Cálculo Ejemplos

$f (x) = \ln (\frac{\sqrt{x^{2} + 1}}{x {(2 x^{3} - 1)}^{2}})$

Paso 1

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

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

Paso 2

Diferencia con la regla de la cadena, que establece que $\frac{d}{d x} [f (g (x))]$ es $f' (g (x)) g' (x)$ donde $f (x) = \ln (x)$ y $g (x) = \frac{{(x^{2} + 1)}^{\frac{1}{2}}}{x {(2 x^{3} - 1)}^{2}}$ .

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Paso 2.1

Para aplicar la regla de la cadena, establece $u_{1}$ como $\frac{{(x^{2} + 1)}^{\frac{1}{2}}}{x {(2 x^{3} - 1)}^{2}}$ .

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

Paso 2.2

La derivada de $\ln (u_{1})$ con respecto a $u_{1}$ es $\frac{1}{u_{1}}$ .

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

Paso 2.3

Reemplaza todos los casos de $u_{1}$ con $\frac{{(x^{2} + 1)}^{\frac{1}{2}}}{x {(2 x^{3} - 1)}^{2}}$ .

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

Paso 3

Multiplica por la recíproca de la fracción para dividir por $\frac{{(x^{2} + 1)}^{\frac{1}{2}}}{x {(2 x^{3} - 1)}^{2}}$ .

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

Paso 4

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

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

Paso 5

Diferencia con la regla del cociente, que establece que $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ es $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ donde $f (x) = {(x^{2} + 1)}^{\frac{1}{2}}$ y $g (x) = x {(2 x^{3} - 1)}^{2}$ .

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

Paso 6

Diferencia con la regla de la cadena, que establece que $\frac{d}{d x} [f (g (x))]$ es $f' (g (x)) g' (x)$ donde $f (x) = x^{\frac{1}{2}}$ y $g (x) = x^{2} + 1$ .

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Paso 6.1

Para aplicar la regla de la cadena, establece $u_{2}$ como $x^{2} + 1$ .

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

Paso 6.2

Diferencia con la regla de la potencia, que establece que $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ es $n {u_{2}}^{n - 1}$ donde $n = \frac{1}{2}$ .

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

Paso 6.3

Reemplaza todos los casos de $u_{2}$ con $x^{2} + 1$ .

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

Paso 7

Para escribir $- 1$ como una fracción con un denominador común, multiplica por $\frac{2}{2}$ .

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

Paso 8

Combina $- 1$ y $\frac{2}{2}$ .

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

Paso 9

Combina los numeradores sobre el denominador común.

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

Paso 10

Simplifica el numerador.

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Paso 10.1

Multiplica $- 1$ por $2$ .

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

Paso 10.2

Resta $2$ de $1$ .

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

Paso 11

Combina fracciones.

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Paso 11.1

Mueve el negativo al frente de la fracción.

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

Paso 11.2

Combina $\frac{1}{2}$ y ${(x^{2} + 1)}^{- \frac{1}{2}}$ .

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

Paso 11.3

Mueve ${(x^{2} + 1)}^{- \frac{1}{2}}$ al denominador mediante la regla del exponente negativo $b^{- n} = \frac{1}{b^{n}}$ .

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

Paso 11.4

Combina $\frac{1}{2 {(x^{2} + 1)}^{\frac{1}{2}}}$ y $x$ .

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

Paso 11.5

Combina $\frac{x}{2 {(x^{2} + 1)}^{\frac{1}{2}}}$ y ${(2 x^{3} - 1)}^{2}$ .

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

Paso 12

Según la regla de la suma, la derivada de $x^{2} + 1$ con respecto a $x$ es $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$ .

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

Paso 13

Diferencia con la regla de la potencia, que establece que $\frac{d}{d x} [x^{n}]$ es $n x^{n - 1}$ donde $n = 2$ .

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

Paso 14

Como $1$ es constante con respecto a $x$ , la derivada de $1$ con respecto a $x$ es $0$ .

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

Paso 15

Combina fracciones.

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Paso 15.1

Suma $2 x$ y $0$ .

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

Paso 15.2

Combina $2$ y $\frac{x {(2 x^{3} - 1)}^{2}}{2 {(x^{2} + 1)}^{\frac{1}{2}}}$ .

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

Paso 15.3

Combina $\frac{2 (x {(2 x^{3} - 1)}^{2})}{2 {(x^{2} + 1)}^{\frac{1}{2}}}$ y $x$ .

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

Paso 16

Eleva $x$ a la potencia de $1$ .

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

Paso 17

Eleva $x$ a la potencia de $1$ .

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

Paso 18

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

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

Paso 19

Suma $1$ y $1$ .

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

Paso 20

Cancela el factor común.

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

Paso 21

Reescribe la expresión.

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

Paso 22

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

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

Paso 23

Combinar.

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

Paso 24

Aplica la propiedad distributiva.

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

Paso 25

Cancela el factor común de ${(x^{2} + 1)}^{\frac{1}{2}}$ .

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Paso 25.1

Cancela el factor común.

Paso 25.2

Reescribe la expresión.

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

Paso 26

Multiplica ${(x^{2} + 1)}^{\frac{1}{2}}$ por ${(x^{2} + 1)}^{\frac{1}{2}}$ sumando los exponentes.

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Paso 26.1

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

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

Paso 26.2

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

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

Paso 26.3

Combina los numeradores sobre el denominador común.

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

Paso 26.4

Suma $1$ y $1$ .

