x^{3} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.19

Aplique a propriedade distributiva.

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 64 x^{6} + 96 x^{4} + 320 x^{3} + 9 (- 8 x^{6}) + 9 (- 28 x^{4}) + 9 (- 160 x^{3}) + 9 (- 6 x^{2}) + 9 (- 80 x) + 9 \cdot 3)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.20

Simplifique.

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Etapa 3.19.3.50.20.1

Multiplique $- 8$ por $9$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 64 x^{6} + 96 x^{4} + 320 x^{3} - 72 x^{6} + 9 (- 28 x^{4}) + 9 (- 160 x^{3}) + 9 (- 6 x^{2}) + 9 (- 80 x) + 9 \cdot 3)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.20.2

Multiplique $- 28$ por $9$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 64 x^{6} + 96 x^{4} + 320 x^{3} - 72 x^{6} - 252 x^{4} + 9 (- 160 x^{3}) + 9 (- 6 x^{2}) + 9 (- 80 x) + 9 \cdot 3)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.20.3

Multiplique $- 160$ por $9$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 64 x^{6} + 96 x^{4} + 320 x^{3} - 72 x^{6} - 252 x^{4} - 1440 x^{3} + 9 (- 6 x^{2}) + 9 (- 80 x) + 9 \cdot 3)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.20.4

Multiplique $- 6$ por $9$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 64 x^{6} + 96 x^{4} + 320 x^{3} - 72 x^{6} - 252 x^{4} - 1440 x^{3} - 54 x^{2} + 9 (- 80 x) + 9 \cdot 3)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.20.5

Multiplique $- 80$ por $9$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 64 x^{6} + 96 x^{4} + 320 x^{3} - 72 x^{6} - 252 x^{4} - 1440 x^{3} - 54 x^{2} - 720 x + 9 \cdot 3)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.20.6

Multiplique $9$ por $3$ .

Etapa 3.19.3.50.21

Some $96 x^{4}$ e $96 x^{4}$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (192 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 64 x^{6} + 320 x^{3} - 72 x^{6} - 252 x^{4} - 1440 x^{3} - 54 x^{2} - 720 x + 27)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.22

Subtraia $252 x^{4}$ de $192 x^{4}$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 60 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 64 x^{6} + 320 x^{3} - 72 x^{6} - 1440 x^{3} - 54 x^{2} - 720 x + 27)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.23

Subtraia $54 x^{2}$ de $144 x^{2}$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 60 x^{4} + 480 x + 320 x^{3} + 480 x + 1600 + 64 x^{6} + 320 x^{3} - 72 x^{6} - 1440 x^{3} + 90 x^{2} - 720 x + 27)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.24

Some $480 x$ e $480 x$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 60 x^{4} + 320 x^{3} + 960 x + 1600 + 64 x^{6} + 320 x^{3} - 72 x^{6} - 1440 x^{3} + 90 x^{2} - 720 x + 27)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.25

Some $320 x^{3}$ e $320 x^{3}$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 60 x^{4} + 640 x^{3} + 960 x + 1600 + 64 x^{6} - 72 x^{6} - 1440 x^{3} + 90 x^{2} - 720 x + 27)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.26

Subtraia $1440 x^{3}$ de $640 x^{3}$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 60 x^{4} - 800 x^{3} + 960 x + 1600 + 64 x^{6} - 72 x^{6} + 90 x^{2} - 720 x + 27)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.27

Subtraia $720 x$ de $960 x$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 60 x^{4} - 800 x^{3} + 1600 + 64 x^{6} - 72 x^{6} + 90 x^{2} + 240 x + 27)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.28

Some $1600$ e $27$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 60 x^{4} - 800 x^{3} + 64 x^{6} - 72 x^{6} + 90 x^{2} + 240 x + 1627)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.29

Subtraia $72 x^{6}$ de $64 x^{6}$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 60 x^{4} - 800 x^{3} - 8 x^{6} + 90 x^{2} + 240 x + 1627)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.3.50.30

Reordene os termos.

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 60 x^{4} - 800 x^{3} + 90 x^{2} + 240 x + 1627)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.4

Combine os termos.

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Etapa 3.19.4.1

Combine $3$ e $\frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 60 x^{4} - 800 x^{3} + 90 x^{2} + 240 x + 1627)}{2}$ .

$\frac{\frac{3 ({(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 60 x^{4} - 800 x^{3} + 90 x^{2} + 240 x + 1627))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.4.2

Reescreva $\frac{\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 60 x^{4} - 800 x^{3} + 90 x^{2} + 240 x + 1627)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$ como um produto.

