Cálculo Ejemplos

$\arccos (x)$

Paso 1

La derivada de $\arccos (x)$ con respecto a $x$ es $- \frac{1}{\sqrt{1 - x^{2}}}$ .

$f' (x) = - \frac{1}{\sqrt{1 - x^{2}}}$

Paso 2

Obtener la segunda derivada.

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Paso 2.1

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

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

Paso 2.2

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 $f (x) = - 1$ y $g (x) = \frac{1}{{(1 - x^{2})}^{\frac{1}{2}}}$ .

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

Paso 2.3

Aplica reglas básicas de exponentes.

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Paso 2.3.1

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

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

Paso 2.3.2

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

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Paso 2.3.2.1

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

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

Paso 2.3.2.2

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

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

Paso 2.3.2.3

Mueve el negativo al frente de la fracción.

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

Paso 2.4

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 $g (x) = 1 - x^{2}$ .

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Paso 2.4.1

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

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

Paso 2.4.2

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

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

Paso 2.4.3

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

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

Paso 2.5

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

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

Paso 2.6

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

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

Paso 2.7

Combina los numeradores sobre el denominador común.

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

Paso 2.8

Simplifica el numerador.

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Paso 2.8.1

Multiplica $- 1$ por $2$ .

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

Paso 2.8.2

Resta $2$ de $- 1$ .

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

Paso 2.9

Combina fracciones.

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Paso 2.9.1

Mueve el negativo al frente de la fracción.

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

Paso 2.9.2

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

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

Paso 2.9.3

Simplifica la expresión.

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Paso 2.9.3.1

Mueve ${(1 - x^{2})}^{- \frac{3}{2}}$ al denominador mediante la regla del exponente negativo $b^{- n} = \frac{1}{b^{n}}$ .

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

Paso 2.9.3.2

Multiplica $- 1$ por $- 1$ .

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

Paso 2.9.3.3

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

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

Paso 2.10

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

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

Paso 2.11

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

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

Paso 2.12

Suma $0$ y $\frac{d}{d x} [- x^{2}]$ .

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

Paso 2.13

Como $- 1$ es constante con respecto a $x$ , la derivada de $- x^{2}$ con respecto a $x$ es $- \frac{d}{d x} [x^{2}]$ .

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

Paso 2.14

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

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

Paso 2.15

Simplifica los términos.

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Paso 2.15.1

Multiplica $2$ por $- 1$ .

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

Paso 2.15.2

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

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

Paso 2.15.3

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

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

Paso 2.15.4

Factoriza $2$ de $- 2 x$ .

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

Paso 2.16

Cancela los factores comunes.

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Paso 2.16.1

Factoriza $2$ de $2 {(1 - x^{2})}^{\frac{3}{2}}$ .

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

Paso 2.16.2

Cancela el factor común.

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

Paso 2.16.3

Reescribe la expresión.

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

Paso 2.17

Mueve el negativo al frente de la fracción.

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

Paso 2.18

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

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

Paso 2.19

Simplifica la expresión.

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Paso 2.19.1

Multiplica $\frac{1}{{(1 - x^{2})}^{\frac{1}{2}}}$ por $0$ .

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

Paso 2.19.2

Suma $- \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}}$ y $0$ .

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

Paso 3

Obtén la tercera derivada.

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Paso 3.1

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

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

Paso 3.2

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

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

Paso 3.3

Diferencia con la regla de la potencia.

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Paso 3.3.1

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

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Paso 3.3.1.1

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

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

Paso 3.3.1.2

Cancela el factor común de $2$ .

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Paso 3.3.1.2.1

Cancela el factor común.

Paso 3.3.1.2.2

Reescribe la expresión.

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

Paso 3.3.2

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

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

Paso 3.3.3

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

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

Paso 3.4

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

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Paso 3.4.1

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

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

Paso 3.4.2

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

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

Paso 3.4.3

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

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

Paso 3.5

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

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

Paso 3.6

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

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

Paso 3.7

Combina los numeradores sobre el denominador común.

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

Paso 3.8

Simplifica el numerador.

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Paso 3.8.1

Multiplica $- 1$ por $2$ .

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

Paso 3.8.2

Resta $2$ de $3$ .

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

Paso 3.9

Combina fracciones.

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Paso 3.9.1

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

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

Paso 3.9.2

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

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

Paso 3.10

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

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

Paso 3.11

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

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

Paso 3.12

Suma $0$ y $\frac{d}{d x} [- x^{2}]$ .

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

Paso 3.13

Como $- 1$ es constante con respecto a $x$ , la derivada de $- x^{2}$ con respecto a $x$ es $- \frac{d}{d x} [x^{2}]$ .

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

Paso 3.14

Multiplica.

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Paso 3.14.1

Multiplica $- 1$ por $- 1$ .

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

Paso 3.14.2

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

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

Paso 3.15

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

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

Paso 3.16

Combina fracciones.

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Paso 3.16.1

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

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

Paso 3.16.2

Multiplica $3$ por $2$ .

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

Paso 3.16.3

Combina $\frac{6 ({(1 - x^{2})}^{\frac{1}{2}} x)}{2}$ y $x$ .