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

Paso 26.5

Divide $2$ por $2$ .

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

Paso 27

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

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

Paso 28

Diferencia con la regla del producto, que establece que $\frac{d}{d x} [f (x) g (x)]$ es $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ donde $f (x) = x$ y $g (x) = {(2 x^{3} - 1)}^{2}$ .

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

Paso 29

Diferencia con la regla de la cadena, que establece que $\frac{d}{d x} [f (g (x))]$ es $f' (g (x)) g' (x)$ donde $f (x) = x^{2}$ y $g (x) = 2 x^{3} - 1$ .

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Paso 29.1

Para aplicar la regla de la cadena, establece $u_{3}$ como $2 x^{3} - 1$ .

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

Paso 29.2

Diferencia con la regla de la potencia, que establece que $\frac{d}{d u_{3}} [{u_{3}}^{n}]$ es $n {u_{3}}^{n - 1}$ donde $n = 2$ .

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

Paso 29.3

Reemplaza todos los casos de $u_{3}$ con $2 x^{3} - 1$ .

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

Paso 30

Diferencia.

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Paso 30.1

Según la regla de la suma, la derivada de $2 x^{3} - 1$ con respecto a $x$ es $\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [- 1]$ .

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

Paso 30.2

Como $2$ es constante con respecto a $x$ , la derivada de $2 x^{3}$ con respecto a $x$ es $2 \frac{d}{d x} [x^{3}]$ .

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

Paso 30.3

Diferencia con la regla de la potencia, que establece que $\frac{d}{d x} [x^{n}]$ es $n x^{n - 1}$ donde $n = 3$ .

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

Paso 30.4

Multiplica $3$ por $2$ .

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

Paso 30.5

Como $- 1$ es constante con respecto a $x$ , la derivada de $- 1$ con respecto a $x$ es $0$ .

$\frac{x {(2 x^{3} - 1)}^{2}}{{(x^{2} + 1)}^{\frac{1}{2}}} \cdot \frac{x^{2} {(2 x^{3} - 1)}^{2} + (x^{2} + 1) (- (x (2 (2 x^{3} - 1) (6 x^{2} + 0)) + {(2 x^{3} - 1)}^{2} \frac{d}{d x} [x]))}{{(x^{2} + 1)}^{\frac{1}{2}} {(x {(2 x^{3} - 1)}^{2})}^{2}}$

Paso 30.6

Simplifica la expresión.

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Paso 30.6.1

Suma $6 x^{2}$ y $0$ .

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

Paso 30.6.2

Multiplica $6$ por $2$ .

$\frac{x {(2 x^{3} - 1)}^{2}}{{(x^{2} + 1)}^{\frac{1}{2}}} \cdot \frac{x^{2} {(2 x^{3} - 1)}^{2} + (x^{2} + 1) (- (x (12 (2 x^{3} - 1) x^{2}) + {(2 x^{3} - 1)}^{2} \frac{d}{d x} [x]))}{{(x^{2} + 1)}^{\frac{1}{2}} {(x {(2 x^{3} - 1)}^{2})}^{2}}$

Paso 31

Eleva $x$ a la potencia de $1$ .

$\frac{x {(2 x^{3} - 1)}^{2}}{{(x^{2} + 1)}^{\frac{1}{2}}} \cdot \frac{x^{2} {(2 x^{3} - 1)}^{2} + (x^{2} + 1) (- (x^{2} x^{1} (12 (2 x^{3} - 1)) + {(2 x^{3} - 1)}^{2} \frac{d}{d x} [x]))}{{(x^{2} + 1)}^{\frac{1}{2}} {(x {(2 x^{3} - 1)}^{2})}^{2}}$

Paso 32

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

$\frac{x {(2 x^{3} - 1)}^{2}}{{(x^{2} + 1)}^{\frac{1}{2}}} \cdot \frac{x^{2} {(2 x^{3} - 1)}^{2} + (x^{2} + 1) (- (x^{2 + 1} (12 (2 x^{3} - 1)) + {(2 x^{3} - 1)}^{2} \frac{d}{d x} [x]))}{{(x^{2} + 1)}^{\frac{1}{2}} {(x {(2 x^{3} - 1)}^{2})}^{2}}$

Paso 33

Suma $2$ y $1$ .

$\frac{x {(2 x^{3} - 1)}^{2}}{{(x^{2} + 1)}^{\frac{1}{2}}} \cdot \frac{x^{2} {(2 x^{3} - 1)}^{2} + (x^{2} + 1) (- (x^{3} (12 (2 x^{3} - 1)) + {(2 x^{3} - 1)}^{2} \frac{d}{d x} [x]))}{{(x^{2} + 1)}^{\frac{1}{2}} {(x {(2 x^{3} - 1)}^{2})}^{2}}$

Paso 34

Diferencia con la regla de la potencia, que establece que $\frac{d}{d x} [x^{n}]$ es $n x^{n - 1}$ donde $n = 1$ .

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

Paso 35

Combina fracciones.

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Paso 35.1

Multiplica ${(2 x^{3} - 1)}^{2}$ por $1$ .

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

Paso 35.2

Multiplica $\frac{x {(2 x^{3} - 1)}^{2}}{{(x^{2} + 1)}^{\frac{1}{2}}}$ por $\frac{x^{2} {(2 x^{3} - 1)}^{2} + (x^{2} + 1) (- (x^{3} (12 (2 x^{3} - 1)) + {(2 x^{3} - 1)}^{2}))}{{(x^{2} + 1)}^{\frac{1}{2}} {(x {(2 x^{3} - 1)}^{2})}^{2}}$ .