$\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 60 x^{4} - 800 x^{3} + 90 x^{2} + 240 x + 1627)}{2} \cdot \frac{1}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.4.3

Multiplique $\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 60 x^{4} - 800 x^{3} + 90 x^{2} + 240 x + 1627)}{2}$ por $\frac{1}{4 {(2 x^{3} + 3 x + 10)}^{3}}$ .

$\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 60 x^{4} - 800 x^{3} + 90 x^{2} + 240 x + 1627)}{2 (4 {(2 x^{3} + 3 x + 10)}^{3})}$

Etapa 3.19.4.4

Multiplique $4$ por $2$ .

$\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 60 x^{4} - 800 x^{3} + 90 x^{2} + 240 x + 1627)}{8 {(2 x^{3} + 3 x + 10)}^{3}}$

Etapa 3.19.4.5

Mova ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ para o denominador usando a regra do expoente negativo $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{3 (- 8 x^{6} - 60 x^{4} - 800 x^{3} + 90 x^{2} + 240 x + 1627)}{8 {(2 x^{3} + 3 x + 10)}^{3} {(2 x^{3} + 3 x + 10)}^{- \frac{1}{2}}}$

Etapa 3.19.4.6

Multiplique ${(2 x^{3} + 3 x + 10)}^{3}$ por ${(2 x^{3} + 3 x + 10)}^{- \frac{1}{2}}$ somando os expoentes.

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Etapa 3.19.4.6.1

Mova ${(2 x^{3} + 3 x + 10)}^{- \frac{1}{2}}$ .

$\frac{3 (- 8 x^{6} - 60 x^{4} - 800 x^{3} + 90 x^{2} + 240 x + 1627)}{8 ({(2 x^{3} + 3 x + 10)}^{- \frac{1}{2}} {(2 x^{3} + 3 x + 10)}^{3})}$

Etapa 3.19.4.6.2

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

$\frac{3 (- 8 x^{6} - 60 x^{4} - 800 x^{3} + 90 x^{2} + 240 x + 1627)}{8 {(2 x^{3} + 3 x + 10)}^{- \frac{1}{2} + 3}}$

Etapa 3.19.4.6.3

Para escrever $3$ como fração com um denominador comum, multiplique por $\frac{2}{2}$ .

$\frac{3 (- 8 x^{6} - 60 x^{4} - 800 x^{3} + 90 x^{2} + 240 x + 1627)}{8 {(2 x^{3} + 3 x + 10)}^{- \frac{1}{2} + 3 \cdot \frac{2}{2}}}$

Etapa 3.19.4.6.4

Combine $3$ e $\frac{2}{2}$ .

$\frac{3 (- 8 x^{6} - 60 x^{4} - 800 x^{3} + 90 x^{2} + 240 x + 1627)}{8 {(2 x^{3} + 3 x + 10)}^{- \frac{1}{2} + \frac{3 \cdot 2}{2}}}$

Etapa 3.19.4.6.5

Combine os numeradores em relação ao denominador comum.

$\frac{3 (- 8 x^{6} - 60 x^{4} - 800 x^{3} + 90 x^{2} + 240 x + 1627)}{8 {(2 x^{3} + 3 x + 10)}^{\frac{- 1 + 3 \cdot 2}{2}}}$

Etapa 3.19.4.6.6

Simplifique o numerador.

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Etapa 3.19.4.6.6.1

Multiplique $3$ por $2$ .

$\frac{3 (- 8 x^{6} - 60 x^{4} - 800 x^{3} + 90 x^{2} + 240 x + 1627)}{8 {(2 x^{3} + 3 x + 10)}^{\frac{- 1 + 6}{2}}}$

Etapa 3.19.4.6.6.2

Some $- 1$ e $6$ .

$\frac{3 (- 8 x^{6} - 60 x^{4} - 800 x^{3} + 90 x^{2} + 240 x + 1627)}{8 {(2 x^{3} + 3 x + 10)}^{\frac{5}{2}}}$

Etapa 3.19.5

Fatore $- 1$ de $- 8 x^{6}$ .

$\frac{3 (- (8 x^{6}) - 60 x^{4} - 800 x^{3} + 90 x^{2} + 240 x + 1627)}{8 {(2 x^{3} + 3 x + 10)}^{\frac{5}{2}}}$

Etapa 3.19.6

Fatore $- 1$ de $- 60 x^{4}$ .

