Cálculo Ejemplos

$f (x) = \cos (\sqrt{\frac{1 - \cos (2 x)}{1 + \cos (2 x)}})$

Paso 1

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

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

Paso 2

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

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Paso 2.1

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

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

Paso 2.2

La derivada de $\cos (u_{1})$ con respecto a $u_{1}$ es $- \sin (u_{1})$ .

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

Paso 2.3

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

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

Paso 3

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 $g (x) = \frac{1 - \cos (2 x)}{1 + \cos (2 x)}$ .

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Paso 3.1

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

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

Paso 3.2

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

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

Paso 3.3

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

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

Paso 4

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

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

Paso 5

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

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

Paso 6

Combina los numeradores sobre el denominador común.

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

Paso 7

Simplifica el numerador.

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Paso 7.1

Multiplica $- 1$ por $2$ .

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

Paso 7.2

Resta $2$ de $1$ .

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

Paso 8

Combina fracciones.

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Paso 8.1

Mueve el negativo al frente de la fracción.

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

Paso 8.2

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

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

Paso 9

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 $f (x) = 1 - \cos (2 x)$ y $g (x) = 1 + \cos (2 x)$ .

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

Paso 10

Diferencia.

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Paso 10.1

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

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

Paso 10.2

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

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

Paso 10.3

Suma $0$ y $\frac{d}{d x} [- \cos (2 x)]$ .

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

Paso 10.4

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

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

Paso 11

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

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Paso 11.1

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

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

Paso 11.2

La derivada de $\cos (u_{3})$ con respecto a $u_{3}$ es $- \sin (u_{3})$ .

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

Paso 11.3

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

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

Paso 12

Diferencia.

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Paso 12.1

Multiplica $- 1$ por $- 1$ .

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

Paso 12.2

Multiplica $\sin (2 x)$ por $1$ .

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

Paso 12.3

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

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

Paso 12.4

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

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

Paso 12.5

Simplifica la expresión.

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Paso 12.5.1

Multiplica $2$ por $1$ .

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

Paso 12.5.2

Mueve $2$ a la izquierda de $(1 + \cos (2 x)) \sin (2 x)$ .

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

Paso 12.6

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

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

Paso 12.7

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

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

Paso 12.8

Suma $0$ y $\frac{d}{d x} [\cos (2 x)]$ .

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

Paso 13

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

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Paso 13.1

Para aplicar la regla de la cadena, establece $u_{4}$ como $2 x$ .

$- \frac{\sin ({(\frac{1 - \cos (2 x)}{1 + \cos (2 x)})}^{\frac{1}{2}})}{2} {(\frac{1 - \cos (2 x)}{1 + \cos (2 x)})}^{- \frac{1}{2}} \frac{2 (1 + \cos (2 x)) \sin (2 x) - (1 - \cos (2 x)) (\frac{d}{d u_{4}} [\cos (u_{4})] \frac{d}{d x} [2 x])}{{(1 + \cos (2 x))}^{2}}$

Paso 13.2

La derivada de $\cos (u_{4})$ con respecto a $u_{4}$ es $- \sin (u_{4})$ .

$- \frac{\sin ({(\frac{1 - \cos (2 x)}{1 + \cos (2 x)})}^{\frac{1}{2}})}{2} {(\frac{1 - \cos (2 x)}{1 + \cos (2 x)})}^{- \frac{1}{2}} \frac{2 (1 + \cos (2 x)) \sin (2 x) - (1 - \cos (2 x)) (- \sin (u_{4}) \frac{d}{d x} [2 x])}{{(1 + \cos (2 x))}^{2}}$

Paso 13.3

Reemplaza todos los casos de $u_{4}$ con $2 x$ .

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

Paso 14

Diferencia.

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Paso 14.1

Multiplica $- 1$ por $- 1$ .

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

Paso 14.2

Multiplica $1 - \cos (2 x)$ por $1$ .

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

Paso 14.3

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

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

Paso 14.4

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

Paso 14.5

Combina fracciones.

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Paso 14.5.1

Multiplica $2$ por $1$ .

Paso 14.5.2

Mueve $2$ a la izquierda de $(1 - \cos (2 x)) \sin (2 x)$ .