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

Paso 3.17

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

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

Paso 3.18

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

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

Paso 3.19

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

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

Paso 3.20

Suma $1$ y $1$ .

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

Paso 3.21

Factoriza $2$ de $6 {(1 - x^{2})}^{\frac{1}{2}} x^{2}$ .

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

Paso 3.22

Cancela los factores comunes.

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Paso 3.22.1

Factoriza $2$ de $2$ .

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

Paso 3.22.2

Cancela el factor común.

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

Paso 3.22.3

Reescribe la expresión.

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

Paso 3.22.4

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

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

Paso 3.23

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

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

Paso 3.24

Simplifica la expresión.

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Paso 3.24.1

Multiplica $\frac{x}{{(1 - x^{2})}^{\frac{3}{2}}}$ por $0$ .

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

Paso 3.24.2

Suma $- \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}}$ y $0$ .

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

Paso 4

Obtén la cuarta derivada.

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Paso 4.1

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

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

Paso 4.2

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

Paso 4.3

Diferencia con la regla de la suma.

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Paso 4.3.1

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

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Paso 4.3.1.1

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

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

Paso 4.3.1.2

Multiplica $3$ por $2$ .

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

Paso 4.3.2

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

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

Paso 4.4

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

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Paso 4.4.1

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

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

Paso 4.4.2

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

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

Paso 4.4.3

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

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

Paso 4.5

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

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

Paso 4.6

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

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

Paso 4.7

Combina los numeradores sobre el denominador común.

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

Paso 4.8

Simplifica el numerador.

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Paso 4.8.1

Multiplica $- 1$ por $2$ .

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

Paso 4.8.2

Resta $2$ de $3$ .

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

Paso 4.9

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

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

Paso 4.10

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

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

Paso 4.11

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

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

Paso 4.12

Suma $0$ y $\frac{d}{d x} [- x^{2}]$ .

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

Paso 4.13

Como $- 1$ es constante con respecto a $x$ , la derivada de $- x^{2}$ con respecto a $x$ es $- \frac{d}{d x} [x^{2}]$ .

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

Paso 4.14

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

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

Paso 4.15

Simplifica los términos.

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Paso 4.15.1

Multiplica $2$ por $- 1$ .

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

Paso 4.15.2

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

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

Paso 4.15.3

Multiplica $3$ por $- 2$ .

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

Paso 4.15.4

Combina $\frac{- 6 {(1 - x^{2})}^{\frac{1}{2}}}{2}$ y $x$ .

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

Paso 4.15.5

Factoriza $2$ de $- 6 {(1 - x^{2})}^{\frac{1}{2}} x$ .

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

Paso 4.16

Cancela los factores comunes.

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Paso 4.16.1

Factoriza $2$ de $2$ .

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

Paso 4.16.2

Cancela el factor común.

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

Paso 4.16.3

Reescribe la expresión.

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

Paso 4.16.4

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

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

Paso 4.17

Como $3$ es constante con respecto a $x$ , la derivada de $3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}$ con respecto a $x$ es $3 \frac{d}{d x} [{(1 - x^{2})}^{\frac{1}{2}} x^{2}]$ .

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

Paso 4.18

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 $f (x) = {(1 - x^{2})}^{\frac{1}{2}}$ y $g (x) = x^{2}$ .

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

Paso 4.19

Diferencia con la regla de la potencia.

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Paso 4.19.1

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

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

Paso 4.19.2

Mueve $2$ a la izquierda de ${(1 - x^{2})}^{\frac{1}{2}}$ .

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

Paso 4.20

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 $g (x) = 1 - x^{2}$ .

Toca para ver más pasos...

Paso 4.20.1

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

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

Paso 4.20.2

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 $n = \frac{1}{2}$ .

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

Paso 4.20.3

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

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

Paso 4.21

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

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

Paso 4.22

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

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

Paso 4.23

Combina los numeradores sobre el denominador común.

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

Paso 4.24

Simplifica el numerador.

Toca para ver más pasos...

Paso 4.24.1

Multiplica $- 1$ por $2$ .

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

Paso 4.24.2

Resta $2$ de $1$ .

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

Paso 4.25

Combina fracciones.

Toca para ver más pasos...

Paso 4.25.1

Mueve el negativo al frente de la fracción.

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

Paso 4.25.2

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

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

Paso 4.25.3

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

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

Paso 4.25.4

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

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

Paso 4.26

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

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

Paso 4.27

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

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

Paso 4.28

Suma $0$ y $\frac{d}{d x} [- x^{2}]$ .

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

Paso 4.29

Como $- 1$ es constante con respecto a $x$ , la derivada de $- x^{2}$ con respecto a $x$ es $- \frac{d}{d x} [x^{2}]$ .

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

Paso 4.30

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

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

Paso 4.31

Combina fracciones.

Toca para ver más pasos...

Paso 4.31.1

Multiplica $2$ por $- 1$ .

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

Paso 4.31.2

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

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

Paso 4.31.3

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

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

Paso 4.32

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

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

Paso 4.33

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

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

Paso 4.34

Suma $1$ y $2$ .

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

Paso 4.35

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

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

Paso 4.36

Cancela los factores comunes.

Toca para ver más pasos...

Paso 4.36.1

Factoriza $2$ de $2 {(1 - x^{2})}^{\frac{1}{2}}$ .