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

Paso 36

Multiplica ${(x^{2} + 1)}^{\frac{1}{2}}$ por ${(x^{2} + 1)}^{\frac{1}{2}}$ sumando los exponentes.

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Paso 36.1

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

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

Paso 36.2

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

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

Paso 36.3

Combina los numeradores sobre el denominador común.

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

Paso 36.4

Suma $1$ y $1$ .

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

Paso 36.5

Divide $2$ por $2$ .

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

Paso 37

Simplifica ${(x^{2} + 1)}^{1} {(x {(2 x^{3} - 1)}^{2})}^{2}$ .

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

Paso 38

Simplifica.

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Paso 38.1

Aplica la regla del producto a $x {(2 x^{3} - 1)}^{2}$ .

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

Paso 38.2

Aplica la propiedad distributiva.

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

Paso 38.3

Aplica la propiedad distributiva.

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

Paso 38.4

Aplica la propiedad distributiva.

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

Paso 38.5

Simplifica el numerador.

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Paso 38.5.1

Combina exponentes.

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Paso 38.5.1.1

Multiplica $12$ por $2$ .

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

Paso 38.5.1.2

Multiplica $24$ por $- 1$ .

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

Paso 38.5.1.3

Multiplica $x^{3}$ por $x^{3}$ sumando los exponentes.

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Paso 38.5.1.3.1

Mueve $x^{3}$ .

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

Paso 38.5.1.3.2

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

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

Paso 38.5.1.3.3

Suma $3$ y $3$ .

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

Paso 38.5.1.4

Multiplica $12$ por $- 1$ .

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

Paso 38.5.1.5

Multiplica $- 1$ por $- 12$ .

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

Paso 38.5.2

Simplifica cada término.

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Paso 38.5.2.1

Reescribe ${(2 x^{3} - 1)}^{2}$ como $(2 x^{3} - 1) (2 x^{3} - 1)$ .

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

Paso 38.5.2.2

Expande $(2 x^{3} - 1) (2 x^{3} - 1)$ con el método PEIU (primero, exterior, interior, ultimo).

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Paso 38.5.2.2.1

Aplica la propiedad distributiva.

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

Paso 38.5.2.2.2

Aplica la propiedad distributiva.

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

Paso 38.5.2.2.3

Aplica la propiedad distributiva.

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

Paso 38.5.2.3

Simplifica y combina los términos similares.

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Paso 38.5.2.3.1

Simplifica cada término.

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Paso 38.5.2.3.1.1

Reescribe con la propiedad conmutativa de la multiplicación.

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

Paso 38.5.2.3.1.2

Multiplica $x^{3}$ por $x^{3}$ sumando los exponentes.

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Paso 38.5.2.3.1.2.1

Mueve $x^{3}$ .

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

Paso 38.5.2.3.1.2.2

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

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

Paso 38.5.2.3.1.2.3

Suma $3$ y $3$ .

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

Paso 38.5.2.3.1.3

Multiplica $2$ por $2$ .

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

Paso 38.5.2.3.1.4

Multiplica $- 1$ por $2$ .

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

Paso 38.5.2.3.1.5

Multiplica $2$ por $- 1$ .

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

Paso 38.5.2.3.1.6

Multiplica $- 1$ por $- 1$ .

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

Paso 38.5.2.3.2

Resta $2 x^{3}$ de $- 2 x^{3}$ .

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

Paso 38.5.2.4

Aplica la propiedad distributiva.

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

Paso 38.5.2.5

Simplifica.

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Paso 38.5.2.5.1

Reescribe con la propiedad conmutativa de la multiplicación.

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

Paso 38.5.2.5.2

Reescribe con la propiedad conmutativa de la multiplicación.

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

Paso 38.5.2.5.3

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

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

Paso 38.5.2.6

Simplifica cada término.

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Paso 38.5.2.6.1

Multiplica $x^{2}$ por $x^{6}$ sumando los exponentes.

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Paso 38.5.2.6.1.1

Mueve $x^{6}$ .

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

Paso 38.5.2.6.1.2

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

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

Paso 38.5.2.6.1.3

Suma $6$ y $2$ .

$\frac{x {(2 x^{3} - 1)}^{2} (4 x^{8} - 4 x^{2} x^{3} + x^{2} + (x^{2} + 1) (- 24 x^{6} + 12 x^{3} - {(2 x^{3} - 1)}^{2}))}{(x^{2} + 1) (x^{2} {({(2 x^{3} - 1)}^{2})}^{2})}$

Paso 38.5.2.6.2

Multiplica $x^{2}$ por $x^{3}$ sumando los exponentes.

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Paso 38.5.2.6.2.1

Mueve $x^{3}$ .

$\frac{x {(2 x^{3} - 1)}^{2} (4 x^{8} - 4 (x^{3} x^{2}) + x^{2} + (x^{2} + 1) (- 24 x^{6} + 12 x^{3} - {(2 x^{3} - 1)}^{2}))}{(x^{2} + 1) (x^{2} {({(2 x^{3} - 1)}^{2})}^{2})}$

Paso 38.5.2.6.2.2

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

$\frac{x {(2 x^{3} - 1)}^{2} (4 x^{8} - 4 x^{3 + 2} + x^{2} + (x^{2} + 1) (- 24 x^{6} + 12 x^{3} - {(2 x^{3} - 1)}^{2}))}{(x^{2} + 1) (x^{2} {({(2 x^{3} - 1)}^{2})}^{2})}$

Paso 38.5.2.6.2.3

Suma $3$ y $2$ .