$- \frac{\sin ({(\frac{1 - \cos (2 x)}{1 + \cos (2 x)})}^{\frac{1}{2}})}{2} {(\frac{1 - \cos (2 x)}{1 + \cos (2 x)})}^{- \frac{1}{2}} \frac{2 (1 + \cos (2 x)) \sin (2 x) + 2 \cdot ((1 - \cos (2 x)) \sin (2 x))}{{(1 + \cos (2 x))}^{2}}$

Paso 14.5.3

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

$- \frac{(2 (1 + \cos (2 x)) \sin (2 x) + 2 (1 - \cos (2 x)) \sin (2 x)) \sin ({(\frac{1 - \cos (2 x)}{1 + \cos (2 x)})}^{\frac{1}{2}})}{{(1 + \cos (2 x))}^{2} \cdot 2} {(\frac{1 - \cos (2 x)}{1 + \cos (2 x)})}^{- \frac{1}{2}}$

Paso 14.5.4

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

$- \frac{(2 (1 + \cos (2 x)) \sin (2 x) + 2 (1 - \cos (2 x)) \sin (2 x)) \sin ({(\frac{1 - \cos (2 x)}{1 + \cos (2 x)})}^{\frac{1}{2}})}{2 \cdot {(1 + \cos (2 x))}^{2}} {(\frac{1 - \cos (2 x)}{1 + \cos (2 x)})}^{- \frac{1}{2}}$

Paso 15

Simplifica.

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Paso 15.1

Cambia el signo del exponente; para ello, reescribe la base como su recíproca.

$- \frac{(2 (1 + \cos (2 x)) \sin (2 x) + 2 (1 - \cos (2 x)) \sin (2 x)) \sin ({(\frac{1 - \cos (2 x)}{1 + \cos (2 x)})}^{\frac{1}{2}})}{2 {(1 + \cos (2 x))}^{2}} {(\frac{1 + \cos (2 x)}{1 - \cos (2 x)})}^{\frac{1}{2}}$

Paso 15.2

Aplica la regla del producto a $\frac{1 - \cos (2 x)}{1 + \cos (2 x)}$ .

$- \frac{(2 (1 + \cos (2 x)) \sin (2 x) + 2 (1 - \cos (2 x)) \sin (2 x)) \sin (\frac{{(1 - \cos (2 x))}^{\frac{1}{2}}}{{(1 + \cos (2 x))}^{\frac{1}{2}}})}{2 {(1 + \cos (2 x))}^{2}} {(\frac{1 + \cos (2 x)}{1 - \cos (2 x)})}^{\frac{1}{2}}$

Paso 15.3

Aplica la regla del producto a $\frac{1 + \cos (2 x)}{1 - \cos (2 x)}$ .

$- \frac{(2 (1 + \cos (2 x)) \sin (2 x) + 2 (1 - \cos (2 x)) \sin (2 x)) \sin (\frac{{(1 - \cos (2 x))}^{\frac{1}{2}}}{{(1 + \cos (2 x))}^{\frac{1}{2}}})}{2 {(1 + \cos (2 x))}^{2}} \cdot \frac{{(1 + \cos (2 x))}^{\frac{1}{2}}}{{(1 - \cos (2 x))}^{\frac{1}{2}}}$

Paso 15.4

Aplica la propiedad distributiva.

$- \frac{((2 \cdot 1 + 2 \cos (2 x)) \sin (2 x) + 2 (1 - \cos (2 x)) \sin (2 x)) \sin (\frac{{(1 - \cos (2 x))}^{\frac{1}{2}}}{{(1 + \cos (2 x))}^{\frac{1}{2}}})}{2 {(1 + \cos (2 x))}^{2}} \cdot \frac{{(1 + \cos (2 x))}^{\frac{1}{2}}}{{(1 - \cos (2 x))}^{\frac{1}{2}}}$

Paso 15.5

Aplica la propiedad distributiva.

$- \frac{(2 \cdot 1 \sin (2 x) + 2 \cos (2 x) \sin (2 x) + 2 (1 - \cos (2 x)) \sin (2 x)) \sin (\frac{{(1 - \cos (2 x))}^{\frac{1}{2}}}{{(1 + \cos (2 x))}^{\frac{1}{2}}})}{2 {(1 + \cos (2 x))}^{2}} \cdot \frac{{(1 + \cos (2 x))}^{\frac{1}{2}}}{{(1 - \cos (2 x))}^{\frac{1}{2}}}$

Paso 15.6

Aplica la propiedad distributiva.

$- \frac{(2 \cdot 1 \sin (2 x) + 2 \cos (2 x) \sin (2 x) + (2 \cdot 1 + 2 (- \cos (2 x))) \sin (2 x)) \sin (\frac{{(1 - \cos (2 x))}^{\frac{1}{2}}}{{(1 + \cos (2 x))}^{\frac{1}{2}}})}{2 {(1 + \cos (2 x))}^{2}} \cdot \frac{{(1 + \cos (2 x))}^{\frac{1}{2}}}{{(1 - \cos (2 x))}^{\frac{1}{2}}}$

Paso 15.7

Aplica la propiedad distributiva.