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

Paso 4.36.2

Cancela el factor común.

Paso 4.36.3

Reescribe la expresión.

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

Paso 4.37

Mueve el negativo al frente de la fracción.

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

Paso 4.38

Reordena $1$ y $- x^{2}$ .

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

Paso 4.39

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

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

Paso 4.40

Combina los numeradores sobre el denominador común.

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

Paso 4.41

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

Toca para ver más pasos...

Paso 4.41.1

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

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

Paso 4.41.2

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

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

Paso 4.41.3

Combina los numeradores sobre el denominador común.

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

Paso 4.41.4

Suma $1$ y $1$ .

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

Paso 4.41.5

Divide $2$ por $2$ .

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

Paso 4.42

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

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

Paso 4.43

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

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

Paso 4.44

Para escribir $- 3 {(1 - x^{2})}^{\frac{1}{2}} x$ como una fracción con un denominador común, multiplica por $\frac{{(- x^{2} + 1)}^{\frac{1}{2}}}{{(- x^{2} + 1)}^{\frac{1}{2}}}$ .

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

Paso 4.45

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

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

Paso 4.46

Combina los numeradores sobre el denominador común.

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

Paso 4.47

Reordena los términos.

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

Paso 4.48

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

Toca para ver más pasos...

Paso 4.48.1

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

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

Paso 4.48.2

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

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

Paso 4.48.3

Combina los numeradores sobre el denominador común.

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

Paso 4.48.4

Suma $1$ y $1$ .

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

Paso 4.48.5

Divide $2$ por $2$ .

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

Paso 4.49

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

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

Paso 4.50

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

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

Paso 4.51

Reordena los términos.

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

Paso 4.52

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

$- \frac{\frac{{(- x^{2} + 1)}^{\frac{1}{2}} ({(- x^{2} + 1)}^{\frac{5}{2}} (- 3 (- x^{2} + 1) x + 3 (2 x (- x^{2} + 1) - x^{3})))}{{(- x^{2} + 1)}^{\frac{1}{2}}} - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

Paso 4.53

Cancela los factores comunes.

Toca para ver más pasos...

Paso 4.53.1

Multiplica por $1$ .

$- \frac{\frac{{(- x^{2} + 1)}^{\frac{1}{2}} ({(- x^{2} + 1)}^{\frac{5}{2}} (- 3 (- x^{2} + 1) x + 3 (2 x (- x^{2} + 1) - x^{3})))}{{(- x^{2} + 1)}^{\frac{1}{2}} \cdot 1} - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

Paso 4.53.2

Cancela el factor común.

Paso 4.53.3

Reescribe la expresión.

$- \frac{\frac{{(- x^{2} + 1)}^{\frac{5}{2}} (- 3 (- x^{2} + 1) x + 3 (2 x (- x^{2} + 1) - x^{3}))}{1} - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

Paso 4.53.4

Divide ${(- x^{2} + 1)}^{\frac{5}{2}} (- 3 (- x^{2} + 1) x + 3 (2 x (- x^{2} + 1) - x^{3}))$ por $1$ .

$- \frac{{(- x^{2} + 1)}^{\frac{5}{2}} (- 3 (- x^{2} + 1) x + 3 (2 x (- x^{2} + 1) - x^{3})) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

Paso 4.54

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

Toca para ver más pasos...

Paso 4.54.1

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

$- \frac{{(- x^{2} + 1)}^{\frac{5}{2}} (- 3 (- x^{2} + 1) x + 3 (2 x (- x^{2} + 1) - x^{3})) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) (\frac{d}{d u_{3}} [{u_{3}}^{3}] \frac{d}{d x} [1 - x^{2}])}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

Paso 4.54.2

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 $n = 3$ .

$- \frac{{(- x^{2} + 1)}^{\frac{5}{2}} (- 3 (- x^{2} + 1) x + 3 (2 x (- x^{2} + 1) - x^{3})) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) (3 {u_{3}}^{2} \frac{d}{d x} [1 - x^{2}])}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

Paso 4.54.3

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

$- \frac{{(- x^{2} + 1)}^{\frac{5}{2}} (- 3 (- x^{2} + 1) x + 3 (2 x (- x^{2} + 1) - x^{3})) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) (3 {(1 - x^{2})}^{2} \frac{d}{d x} [1 - x^{2}])}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

Paso 4.55

Diferencia.

Toca para ver más pasos...

Paso 4.55.1

Multiplica $3$ por $- 1$ .

$- \frac{{(- x^{2} + 1)}^{\frac{5}{2}} (- 3 (- x^{2} + 1) x + 3 (2 x (- x^{2} + 1) - x^{3})) - 3 ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) ({(1 - x^{2})}^{2} \frac{d}{d x} [1 - x^{2}])}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

Paso 4.55.2

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

$- \frac{{(- x^{2} + 1)}^{\frac{5}{2}} (- 3 (- x^{2} + 1) x + 3 (2 x (- x^{2} + 1) - x^{3})) - 3 ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) ({(1 - x^{2})}^{2} (\frac{d}{d x} [1] + \frac{d}{d x} [- x^{2}]))}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

Paso 4.55.3

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

$- \frac{{(- x^{2} + 1)}^{\frac{5}{2}} (- 3 (- x^{2} + 1) x + 3 (2 x (- x^{2} + 1) - x^{3})) - 3 ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) ({(1 - x^{2})}^{2} (0 + \frac{d}{d x} [- x^{2}]))}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

Paso 4.55.4

Suma $0$ y $\frac{d}{d x} [- x^{2}]$ .