$\frac{x {(2 x^{3} - 1)}^{2} (4 x^{8} - 4 x^{5} + x^{2} + (x^{2} + 1) (- 24 x^{6} + 12 x^{3} - {(2 x^{3} - 1)}^{2}))}{(x^{2} + 1) (x^{2} {({(2 x^{3} - 1)}^{2})}^{2})}$

Paso 38.5.2.7

Simplifica cada término.

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Paso 38.5.2.7.1

Reescribe ${(2 x^{3} - 1)}^{2}$ como $(2 x^{3} - 1) (2 x^{3} - 1)$ .

$\frac{x {(2 x^{3} - 1)}^{2} (4 x^{8} - 4 x^{5} + x^{2} + (x^{2} + 1) (- 24 x^{6} + 12 x^{3} - ((2 x^{3} - 1) (2 x^{3} - 1))))}{(x^{2} + 1) (x^{2} {({(2 x^{3} - 1)}^{2})}^{2})}$

Paso 38.5.2.7.2

Expande $(2 x^{3} - 1) (2 x^{3} - 1)$ con el método PEIU (primero, exterior, interior, ultimo).

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Paso 38.5.2.7.2.1

Aplica la propiedad distributiva.

$\frac{x {(2 x^{3} - 1)}^{2} (4 x^{8} - 4 x^{5} + x^{2} + (x^{2} + 1) (- 24 x^{6} + 12 x^{3} - (2 x^{3} (2 x^{3} - 1) - 1 (2 x^{3} - 1))))}{(x^{2} + 1) (x^{2} {({(2 x^{3} - 1)}^{2})}^{2})}$

Paso 38.5.2.7.2.2

Aplica la propiedad distributiva.

$\frac{x {(2 x^{3} - 1)}^{2} (4 x^{8} - 4 x^{5} + x^{2} + (x^{2} + 1) (- 24 x^{6} + 12 x^{3} - (2 x^{3} (2 x^{3}) + 2 x^{3} \cdot - 1 - 1 (2 x^{3} - 1))))}{(x^{2} + 1) (x^{2} {({(2 x^{3} - 1)}^{2})}^{2})}$

Paso 38.5.2.7.2.3

Aplica la propiedad distributiva.

$\frac{x {(2 x^{3} - 1)}^{2} (4 x^{8} - 4 x^{5} + x^{2} + (x^{2} + 1) (- 24 x^{6} + 12 x^{3} - (2 x^{3} (2 x^{3}) + 2 x^{3} \cdot - 1 - 1 (2 x^{3}) - 1 \cdot - 1)))}{(x^{2} + 1) (x^{2} {({(2 x^{3} - 1)}^{2})}^{2})}$

Paso 38.5.2.7.3

Simplifica y combina los términos similares.

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Paso 38.5.2.7.3.1

Simplifica cada término.

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Paso 38.5.2.7.3.1.1

Reescribe con la propiedad conmutativa de la multiplicación.

$\frac{x {(2 x^{3} - 1)}^{2} (4 x^{8} - 4 x^{5} + x^{2} + (x^{2} + 1) (- 24 x^{6} + 12 x^{3} - (2 \cdot 2 x^{3} x^{3} + 2 x^{3} \cdot - 1 - 1 (2 x^{3}) - 1 \cdot - 1)))}{(x^{2} + 1) (x^{2} {({(2 x^{3} - 1)}^{2})}^{2})}$

Paso 38.5.2.7.3.1.2

Multiplica $x^{3}$ por $x^{3}$ sumando los exponentes.

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Paso 38.5.2.7.3.1.2.1

Mueve $x^{3}$ .

$\frac{x {(2 x^{3} - 1)}^{2} (4 x^{8} - 4 x^{5} + x^{2} + (x^{2} + 1) (- 24 x^{6} + 12 x^{3} - (2 \cdot 2 (x^{3} x^{3}) + 2 x^{3} \cdot - 1 - 1 (2 x^{3}) - 1 \cdot - 1)))}{(x^{2} + 1) (x^{2} {({(2 x^{3} - 1)}^{2})}^{2})}$

Paso 38.5.2.7.3.1.2.2

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

$\frac{x {(2 x^{3} - 1)}^{2} (4 x^{8} - 4 x^{5} + x^{2} + (x^{2} + 1) (- 24 x^{6} + 12 x^{3} - (2 \cdot 2 x^{3 + 3} + 2 x^{3} \cdot - 1 - 1 (2 x^{3}) - 1 \cdot - 1)))}{(x^{2} + 1) (x^{2} {({(2 x^{3} - 1)}^{2})}^{2})}$

Paso 38.5.2.7.3.1.2.3

Suma $3$ y $3$ .

$\frac{x {(2 x^{3} - 1)}^{2} (4 x^{8} - 4 x^{5} + x^{2} + (x^{2} + 1) (- 24 x^{6} + 12 x^{3} - (2 \cdot 2 x^{6} + 2 x^{3} \cdot - 1 - 1 (2 x^{3}) - 1 \cdot - 1)))}{(x^{2} + 1) (x^{2} {({(2 x^{3} - 1)}^{2})}^{2})}$

Paso 38.5.2.7.3.1.3

Multiplica $2$ por $2$ .