$- \frac{(2 \cdot 1 \sin (2 x) + 2 \cos (2 x) \sin (2 x) + 2 \cdot 1 \sin (2 x) + 2 (- \cos (2 x)) \sin (2 x)) \sin (\frac{{(1 - \cos (2 x))}^{\frac{1}{2}}}{{(1 + \cos (2 x))}^{\frac{1}{2}}})}{2 {(1 + \cos (2 x))}^{2}} \cdot \frac{{(1 + \cos (2 x))}^{\frac{1}{2}}}{{(1 - \cos (2 x))}^{\frac{1}{2}}}$

Paso 15.8

Combina los términos.

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Paso 15.8.1

Multiplica $2$ por $1$ .

$- \frac{(2 \sin (2 x) + 2 \cos (2 x) \sin (2 x) + 2 \cdot 1 \sin (2 x) + 2 (- \cos (2 x)) \sin (2 x)) \sin (\frac{{(1 - \cos (2 x))}^{\frac{1}{2}}}{{(1 + \cos (2 x))}^{\frac{1}{2}}})}{2 {(1 + \cos (2 x))}^{2}} \cdot \frac{{(1 + \cos (2 x))}^{\frac{1}{2}}}{{(1 - \cos (2 x))}^{\frac{1}{2}}}$

Paso 15.8.2

Multiplica $2$ por $1$ .

$- \frac{(2 \sin (2 x) + 2 \cos (2 x) \sin (2 x) + 2 \sin (2 x) + 2 (- \cos (2 x)) \sin (2 x)) \sin (\frac{{(1 - \cos (2 x))}^{\frac{1}{2}}}{{(1 + \cos (2 x))}^{\frac{1}{2}}})}{2 {(1 + \cos (2 x))}^{2}} \cdot \frac{{(1 + \cos (2 x))}^{\frac{1}{2}}}{{(1 - \cos (2 x))}^{\frac{1}{2}}}$

Paso 15.8.3

Multiplica $- 1$ por $2$ .

$- \frac{(2 \sin (2 x) + 2 \cos (2 x) \sin (2 x) + 2 \sin (2 x) - 2 \cos (2 x) \sin (2 x)) \sin (\frac{{(1 - \cos (2 x))}^{\frac{1}{2}}}{{(1 + \cos (2 x))}^{\frac{1}{2}}})}{2 {(1 + \cos (2 x))}^{2}} \cdot \frac{{(1 + \cos (2 x))}^{\frac{1}{2}}}{{(1 - \cos (2 x))}^{\frac{1}{2}}}$

Paso 15.8.4

Suma $2 \sin (2 x)$ y $2 \sin (2 x)$ .

$- \frac{(2 \cos (2 x) \sin (2 x) + 4 \sin (2 x) - 2 \cos (2 x) \sin (2 x)) \sin (\frac{{(1 - \cos (2 x))}^{\frac{1}{2}}}{{(1 + \cos (2 x))}^{\frac{1}{2}}})}{2 {(1 + \cos (2 x))}^{2}} \cdot \frac{{(1 + \cos (2 x))}^{\frac{1}{2}}}{{(1 - \cos (2 x))}^{\frac{1}{2}}}$

Paso 15.8.5

Resta $2 \cos (2 x) \sin (2 x)$ de $2 \cos (2 x) \sin (2 x)$ .

$- \frac{(0 + 4 \sin (2 x)) \sin (\frac{{(1 - \cos (2 x))}^{\frac{1}{2}}}{{(1 + \cos (2 x))}^{\frac{1}{2}}})}{2 {(1 + \cos (2 x))}^{2}} \cdot \frac{{(1 + \cos (2 x))}^{\frac{1}{2}}}{{(1 - \cos (2 x))}^{\frac{1}{2}}}$

Paso 15.8.6

Suma $0$ y $4 \sin (2 x)$ .

$- \frac{4 \sin (2 x) \sin (\frac{{(1 - \cos (2 x))}^{\frac{1}{2}}}{{(1 + \cos (2 x))}^{\frac{1}{2}}})}{2 {(1 + \cos (2 x))}^{2}} \cdot \frac{{(1 + \cos (2 x))}^{\frac{1}{2}}}{{(1 - \cos (2 x))}^{\frac{1}{2}}}$

Paso 15.8.7

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

$- \frac{2 (2 \sin (2 x) \sin (\frac{{(1 - \cos (2 x))}^{\frac{1}{2}}}{{(1 + \cos (2 x))}^{\frac{1}{2}}}))}{2 {(1 + \cos (2 x))}^{2}} \cdot \frac{{(1 + \cos (2 x))}^{\frac{1}{2}}}{{(1 - \cos (2 x))}^{\frac{1}{2}}}$

Paso 15.8.8

Cancela los factores comunes.

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Paso 15.8.8.1

Factoriza $2$ de $2 {(1 + \cos (2 x))}^{2}$ .