$- \frac{{(- x^{2} + 1)}^{\frac{5}{2}} (- 3 (- x^{2} + 1) x + 3 (2 x (- x^{2} + 1) - x^{3})) - 3 ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) ({(1 - x^{2})}^{2} \frac{d}{d x} [- x^{2}])}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

Paso 4.55.5

Como $- 1$ es constante con respecto a $x$ , la derivada de $- x^{2}$ con respecto a $x$ es $- \frac{d}{d x} [x^{2}]$ .

$- \frac{{(- x^{2} + 1)}^{\frac{5}{2}} (- 3 (- x^{2} + 1) x + 3 (2 x (- x^{2} + 1) - x^{3})) - 3 ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) ({(1 - x^{2})}^{2} (- \frac{d}{d x} [x^{2}]))}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

Paso 4.55.6

Multiplica $- 1$ por $- 3$ .

$- \frac{{(- x^{2} + 1)}^{\frac{5}{2}} (- 3 (- x^{2} + 1) x + 3 (2 x (- x^{2} + 1) - x^{3})) + 3 ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) ({(1 - x^{2})}^{2} (\frac{d}{d x} [x^{2}]))}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

Paso 4.55.7

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

$- \frac{{(- x^{2} + 1)}^{\frac{5}{2}} (- 3 (- x^{2} + 1) x + 3 (2 x (- x^{2} + 1) - x^{3})) + 3 ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) ({(1 - x^{2})}^{2} (2 x))}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

Paso 4.55.8

Simplifica la expresión.

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Paso 4.55.8.1

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

$- \frac{{(- x^{2} + 1)}^{\frac{5}{2}} (- 3 (- x^{2} + 1) x + 3 (2 x (- x^{2} + 1) - x^{3})) + 3 ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) (2 \cdot {(1 - x^{2})}^{2} x)}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

Paso 4.55.8.2

Multiplica $2$ por $3$ .

$- \frac{{(- x^{2} + 1)}^{\frac{5}{2}} (- 3 (- x^{2} + 1) x + 3 (2 x (- x^{2} + 1) - x^{3})) + 6 ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) ({(1 - x^{2})}^{2} x)}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

Paso 4.55.9

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

Paso 4.55.10

Simplifica la expresión.

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Paso 4.55.10.1

Multiplica $\frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}}$ por $0$ .

Paso 4.55.10.2

Suma $- \frac{{(- x^{2} + 1)}^{\frac{5}{2}} (- 3 (- x^{2} + 1) x + 3 (2 x (- x^{2} + 1) - x^{3})) + 6 ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) ({(1 - x^{2})}^{2} x)}{{(1 - x^{2})}^{6}}$ y $0$ .

Paso 4.56

Simplifica.

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Paso 4.56.1

Aplica la propiedad distributiva.

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

Paso 4.56.2

Aplica la propiedad distributiva.

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

Paso 4.56.3

Aplica la propiedad distributiva.

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

Paso 4.56.4

Aplica la propiedad distributiva.

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

Paso 4.56.5

Aplica la propiedad distributiva.

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

Paso 4.56.6

Simplifica el numerador.

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Paso 4.56.6.1

Simplifica cada término.

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Paso 4.56.6.1.1

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

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Paso 4.56.6.1.1.1

Mueve $x$ .

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

Paso 4.56.6.1.1.2

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

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Paso 4.56.6.1.1.2.1

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

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

Paso 4.56.6.1.1.2.2

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

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

Paso 4.56.6.1.1.3

Suma $1$ y $2$ .

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

Paso 4.56.6.1.2

Multiplica $- 1$ por $- 3$ .

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

Paso 4.56.6.1.3

Multiplica $- 3$ por $1$ .

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

Paso 4.56.6.1.4

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

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Paso 4.56.6.1.4.1

Mueve $x^{2}$ .

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

Paso 4.56.6.1.4.2

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

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Paso 4.56.6.1.4.2.1

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

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

Paso 4.56.6.1.4.2.2

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

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

Paso 4.56.6.1.4.3

Suma $2$ y $1$ .

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

Paso 4.56.6.1.5

Multiplica $- 1$ por $2$ .

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

Paso 4.56.6.1.6

Multiplica $- 2$ por $3$ .

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

Paso 4.56.6.1.7

Multiplica $2$ por $1$ .

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

Paso 4.56.6.1.8

Multiplica $2$ por $3$ .

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

Paso 4.56.6.1.9

Multiplica $- 1$ por $3$ .

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

Paso 4.56.6.2

Combina los términos opuestos en $3 x^{3} - 3 x - 6 x^{3} + 6 x - 3 x^{3}$ .

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Paso 4.56.6.2.1

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

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

Paso 4.56.6.2.2

Suma $- 3 x - 6 x^{3} + 6 x$ y $0$ .

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

Paso 4.56.6.3

Suma $- 3 x$ y $6 x$ .