$\frac{x {(2 x^{3} - 1)}^{2} (4 x^{8} - 4 x^{5} + x^{2} + (x^{2} + 1) (- 24 x^{6} + 12 x^{3} - (4 x^{6} + 2 x^{3} \cdot - 1 - 1 (2 x^{3}) - 1 \cdot - 1)))}{(x^{2} + 1) (x^{2} {({(2 x^{3} - 1)}^{2})}^{2})}$

Paso 38.5.2.7.3.1.4

Multiplica $- 1$ por $2$ .

$\frac{x {(2 x^{3} - 1)}^{2} (4 x^{8} - 4 x^{5} + x^{2} + (x^{2} + 1) (- 24 x^{6} + 12 x^{3} - (4 x^{6} - 2 x^{3} - 1 (2 x^{3}) - 1 \cdot - 1)))}{(x^{2} + 1) (x^{2} {({(2 x^{3} - 1)}^{2})}^{2})}$

Paso 38.5.2.7.3.1.5

Multiplica $2$ por $- 1$ .

$\frac{x {(2 x^{3} - 1)}^{2} (4 x^{8} - 4 x^{5} + x^{2} + (x^{2} + 1) (- 24 x^{6} + 12 x^{3} - (4 x^{6} - 2 x^{3} - 2 x^{3} - 1 \cdot - 1)))}{(x^{2} + 1) (x^{2} {({(2 x^{3} - 1)}^{2})}^{2})}$

Paso 38.5.2.7.3.1.6

Multiplica $- 1$ por $- 1$ .

$\frac{x {(2 x^{3} - 1)}^{2} (4 x^{8} - 4 x^{5} + x^{2} + (x^{2} + 1) (- 24 x^{6} + 12 x^{3} - (4 x^{6} - 2 x^{3} - 2 x^{3} + 1)))}{(x^{2} + 1) (x^{2} {({(2 x^{3} - 1)}^{2})}^{2})}$

Paso 38.5.2.7.3.2

Resta $2 x^{3}$ de $- 2 x^{3}$ .

$\frac{x {(2 x^{3} - 1)}^{2} (4 x^{8} - 4 x^{5} + x^{2} + (x^{2} + 1) (- 24 x^{6} + 12 x^{3} - (4 x^{6} - 4 x^{3} + 1)))}{(x^{2} + 1) (x^{2} {({(2 x^{3} - 1)}^{2})}^{2})}$

Paso 38.5.2.7.4

Aplica la propiedad distributiva.

$\frac{x {(2 x^{3} - 1)}^{2} (4 x^{8} - 4 x^{5} + x^{2} + (x^{2} + 1) (- 24 x^{6} + 12 x^{3} - (4 x^{6}) - (- 4 x^{3}) - 1 \cdot 1))}{(x^{2} + 1) (x^{2} {({(2 x^{3} - 1)}^{2})}^{2})}$

Paso 38.5.2.7.5

Simplifica.

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Paso 38.5.2.7.5.1

Multiplica $4$ por $- 1$ .

$\frac{x {(2 x^{3} - 1)}^{2} (4 x^{8} - 4 x^{5} + x^{2} + (x^{2} + 1) (- 24 x^{6} + 12 x^{3} - 4 x^{6} - (- 4 x^{3}) - 1 \cdot 1))}{(x^{2} + 1) (x^{2} {({(2 x^{3} - 1)}^{2})}^{2})}$

Paso 38.5.2.7.5.2

Multiplica $- 4$ por $- 1$ .

$\frac{x {(2 x^{3} - 1)}^{2} (4 x^{8} - 4 x^{5} + x^{2} + (x^{2} + 1) (- 24 x^{6} + 12 x^{3} - 4 x^{6} + 4 x^{3} - 1 \cdot 1))}{(x^{2} + 1) (x^{2} {({(2 x^{3} - 1)}^{2})}^{2})}$

Paso 38.5.2.7.5.3

Multiplica $- 1$ por $1$ .

$\frac{x {(2 x^{3} - 1)}^{2} (4 x^{8} - 4 x^{5} + x^{2} + (x^{2} + 1) (- 24 x^{6} + 12 x^{3} - 4 x^{6} + 4 x^{3} - 1))}{(x^{2} + 1) (x^{2} {({(2 x^{3} - 1)}^{2})}^{2})}$

Paso 38.5.2.8

Resta $4 x^{6}$ de $- 24 x^{6}$ .

$\frac{x {(2 x^{3} - 1)}^{2} (4 x^{8} - 4 x^{5} + x^{2} + (x^{2} + 1) (- 28 x^{6} + 12 x^{3} + 4 x^{3} - 1))}{(x^{2} + 1) (x^{2} {({(2 x^{3} - 1)}^{2})}^{2})}$

Paso 38.5.2.9

Suma $12 x^{3}$ y $4 x^{3}$ .

$\frac{x {(2 x^{3} - 1)}^{2} (4 x^{8} - 4 x^{5} + x^{2} + (x^{2} + 1) (- 28 x^{6} + 16 x^{3} - 1))}{(x^{2} + 1) (x^{2} {({(2 x^{3} - 1)}^{2})}^{2})}$

Paso 38.5.2.10

Expande $(x^{2} + 1) (- 28 x^{6} + 16 x^{3} - 1)$ mediante la multiplicación de cada término de la primera expresión por cada término de la segunda expresión.

$\frac{x {(2 x^{3} - 1)}^{2} (4 x^{8} - 4 x^{5} + x^{2} + x^{2} (- 28 x^{6}) + x^{2} (16 x^{3}) + x^{2} \cdot - 1 + 1 (- 28 x^{6}) + 1 (16 x^{3}) + 1 \cdot - 1)}{(x^{2} + 1) (x^{2} {({(2 x^{3} - 1)}^{2})}^{2})}$

Paso 38.5.2.11

Simplifica cada término.