$- \frac{2 (2 \sin (2 x) \sin (\frac{{(1 - \cos (2 x))}^{\frac{1}{2}}}{{(1 + \cos (2 x))}^{\frac{1}{2}}}))}{2 ({(1 + \cos (2 x))}^{2})} \cdot \frac{{(1 + \cos (2 x))}^{\frac{1}{2}}}{{(1 - \cos (2 x))}^{\frac{1}{2}}}$

Paso 15.8.8.2

Cancela el factor común.

Paso 15.8.8.3

Reescribe la expresión.

$- \frac{2 \sin (2 x) \sin (\frac{{(1 - \cos (2 x))}^{\frac{1}{2}}}{{(1 + \cos (2 x))}^{\frac{1}{2}}})}{{(1 + \cos (2 x))}^{2}} \cdot \frac{{(1 + \cos (2 x))}^{\frac{1}{2}}}{{(1 - \cos (2 x))}^{\frac{1}{2}}}$

Paso 15.8.9

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

$- \frac{{(1 + \cos (2 x))}^{\frac{1}{2}} (2 \sin (2 x) \sin (\frac{{(1 - \cos (2 x))}^{\frac{1}{2}}}{{(1 + \cos (2 x))}^{\frac{1}{2}}}))}{{(1 - \cos (2 x))}^{\frac{1}{2}} {(1 + \cos (2 x))}^{2}}$

Paso 15.8.10

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

$- \frac{2 \cdot {(1 + \cos (2 x))}^{\frac{1}{2}} \sin (2 x) \sin (\frac{{(1 - \cos (2 x))}^{\frac{1}{2}}}{{(1 + \cos (2 x))}^{\frac{1}{2}}})}{{(1 - \cos (2 x))}^{\frac{1}{2}} {(1 + \cos (2 x))}^{2}}$

Paso 15.8.11

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

$- \frac{2 \sin (2 x) \sin (\frac{{(1 - \cos (2 x))}^{\frac{1}{2}}}{{(1 + \cos (2 x))}^{\frac{1}{2}}})}{{(1 - \cos (2 x))}^{\frac{1}{2}} {(1 + \cos (2 x))}^{2} {(1 + \cos (2 x))}^{- \frac{1}{2}}}$

Paso 15.8.12

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

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Paso 15.8.12.1

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

$- \frac{2 \sin (2 x) \sin (\frac{{(1 - \cos (2 x))}^{\frac{1}{2}}}{{(1 + \cos (2 x))}^{\frac{1}{2}}})}{{(1 - \cos (2 x))}^{\frac{1}{2}} ({(1 + \cos (2 x))}^{- \frac{1}{2}} {(1 + \cos (2 x))}^{2})}$

Paso 15.8.12.2

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

$- \frac{2 \sin (2 x) \sin (\frac{{(1 - \cos (2 x))}^{\frac{1}{2}}}{{(1 + \cos (2 x))}^{\frac{1}{2}}})}{{(1 - \cos (2 x))}^{\frac{1}{2}} {(1 + \cos (2 x))}^{- \frac{1}{2} + 2}}$

Paso 15.8.12.3

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

$- \frac{2 \sin (2 x) \sin (\frac{{(1 - \cos (2 x))}^{\frac{1}{2}}}{{(1 + \cos (2 x))}^{\frac{1}{2}}})}{{(1 - \cos (2 x))}^{\frac{1}{2}} {(1 + \cos (2 x))}^{- \frac{1}{2} + 2 \cdot \frac{2}{2}}}$

Paso 15.8.12.4

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

$- \frac{2 \sin (2 x) \sin (\frac{{(1 - \cos (2 x))}^{\frac{1}{2}}}{{(1 + \cos (2 x))}^{\frac{1}{2}}})}{{(1 - \cos (2 x))}^{\frac{1}{2}} {(1 + \cos (2 x))}^{- \frac{1}{2} + \frac{2 \cdot 2}{2}}}$

Paso 15.8.12.5

Combina los numeradores sobre el denominador común.

$- \frac{2 \sin (2 x) \sin (\frac{{(1 - \cos (2 x))}^{\frac{1}{2}}}{{(1 + \cos (2 x))}^{\frac{1}{2}}})}{{(1 - \cos (2 x))}^{\frac{1}{2}} {(1 + \cos (2 x))}^{\frac{- 1 + 2 \cdot 2}{2}}}$

Paso 15.8.12.6

Simplifica el numerador.

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Paso 15.8.12.6.1

Multiplica $2$ por $2$ .

$- \frac{2 \sin (2 x) \sin (\frac{{(1 - \cos (2 x))}^{\frac{1}{2}}}{{(1 + \cos (2 x))}^{\frac{1}{2}}})}{{(1 - \cos (2 x))}^{\frac{1}{2}} {(1 + \cos (2 x))}^{\frac{- 1 + 4}{2}}}$

Paso 15.8.12.6.2

Suma $- 1$ y $4$ .

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