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

Paso 4.56.6.4

Aplica la propiedad distributiva.

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

Paso 4.56.6.5

Reescribe con la propiedad conmutativa de la multiplicación.

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

Paso 4.56.6.6

Reescribe con la propiedad conmutativa de la multiplicación.

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

Paso 4.56.6.7

Multiplica $3$ por $6$ .

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + (6 {(1 - x^{2})}^{\frac{3}{2}} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) ({(1 - x^{2})}^{2} x)}{{(1 - x^{2})}^{6}}$

Paso 4.56.6.8

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

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + (6 {(1 - x^{2})}^{\frac{3}{2}} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) ((1 - x^{2}) (1 - x^{2}) x)}{{(1 - x^{2})}^{6}}$

Paso 4.56.6.9

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

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Paso 4.56.6.9.1

Aplica la propiedad distributiva.

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + (6 {(1 - x^{2})}^{\frac{3}{2}} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) ((1 (1 - x^{2}) - x^{2} (1 - x^{2})) x)}{{(1 - x^{2})}^{6}}$

Paso 4.56.6.9.2

Aplica la propiedad distributiva.

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + (6 {(1 - x^{2})}^{\frac{3}{2}} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) ((1 \cdot 1 + 1 (- x^{2}) - x^{2} (1 - x^{2})) x)}{{(1 - x^{2})}^{6}}$

Paso 4.56.6.9.3

Aplica la propiedad distributiva.

Paso 4.56.6.10

Simplifica y combina los términos similares.

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Paso 4.56.6.10.1

Simplifica cada término.

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Paso 4.56.6.10.1.1

Multiplica $1$ por $1$ .

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + (6 {(1 - x^{2})}^{\frac{3}{2}} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) ((1 + 1 (- x^{2}) - x^{2} \cdot 1 - x^{2} (- x^{2})) x)}{{(1 - x^{2})}^{6}}$

Paso 4.56.6.10.1.2

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

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + (6 {(1 - x^{2})}^{\frac{3}{2}} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) ((1 - x^{2} - x^{2} \cdot 1 - x^{2} (- x^{2})) x)}{{(1 - x^{2})}^{6}}$

Paso 4.56.6.10.1.3

Multiplica $- 1$ por $1$ .

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + (6 {(1 - x^{2})}^{\frac{3}{2}} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) ((1 - x^{2} - x^{2} - x^{2} (- x^{2})) x)}{{(1 - x^{2})}^{6}}$

Paso 4.56.6.10.1.4

Reescribe con la propiedad conmutativa de la multiplicación.

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + (6 {(1 - x^{2})}^{\frac{3}{2}} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) ((1 - x^{2} - x^{2} - 1 \cdot - 1 x^{2} x^{2}) x)}{{(1 - x^{2})}^{6}}$

Paso 4.56.6.10.1.5

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

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Paso 4.56.6.10.1.5.1

Mueve $x^{2}$ .

Paso 4.56.6.10.1.5.2

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

Paso 4.56.6.10.1.5.3

Suma $2$ y $2$ .

Paso 4.56.6.10.1.6

Multiplica $- 1$ por $- 1$ .

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + (6 {(1 - x^{2})}^{\frac{3}{2}} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) ((1 - x^{2} - x^{2} + 1 x^{4}) x)}{{(1 - x^{2})}^{6}}$

Paso 4.56.6.10.1.7

Multiplica $x^{4}$ por $1$ .

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + (6 {(1 - x^{2})}^{\frac{3}{2}} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) ((1 - x^{2} - x^{2} + x^{4}) x)}{{(1 - x^{2})}^{6}}$

Paso 4.56.6.10.2

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

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + (6 {(1 - x^{2})}^{\frac{3}{2}} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) ((1 - 2 x^{2} + x^{4}) x)}{{(1 - x^{2})}^{6}}$

Paso 4.56.6.11

Aplica la propiedad distributiva.

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + (6 {(1 - x^{2})}^{\frac{3}{2}} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) (1 x - 2 x^{2} x + x^{4} x)}{{(1 - x^{2})}^{6}}$

Paso 4.56.6.12

Simplifica.

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Paso 4.56.6.12.1

Multiplica $x$ por $1$ .

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + (6 {(1 - x^{2})}^{\frac{3}{2}} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) (x - 2 x^{2} x + x^{4} x)}{{(1 - x^{2})}^{6}}$

Paso 4.56.6.12.2

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

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Paso 4.56.6.12.2.1

Mueve $x$ .

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + (6 {(1 - x^{2})}^{\frac{3}{2}} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) (x - 2 (x \cdot x^{2}) + x^{4} x)}{{(1 - x^{2})}^{6}}$

Paso 4.56.6.12.2.2

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

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Paso 4.56.6.12.2.2.1

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

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + (6 {(1 - x^{2})}^{\frac{3}{2}} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) (x - 2 (x^{1} x^{2}) + x^{4} x)}{{(1 - x^{2})}^{6}}$

Paso 4.56.6.12.2.2.2

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

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + (6 {(1 - x^{2})}^{\frac{3}{2}} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) (x - 2 x^{1 + 2} + x^{4} x)}{{(1 - x^{2})}^{6}}$

Paso 4.56.6.12.2.3

Suma $1$ y $2$ .