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Paso 38.5.2.11.1

Reescribe con la propiedad conmutativa de la multiplicación.

$\frac{x {(2 x^{3} - 1)}^{2} (4 x^{8} - 4 x^{5} + x^{2} - 28 x^{2} x^{6} + x^{2} (16 x^{3}) + x^{2} \cdot - 1 + 1 (- 28 x^{6}) + 1 (16 x^{3}) + 1 \cdot - 1)}{(x^{2} + 1) (x^{2} {({(2 x^{3} - 1)}^{2})}^{2})}$

Paso 38.5.2.11.2

Multiplica $x^{2}$ por $x^{6}$ sumando los exponentes.

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Paso 38.5.2.11.2.1

Mueve $x^{6}$ .

$\frac{x {(2 x^{3} - 1)}^{2} (4 x^{8} - 4 x^{5} + x^{2} - 28 (x^{6} x^{2}) + x^{2} (16 x^{3}) + x^{2} \cdot - 1 + 1 (- 28 x^{6}) + 1 (16 x^{3}) + 1 \cdot - 1)}{(x^{2} + 1) (x^{2} {({(2 x^{3} - 1)}^{2})}^{2})}$

Paso 38.5.2.11.2.2

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

$\frac{x {(2 x^{3} - 1)}^{2} (4 x^{8} - 4 x^{5} + x^{2} - 28 x^{6 + 2} + x^{2} (16 x^{3}) + x^{2} \cdot - 1 + 1 (- 28 x^{6}) + 1 (16 x^{3}) + 1 \cdot - 1)}{(x^{2} + 1) (x^{2} {({(2 x^{3} - 1)}^{2})}^{2})}$

Paso 38.5.2.11.2.3

Suma $6$ y $2$ .

$\frac{x {(2 x^{3} - 1)}^{2} (4 x^{8} - 4 x^{5} + x^{2} - 28 x^{8} + x^{2} (16 x^{3}) + x^{2} \cdot - 1 + 1 (- 28 x^{6}) + 1 (16 x^{3}) + 1 \cdot - 1)}{(x^{2} + 1) (x^{2} {({(2 x^{3} - 1)}^{2})}^{2})}$

Paso 38.5.2.11.3

Reescribe con la propiedad conmutativa de la multiplicación.

$\frac{x {(2 x^{3} - 1)}^{2} (4 x^{8} - 4 x^{5} + x^{2} - 28 x^{8} + 16 x^{2} x^{3} + x^{2} \cdot - 1 + 1 (- 28 x^{6}) + 1 (16 x^{3}) + 1 \cdot - 1)}{(x^{2} + 1) (x^{2} {({(2 x^{3} - 1)}^{2})}^{2})}$

Paso 38.5.2.11.4

Multiplica $x^{2}$ por $x^{3}$ sumando los exponentes.

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Paso 38.5.2.11.4.1

Mueve $x^{3}$ .

$\frac{x {(2 x^{3} - 1)}^{2} (4 x^{8} - 4 x^{5} + x^{2} - 28 x^{8} + 16 (x^{3} x^{2}) + x^{2} \cdot - 1 + 1 (- 28 x^{6}) + 1 (16 x^{3}) + 1 \cdot - 1)}{(x^{2} + 1) (x^{2} {({(2 x^{3} - 1)}^{2})}^{2})}$

Paso 38.5.2.11.4.2

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

$\frac{x {(2 x^{3} - 1)}^{2} (4 x^{8} - 4 x^{5} + x^{2} - 28 x^{8} + 16 x^{3 + 2} + x^{2} \cdot - 1 + 1 (- 28 x^{6}) + 1 (16 x^{3}) + 1 \cdot - 1)}{(x^{2} + 1) (x^{2} {({(2 x^{3} - 1)}^{2})}^{2})}$

Paso 38.5.2.11.4.3

Suma $3$ y $2$ .

$\frac{x {(2 x^{3} - 1)}^{2} (4 x^{8} - 4 x^{5} + x^{2} - 28 x^{8} + 16 x^{5} + x^{2} \cdot - 1 + 1 (- 28 x^{6}) + 1 (16 x^{3}) + 1 \cdot - 1)}{(x^{2} + 1) (x^{2} {({(2 x^{3} - 1)}^{2})}^{2})}$

Paso 38.5.2.11.5

Mueve $- 1$ a la izquierda de $x^{2}$ .

$\frac{x {(2 x^{3} - 1)}^{2} (4 x^{8} - 4 x^{5} + x^{2} - 28 x^{8} + 16 x^{5} - 1 \cdot x^{2} + 1 (- 28 x^{6}) + 1 (16 x^{3}) + 1 \cdot - 1)}{(x^{2} + 1) (x^{2} {({(2 x^{3} - 1)}^{2})}^{2})}$

Paso 38.5.2.11.6

Reescribe $- 1 x^{2}$ como $- x^{2}$ .

$\frac{x {(2 x^{3} - 1)}^{2} (4 x^{8} - 4 x^{5} + x^{2} - 28 x^{8} + 16 x^{5} - x^{2} + 1 (- 28 x^{6}) + 1 (16 x^{3}) + 1 \cdot - 1)}{(x^{2} + 1) (x^{2} {({(2 x^{3} - 1)}^{2})}^{2})}$

Paso 38.5.2.11.7

Multiplica $- 28 x^{6}$ por $1$ .