$\frac{3 (- (8 x^{6}) - (60 x^{4}) - 800 x^{3} + 90 x^{2} + 240 x + 1627)}{8 {(2 x^{3} + 3 x + 10)}^{\frac{5}{2}}}$

Etapa 3.19.7

Fatore $- 1$ de $- (8 x^{6}) - (60 x^{4})$ .

$\frac{3 (- (8 x^{6} + 60 x^{4}) - 800 x^{3} + 90 x^{2} + 240 x + 1627)}{8 {(2 x^{3} + 3 x + 10)}^{\frac{5}{2}}}$

Etapa 3.19.8

Fatore $- 1$ de $- 800 x^{3}$ .

$\frac{3 (- (8 x^{6} + 60 x^{4}) - (800 x^{3}) + 90 x^{2} + 240 x + 1627)}{8 {(2 x^{3} + 3 x + 10)}^{\frac{5}{2}}}$

Etapa 3.19.9

Fatore $- 1$ de $- (8 x^{6} + 60 x^{4}) - (800 x^{3})$ .

$\frac{3 (- (8 x^{6} + 60 x^{4} + 800 x^{3}) + 90 x^{2} + 240 x + 1627)}{8 {(2 x^{3} + 3 x + 10)}^{\frac{5}{2}}}$

Etapa 3.19.10

Fatore $- 1$ de $90 x^{2}$ .

$\frac{3 (- (8 x^{6} + 60 x^{4} + 800 x^{3}) - (- 90 x^{2}) + 240 x + 1627)}{8 {(2 x^{3} + 3 x + 10)}^{\frac{5}{2}}}$

Etapa 3.19.11

Fatore $- 1$ de $- (8 x^{6} + 60 x^{4} + 800 x^{3}) - (- 90 x^{2})$ .

$\frac{3 (- (8 x^{6} + 60 x^{4} + 800 x^{3} - 90 x^{2}) + 240 x + 1627)}{8 {(2 x^{3} + 3 x + 10)}^{\frac{5}{2}}}$

Etapa 3.19.12

Fatore $- 1$ de $240 x$ .

$\frac{3 (- (8 x^{6} + 60 x^{4} + 800 x^{3} - 90 x^{2}) - (- 240 x) + 1627)}{8 {(2 x^{3} + 3 x + 10)}^{\frac{5}{2}}}$

Etapa 3.19.13

Fatore $- 1$ de $- (8 x^{6} + 60 x^{4} + 800 x^{3} - 90 x^{2}) - (- 240 x)$ .

$\frac{3 (- (8 x^{6} + 60 x^{4} + 800 x^{3} - 90 x^{2} - 240 x) + 1627)}{8 {(2 x^{3} + 3 x + 10)}^{\frac{5}{2}}}$

Etapa 3.19.14

Reescreva $1627$ como $- 1 (- 1627)$ .

$\frac{3 (- (8 x^{6} + 60 x^{4} + 800 x^{3} - 90 x^{2} - 240 x) - 1 (- 1627))}{8 {(2 x^{3} + 3 x + 10)}^{\frac{5}{2}}}$

Etapa 3.19.15

Fatore $- 1$ de $- (8 x^{6} + 60 x^{4} + 800 x^{3} - 90 x^{2} - 240 x) - 1 (- 1627)$ .

$\frac{3 (- (8 x^{6} + 60 x^{4} + 800 x^{3} - 90 x^{2} - 240 x - 1627))}{8 {(2 x^{3} + 3 x + 10)}^{\frac{5}{2}}}$

Etapa 3.19.16

Reescreva $- (8 x^{6} + 60 x^{4} + 800 x^{3} - 90 x^{2} - 240 x - 1627)$ como $- 1 (8 x^{6} + 60 x^{4} + 800 x^{3} - 90 x^{2} - 240 x - 1627)$ .

$\frac{3 (- 1 (8 x^{6} + 60 x^{4} + 800 x^{3} - 90 x^{2} - 240 x - 1627))}{8 {(2 x^{3} + 3 x + 10)}^{\frac{5}{2}}}$

Etapa 3.19.17

Mova o número negativo para a frente da fração.

$f''' (x) = - \frac{3 (8 x^{6} + 60 x^{4} + 800 x^{3} - 90 x^{2} - 240 x - 1627)}{8 {(2 x^{3} + 3 x + 10)}^{\frac{5}{2}}}$