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + (6 {(1 - x^{2})}^{\frac{3}{2}} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) (x - 2 x^{3} + x^{4} x)}{{(1 - x^{2})}^{6}}$

Paso 4.56.6.12.3

Multiplica $x^{4}$ por $x$ sumando los exponentes.

Toca para ver más pasos...

Paso 4.56.6.12.3.1

Multiplica $x^{4}$ por $x$ .

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Paso 4.56.6.12.3.1.1

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

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + (6 {(1 - x^{2})}^{\frac{3}{2}} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) (x - 2 x^{3} + x^{4} x^{1})}{{(1 - x^{2})}^{6}}$

Paso 4.56.6.12.3.1.2

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

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + (6 {(1 - x^{2})}^{\frac{3}{2}} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) (x - 2 x^{3} + x^{4 + 1})}{{(1 - x^{2})}^{6}}$

Paso 4.56.6.12.3.2

Suma $4$ y $1$ .

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + (6 {(1 - x^{2})}^{\frac{3}{2}} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) (x - 2 x^{3} + x^{5})}{{(1 - x^{2})}^{6}}$

Paso 4.56.6.13

Expande $(6 {(1 - x^{2})}^{\frac{3}{2}} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) (x - 2 x^{3} + x^{5})$ mediante la multiplicación de cada término de la primera expresión por cada término de la segunda expresión.

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + 6 {(1 - x^{2})}^{\frac{3}{2}} x + 6 {(1 - x^{2})}^{\frac{3}{2}} (- 2 x^{3}) + 6 {(1 - x^{2})}^{\frac{3}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2} x + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2} (- 2 x^{3}) + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2} x^{5}}{{(1 - x^{2})}^{6}}$

Paso 4.56.6.14

Simplifica cada término.

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Paso 4.56.6.14.1

Reescribe con la propiedad conmutativa de la multiplicación.

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + 6 {(1 - x^{2})}^{\frac{3}{2}} x + 6 \cdot - 2 {(1 - x^{2})}^{\frac{3}{2}} x^{3} + 6 {(1 - x^{2})}^{\frac{3}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2} x + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2} (- 2 x^{3}) + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2} x^{5}}{{(1 - x^{2})}^{6}}$

Paso 4.56.6.14.2

Multiplica $6$ por $- 2$ .

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + 6 {(1 - x^{2})}^{\frac{3}{2}} x - 12 {(1 - x^{2})}^{\frac{3}{2}} x^{3} + 6 {(1 - x^{2})}^{\frac{3}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2} x + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2} (- 2 x^{3}) + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2} x^{5}}{{(1 - x^{2})}^{6}}$

Paso 4.56.6.14.3

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

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Paso 4.56.6.14.3.1

Mueve $x$ .

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + 6 {(1 - x^{2})}^{\frac{3}{2}} x - 12 {(1 - x^{2})}^{\frac{3}{2}} x^{3} + 6 {(1 - x^{2})}^{\frac{3}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} (x \cdot x^{2}) + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2} (- 2 x^{3}) + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2} x^{5}}{{(1 - x^{2})}^{6}}$

Paso 4.56.6.14.3.2

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

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Paso 4.56.6.14.3.2.1

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

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + 6 {(1 - x^{2})}^{\frac{3}{2}} x - 12 {(1 - x^{2})}^{\frac{3}{2}} x^{3} + 6 {(1 - x^{2})}^{\frac{3}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} (x^{1} x^{2}) + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2} (- 2 x^{3}) + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2} x^{5}}{{(1 - x^{2})}^{6}}$

Paso 4.56.6.14.3.2.2

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

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + 6 {(1 - x^{2})}^{\frac{3}{2}} x - 12 {(1 - x^{2})}^{\frac{3}{2}} x^{3} + 6 {(1 - x^{2})}^{\frac{3}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{1 + 2} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2} (- 2 x^{3}) + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2} x^{5}}{{(1 - x^{2})}^{6}}$

Paso 4.56.6.14.3.3

Suma $1$ y $2$ .

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + 6 {(1 - x^{2})}^{\frac{3}{2}} x - 12 {(1 - x^{2})}^{\frac{3}{2}} x^{3} + 6 {(1 - x^{2})}^{\frac{3}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{3} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2} (- 2 x^{3}) + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2} x^{5}}{{(1 - x^{2})}^{6}}$

Paso 4.56.6.14.4

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

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Paso 4.56.6.14.4.1

Mueve $x^{3}$ .

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + 6 {(1 - x^{2})}^{\frac{3}{2}} x - 12 {(1 - x^{2})}^{\frac{3}{2}} x^{3} + 6 {(1 - x^{2})}^{\frac{3}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{3} + 18 {(1 - x^{2})}^{\frac{1}{2}} (x^{3} x^{2}) \cdot - 2 + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2} x^{5}}{{(1 - x^{2})}^{6}}$

Paso 4.56.6.14.4.2

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

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + 6 {(1 - x^{2})}^{\frac{3}{2}} x - 12 {(1 - x^{2})}^{\frac{3}{2}} x^{3} + 6 {(1 - x^{2})}^{\frac{3}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{3} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{3 + 2} \cdot - 2 + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2} x^{5}}{{(1 - x^{2})}^{6}}$

Paso 4.56.6.14.4.3

Suma $3$ y $2$ .