$\frac{x {(2 x^{3} - 1)}^{2} (4 x^{8} - 4 x^{5} + x^{2} - 28 x^{8} + 16 x^{5} - x^{2} - 28 x^{6} + 1 (16 x^{3}) + 1 \cdot - 1)}{(x^{2} + 1) (x^{2} {({(2 x^{3} - 1)}^{2})}^{2})}$

Paso 38.5.2.11.8

Multiplica $16 x^{3}$ por $1$ .

$\frac{x {(2 x^{3} - 1)}^{2} (4 x^{8} - 4 x^{5} + x^{2} - 28 x^{8} + 16 x^{5} - x^{2} - 28 x^{6} + 16 x^{3} + 1 \cdot - 1)}{(x^{2} + 1) (x^{2} {({(2 x^{3} - 1)}^{2})}^{2})}$

Paso 38.5.2.11.9

Multiplica $- 1$ por $1$ .

$\frac{x {(2 x^{3} - 1)}^{2} (4 x^{8} - 4 x^{5} + x^{2} - 28 x^{8} + 16 x^{5} - x^{2} - 28 x^{6} + 16 x^{3} - 1)}{(x^{2} + 1) (x^{2} {({(2 x^{3} - 1)}^{2})}^{2})}$

Paso 38.5.3

Combina los términos opuestos en $4 x^{8} - 4 x^{5} + x^{2} - 28 x^{8} + 16 x^{5} - x^{2} - 28 x^{6} + 16 x^{3} - 1$ .

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Paso 38.5.3.1

Resta $x^{2}$ de $x^{2}$ .

$\frac{x {(2 x^{3} - 1)}^{2} (4 x^{8} - 4 x^{5} - 28 x^{8} + 16 x^{5} + 0 - 28 x^{6} + 16 x^{3} - 1)}{(x^{2} + 1) (x^{2} {({(2 x^{3} - 1)}^{2})}^{2})}$

Paso 38.5.3.2

Suma $4 x^{8} - 4 x^{5} - 28 x^{8} + 16 x^{5}$ y $0$ .

$\frac{x {(2 x^{3} - 1)}^{2} (4 x^{8} - 4 x^{5} - 28 x^{8} + 16 x^{5} - 28 x^{6} + 16 x^{3} - 1)}{(x^{2} + 1) (x^{2} {({(2 x^{3} - 1)}^{2})}^{2})}$

Paso 38.5.4

Resta $28 x^{8}$ de $4 x^{8}$ .

$\frac{x {(2 x^{3} - 1)}^{2} (- 24 x^{8} - 4 x^{5} + 16 x^{5} - 28 x^{6} + 16 x^{3} - 1)}{(x^{2} + 1) (x^{2} {({(2 x^{3} - 1)}^{2})}^{2})}$

Paso 38.5.5

Suma $- 4 x^{5}$ y $16 x^{5}$ .

$\frac{x {(2 x^{3} - 1)}^{2} (- 24 x^{8} + 12 x^{5} - 28 x^{6} + 16 x^{3} - 1)}{(x^{2} + 1) (x^{2} {({(2 x^{3} - 1)}^{2})}^{2})}$

Paso 38.5.6

Reordena los términos.

$\frac{x {(2 x^{3} - 1)}^{2} (- 24 x^{8} - 28 x^{6} + 12 x^{5} + 16 x^{3} - 1)}{(x^{2} + 1) (x^{2} {({(2 x^{3} - 1)}^{2})}^{2})}$

Paso 38.6

Combina los términos.

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Paso 38.6.1

Multiplica los exponentes en ${({(2 x^{3} - 1)}^{2})}^{2}$ .

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Paso 38.6.1.1

Aplica la regla de la potencia y multiplica los exponentes, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{x {(2 x^{3} - 1)}^{2} (- 24 x^{8} - 28 x^{6} + 12 x^{5} + 16 x^{3} - 1)}{(x^{2} + 1) (x^{2} {(2 x^{3} - 1)}^{2 \cdot 2})}$

Paso 38.6.1.2

Multiplica $2$ por $2$ .

$\frac{x {(2 x^{3} - 1)}^{2} (- 24 x^{8} - 28 x^{6} + 12 x^{5} + 16 x^{3} - 1)}{(x^{2} + 1) (x^{2} {(2 x^{3} - 1)}^{4})}$

Paso 38.6.2

Cancela el factor común de $x$ y $x^{2}$ .

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Paso 38.6.2.1

Factoriza $x$ de $x {(2 x^{3} - 1)}^{2} (- 24 x^{8} - 28 x^{6} + 12 x^{5} + 16 x^{3} - 1)$ .

$\frac{x ({(2 x^{3} - 1)}^{2} (- 24 x^{8} - 28 x^{6} + 12 x^{5} + 16 x^{3} - 1))}{(x^{2} + 1) x^{2} {(2 x^{3} - 1)}^{4}}$

Paso 38.6.2.2

Cancela los factores comunes.

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Paso 38.6.2.2.1

Factoriza $x$ de $(x^{2} + 1) x^{2} {(2 x^{3} - 1)}^{4}$ .

$\frac{x ({(2 x^{3} - 1)}^{2} (- 24 x^{8} - 28 x^{6} + 12 x^{5} + 16 x^{3} - 1))}{x ((x^{2} + 1) x {(2 x^{3} - 1)}^{4})}$

Paso 38.6.2.2.2

Cancela el factor común.