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + 6 {(1 - x^{2})}^{\frac{3}{2}} x - 12 {(1 - x^{2})}^{\frac{3}{2}} x^{3} + 6 {(1 - x^{2})}^{\frac{3}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{3} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{5} \cdot - 2 + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2} x^{5}}{{(1 - x^{2})}^{6}}$

Paso 4.56.6.14.5

Multiplica $- 2$ por $18$ .

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + 6 {(1 - x^{2})}^{\frac{3}{2}} x - 12 {(1 - x^{2})}^{\frac{3}{2}} x^{3} + 6 {(1 - x^{2})}^{\frac{3}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{3} - 36 {(1 - x^{2})}^{\frac{1}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2} x^{5}}{{(1 - x^{2})}^{6}}$

Paso 4.56.6.14.6

Multiplica $x^{2}$ por $x^{5}$ sumando los exponentes.

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Paso 4.56.6.14.6.1

Mueve $x^{5}$ .

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + 6 {(1 - x^{2})}^{\frac{3}{2}} x - 12 {(1 - x^{2})}^{\frac{3}{2}} x^{3} + 6 {(1 - x^{2})}^{\frac{3}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{3} - 36 {(1 - x^{2})}^{\frac{1}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} (x^{5} x^{2})}{{(1 - x^{2})}^{6}}$

Paso 4.56.6.14.6.2

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

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + 6 {(1 - x^{2})}^{\frac{3}{2}} x - 12 {(1 - x^{2})}^{\frac{3}{2}} x^{3} + 6 {(1 - x^{2})}^{\frac{3}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{3} - 36 {(1 - x^{2})}^{\frac{1}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{5 + 2}}{{(1 - x^{2})}^{6}}$

Paso 4.56.6.14.6.3

Suma $5$ y $2$ .

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + 6 {(1 - x^{2})}^{\frac{3}{2}} x - 12 {(1 - x^{2})}^{\frac{3}{2}} x^{3} + 6 {(1 - x^{2})}^{\frac{3}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{3} - 36 {(1 - x^{2})}^{\frac{1}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{7}}{{(1 - x^{2})}^{6}}$

Paso 4.56.6.15

Reescribe $- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + 6 {(1 - x^{2})}^{\frac{3}{2}} x - 12 {(1 - x^{2})}^{\frac{3}{2}} x^{3} + 6 {(1 - x^{2})}^{\frac{3}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{3} - 36 {(1 - x^{2})}^{\frac{1}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{7}$ en forma factorizada.

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Paso 4.56.6.15.1

Factoriza $3 x$ de $- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + 6 {(1 - x^{2})}^{\frac{3}{2}} x - 12 {(1 - x^{2})}^{\frac{3}{2}} x^{3} + 6 {(1 - x^{2})}^{\frac{3}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{3} - 36 {(1 - x^{2})}^{\frac{1}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{7}$ .

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Paso 4.56.6.15.1.1

Factoriza $3 x$ de $- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3}$ .

$- \frac{3 x (- 2 {(- x^{2} + 1)}^{\frac{5}{2}} x^{2}) + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + 6 {(1 - x^{2})}^{\frac{3}{2}} x - 12 {(1 - x^{2})}^{\frac{3}{2}} x^{3} + 6 {(1 - x^{2})}^{\frac{3}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{3} - 36 {(1 - x^{2})}^{\frac{1}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{7}}{{(1 - x^{2})}^{6}}$

Paso 4.56.6.15.1.2

Factoriza $3 x$ de $3 {(- x^{2} + 1)}^{\frac{5}{2}} x$ .

$- \frac{3 x (- 2 {(- x^{2} + 1)}^{\frac{5}{2}} x^{2}) + 3 x ({(- x^{2} + 1)}^{\frac{5}{2}}) + 6 {(1 - x^{2})}^{\frac{3}{2}} x - 12 {(1 - x^{2})}^{\frac{3}{2}} x^{3} + 6 {(1 - x^{2})}^{\frac{3}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{3} - 36 {(1 - x^{2})}^{\frac{1}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{7}}{{(1 - x^{2})}^{6}}$

Paso 4.56.6.15.1.3

Factoriza $3 x$ de $6 {(1 - x^{2})}^{\frac{3}{2}} x$ .

$- \frac{3 x (- 2 {(- x^{2} + 1)}^{\frac{5}{2}} x^{2}) + 3 x ({(- x^{2} + 1)}^{\frac{5}{2}}) + 3 x (2 {(1 - x^{2})}^{\frac{3}{2}}) - 12 {(1 - x^{2})}^{\frac{3}{2}} x^{3} + 6 {(1 - x^{2})}^{\frac{3}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{3} - 36 {(1 - x^{2})}^{\frac{1}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{7}}{{(1 - x^{2})}^{6}}$

Paso 4.56.6.15.1.4

Factoriza $3 x$ de $- 12 {(1 - x^{2})}^{\frac{3}{2}} x^{3}$ .