$\frac{x ({(2 x^{3} - 1)}^{2} (- 24 x^{8} - 28 x^{6} + 12 x^{5} + 16 x^{3} - 1))}{x ((x^{2} + 1) x {(2 x^{3} - 1)}^{4})}$

Paso 38.6.2.2.3

Reescribe la expresión.

$\frac{{(2 x^{3} - 1)}^{2} (- 24 x^{8} - 28 x^{6} + 12 x^{5} + 16 x^{3} - 1)}{(x^{2} + 1) x {(2 x^{3} - 1)}^{4}}$

Paso 38.6.3

Cancela los factores comunes.

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Paso 38.6.3.1

Factoriza ${(2 x^{3} - 1)}^{2}$ de $(x^{2} + 1) x {(2 x^{3} - 1)}^{4}$ .

$\frac{{(2 x^{3} - 1)}^{2} (- 24 x^{8} - 28 x^{6} + 12 x^{5} + 16 x^{3} - 1)}{{(2 x^{3} - 1)}^{2} ((x^{2} + 1) x {(2 x^{3} - 1)}^{2})}$

Paso 38.6.3.2

Cancela el factor común.

$\frac{{(2 x^{3} - 1)}^{2} (- 24 x^{8} - 28 x^{6} + 12 x^{5} + 16 x^{3} - 1)}{{(2 x^{3} - 1)}^{2} ((x^{2} + 1) x {(2 x^{3} - 1)}^{2})}$

Paso 38.6.3.3

Reescribe la expresión.

$\frac{- 24 x^{8} - 28 x^{6} + 12 x^{5} + 16 x^{3} - 1}{(x^{2} + 1) x {(2 x^{3} - 1)}^{2}}$

Paso 38.7

Reordena los términos.

$\frac{- 24 x^{8} - 28 x^{6} + 12 x^{5} + 16 x^{3} - 1}{x {(2 x^{3} - 1)}^{2} (x^{2} + 1)}$

Paso 38.8

Factoriza $- 1$ de $- 24 x^{8}$ .

$\frac{- (24 x^{8}) - 28 x^{6} + 12 x^{5} + 16 x^{3} - 1}{x {(2 x^{3} - 1)}^{2} (x^{2} + 1)}$

Paso 38.9

Factoriza $- 1$ de $- 28 x^{6}$ .

$\frac{- (24 x^{8}) - (28 x^{6}) + 12 x^{5} + 16 x^{3} - 1}{x {(2 x^{3} - 1)}^{2} (x^{2} + 1)}$

Paso 38.10

Factoriza $- 1$ de $- (24 x^{8}) - (28 x^{6})$ .

$\frac{- (24 x^{8} + 28 x^{6}) + 12 x^{5} + 16 x^{3} - 1}{x {(2 x^{3} - 1)}^{2} (x^{2} + 1)}$

Paso 38.11

Factoriza $- 1$ de $12 x^{5}$ .

$\frac{- (24 x^{8} + 28 x^{6}) - (- 12 x^{5}) + 16 x^{3} - 1}{x {(2 x^{3} - 1)}^{2} (x^{2} + 1)}$

Paso 38.12

Factoriza $- 1$ de $- (24 x^{8} + 28 x^{6}) - (- 12 x^{5})$ .

$\frac{- (24 x^{8} + 28 x^{6} - 12 x^{5}) + 16 x^{3} - 1}{x {(2 x^{3} - 1)}^{2} (x^{2} + 1)}$

Paso 38.13

Factoriza $- 1$ de $16 x^{3}$ .

$\frac{- (24 x^{8} + 28 x^{6} - 12 x^{5}) - (- 16 x^{3}) - 1}{x {(2 x^{3} - 1)}^{2} (x^{2} + 1)}$

Paso 38.14

Factoriza $- 1$ de $- (24 x^{8} + 28 x^{6} - 12 x^{5}) - (- 16 x^{3})$ .

$\frac{- (24 x^{8} + 28 x^{6} - 12 x^{5} - 16 x^{3}) - 1}{x {(2 x^{3} - 1)}^{2} (x^{2} + 1)}$

Paso 38.15

Reescribe $- 1$ como $- 1 (1)$ .

$\frac{- (24 x^{8} + 28 x^{6} - 12 x^{5} - 16 x^{3}) - 1 (1)}{x {(2 x^{3} - 1)}^{2} (x^{2} + 1)}$

Paso 38.16

Factoriza $- 1$ de $- (24 x^{8} + 28 x^{6} - 12 x^{5} - 16 x^{3}) - 1 (1)$ .

$\frac{- (24 x^{8} + 28 x^{6} - 12 x^{5} - 16 x^{3} + 1)}{x {(2 x^{3} - 1)}^{2} (x^{2} + 1)}$

Paso 38.17

Reescribe $- (24 x^{8} + 28 x^{6} - 12 x^{5} - 16 x^{3} + 1)$ como $- 1 (24 x^{8} + 28 x^{6} - 12 x^{5} - 16 x^{3} + 1)$ .

$\frac{- 1 (24 x^{8} + 28 x^{6} - 12 x^{5} - 16 x^{3} + 1)}{x {(2 x^{3} - 1)}^{2} (x^{2} + 1)}$

Paso 38.18

Mueve el negativo al frente de la fracción.

$- \frac{24 x^{8} + 28 x^{6} - 12 x^{5} - 16 x^{3} + 1}{x {(2 x^{3} - 1)}^{2} (x^{2} + 1)}$