$- \frac{3 x (- 2 {(- x^{2} + 1)}^{\frac{5}{2}} x^{2}) + 3 x ({(- x^{2} + 1)}^{\frac{5}{2}}) + 3 x (2 {(1 - x^{2})}^{\frac{3}{2}}) + 3 x (- 4 {(1 - x^{2})}^{\frac{3}{2}} x^{2}) + 6 {(1 - x^{2})}^{\frac{3}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{3} - 36 {(1 - x^{2})}^{\frac{1}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{7}}{{(1 - x^{2})}^{6}}$

Paso 4.56.6.15.1.5

Factoriza $3 x$ de $6 {(1 - x^{2})}^{\frac{3}{2}} x^{5}$ .

$- \frac{3 x (- 2 {(- x^{2} + 1)}^{\frac{5}{2}} x^{2}) + 3 x ({(- x^{2} + 1)}^{\frac{5}{2}}) + 3 x (2 {(1 - x^{2})}^{\frac{3}{2}}) + 3 x (- 4 {(1 - x^{2})}^{\frac{3}{2}} x^{2}) + 3 x (2 {(1 - x^{2})}^{\frac{3}{2}} x^{4}) + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{3} - 36 {(1 - x^{2})}^{\frac{1}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{7}}{{(1 - x^{2})}^{6}}$

Paso 4.56.6.15.1.6

Factoriza $3 x$ de $18 {(1 - x^{2})}^{\frac{1}{2}} x^{3}$ .

$- \frac{3 x (- 2 {(- x^{2} + 1)}^{\frac{5}{2}} x^{2}) + 3 x ({(- x^{2} + 1)}^{\frac{5}{2}}) + 3 x (2 {(1 - x^{2})}^{\frac{3}{2}}) + 3 x (- 4 {(1 - x^{2})}^{\frac{3}{2}} x^{2}) + 3 x (2 {(1 - x^{2})}^{\frac{3}{2}} x^{4}) + 3 x (6 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) - 36 {(1 - x^{2})}^{\frac{1}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{7}}{{(1 - x^{2})}^{6}}$

Paso 4.56.6.15.1.7

Factoriza $3 x$ de $- 36 {(1 - x^{2})}^{\frac{1}{2}} x^{5}$ .

$- \frac{3 x (- 2 {(- x^{2} + 1)}^{\frac{5}{2}} x^{2}) + 3 x ({(- x^{2} + 1)}^{\frac{5}{2}}) + 3 x (2 {(1 - x^{2})}^{\frac{3}{2}}) + 3 x (- 4 {(1 - x^{2})}^{\frac{3}{2}} x^{2}) + 3 x (2 {(1 - x^{2})}^{\frac{3}{2}} x^{4}) + 3 x (6 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) + 3 x (- 12 {(1 - x^{2})}^{\frac{1}{2}} x^{4}) + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{7}}{{(1 - x^{2})}^{6}}$

Paso 4.56.6.15.1.8

Factoriza $3 x$ de $18 {(1 - x^{2})}^{\frac{1}{2}} x^{7}$ .

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

Paso 4.56.6.15.1.9

Factoriza $3 x$ de $3 x (- 2 {(- x^{2} + 1)}^{\frac{5}{2}} x^{2}) + 3 x ({(- x^{2} + 1)}^{\frac{5}{2}})$ .

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

Paso 4.56.6.15.1.10

Factoriza $3 x$ de $3 x (- 2 {(- x^{2} + 1)}^{\frac{5}{2}} x^{2} + {(- x^{2} + 1)}^{\frac{5}{2}}) + 3 x (2 {(1 - x^{2})}^{\frac{3}{2}})$ .

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

Paso 4.56.6.15.1.11

Factoriza $3 x$ de $3 x (- 2 {(- x^{2} + 1)}^{\frac{5}{2}} x^{2} + {(- x^{2} + 1)}^{\frac{5}{2}} + 2 {(1 - x^{2})}^{\frac{3}{2}}) + 3 x (- 4 {(1 - x^{2})}^{\frac{3}{2}} x^{2})$ .

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

Paso 4.56.6.15.1.12

Factoriza $3 x$ de $3 x (- 2 {(- x^{2} + 1)}^{\frac{5}{2}} x^{2} + {(- x^{2} + 1)}^{\frac{5}{2}} + 2 {(1 - x^{2})}^{\frac{3}{2}} - 4 {(1 - x^{2})}^{\frac{3}{2}} x^{2}) + 3 x (2 {(1 - x^{2})}^{\frac{3}{2}} x^{4})$ .

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

Paso 4.56.6.15.1.13

Factoriza $3 x$ de $3 x (- 2 {(- x^{2} + 1)}^{\frac{5}{2}} x^{2} + {(- x^{2} + 1)}^{\frac{5}{2}} + 2 {(1 - x^{2})}^{\frac{3}{2}} - 4 {(1 - x^{2})}^{\frac{3}{2}} x^{2} + 2 {(1 - x^{2})}^{\frac{3}{2}} x^{4}) + 3 x (6 {(1 - x^{2})}^{\frac{1}{2}} x^{2})$ .

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

Paso 4.56.6.15.1.14

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

Paso 4.56.6.15.1.15

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

Paso 4.56.6.15.2

Reordena los términos.

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