Cálculo Ejemplos

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

Paso 1

Como $- 4$ es constante con respecto a $x$ , la derivada de $- 4 x^{\frac{1}{2} + 2 x^{5}}$ con respecto a $x$ es $- 4 \frac{d}{d x} [x^{\frac{1}{2} + 2 x^{5}}]$ .

$- 4 \frac{d}{d x} [x^{\frac{1}{2} + 2 x^{5}}]$

Paso 2

Usa las propiedades de los logaritmos para simplificar la diferenciación.

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Paso 2.1

Reescribe $x^{\frac{1}{2} + 2 x^{5}}$ como $e^{\ln (x^{\frac{1}{2} + 2 x^{5}})}$ .

$- 4 \frac{d}{d x} [e^{\ln (x^{\frac{1}{2} + 2 x^{5}})}]$

Paso 2.2

Expande $\ln (x^{\frac{1}{2} + 2 x^{5}})$ ; para ello, mueve $\frac{1}{2} + 2 x^{5}$ fuera del logaritmo.

$- 4 \frac{d}{d x} [e^{(\frac{1}{2} + 2 x^{5}) \ln (x)}]$

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 $f (x) = e^{x}$ y $g (x) = (\frac{1}{2} + 2 x^{5}) \ln (x)$ .

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Paso 3.1

Para aplicar la regla de la cadena, establece $u$ como $(\frac{1}{2} + 2 x^{5}) \ln (x)$ .

$- 4 (\frac{d}{d u} [e^{u}] \frac{d}{d x} [(\frac{1}{2} + 2 x^{5}) \ln (x)])$

Paso 3.2

Diferencia con la regla exponencial, que establece que $\frac{d}{d u} [a^{u}]$ es $a^{u} \ln (a)$ donde $a$ = $e$ .

$- 4 (e^{u} \frac{d}{d x} [(\frac{1}{2} + 2 x^{5}) \ln (x)])$

Paso 3.3

Reemplaza todos los casos de $u$ con $(\frac{1}{2} + 2 x^{5}) \ln (x)$ .

$- 4 (e^{(\frac{1}{2} + 2 x^{5}) \ln (x)} \frac{d}{d x} [(\frac{1}{2} + 2 x^{5}) \ln (x)])$

Paso 4

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 $f (x) = \frac{1}{2} + 2 x^{5}$ y $g (x) = \ln (x)$ .

$- 4 e^{(\frac{1}{2} + 2 x^{5}) \ln (x)} ((\frac{1}{2} + 2 x^{5}) \frac{d}{d x} [\ln (x)] + \ln (x) \frac{d}{d x} [\frac{1}{2} + 2 x^{5}])$

Paso 5

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

$- 4 e^{(\frac{1}{2} + 2 x^{5}) \ln (x)} ((\frac{1}{2} + 2 x^{5}) \frac{1}{x} + \ln (x) \frac{d}{d x} [\frac{1}{2} + 2 x^{5}])$

Paso 6

Diferencia.

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Paso 6.1

Según la regla de la suma, la derivada de $\frac{1}{2} + 2 x^{5}$ con respecto a $x$ es $\frac{d}{d x} [\frac{1}{2}] + \frac{d}{d x} [2 x^{5}]$ .

$- 4 e^{(\frac{1}{2} + 2 x^{5}) \ln (x)} ((\frac{1}{2} + 2 x^{5}) \frac{1}{x} + \ln (x) (\frac{d}{d x} [\frac{1}{2}] + \frac{d}{d x} [2 x^{5}]))$

Paso 6.2

Como $\frac{1}{2}$ es constante con respecto a $x$ , la derivada de $\frac{1}{2}$ con respecto a $x$ es $0$ .

$- 4 e^{(\frac{1}{2} + 2 x^{5}) \ln (x)} ((\frac{1}{2} + 2 x^{5}) \frac{1}{x} + \ln (x) (0 + \frac{d}{d x} [2 x^{5}]))$

Paso 6.3

Suma $0$ y $\frac{d}{d x} [2 x^{5}]$ .

$- 4 e^{(\frac{1}{2} + 2 x^{5}) \ln (x)} ((\frac{1}{2} + 2 x^{5}) \frac{1}{x} + \ln (x) \frac{d}{d x} [2 x^{5}])$

Paso 6.4

Como $2$ es constante con respecto a $x$ , la derivada de $2 x^{5}$ con respecto a $x$ es $2 \frac{d}{d x} [x^{5}]$ .

$- 4 e^{(\frac{1}{2} + 2 x^{5}) \ln (x)} ((\frac{1}{2} + 2 x^{5}) \frac{1}{x} + \ln (x) (2 \frac{d}{d x} [x^{5}]))$

Paso 6.5

Diferencia con la regla de la potencia, que establece que $\frac{d}{d x} [x^{n}]$ es $n x^{n - 1}$ donde $n = 5$ .

$- 4 e^{(\frac{1}{2} + 2 x^{5}) \ln (x)} ((\frac{1}{2} + 2 x^{5}) \frac{1}{x} + \ln (x) (2 (5 x^{4})))$

Paso 6.6

Multiplica $5$ por $2$ .

$- 4 e^{(\frac{1}{2} + 2 x^{5}) \ln (x)} ((\frac{1}{2} + 2 x^{5}) \frac{1}{x} + \ln (x) (10 x^{4}))$

Paso 7

Simplifica.

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Paso 7.1

Aplica la propiedad distributiva.

$- 4 e^{\frac{1}{2} \ln (x) + 2 x^{5} \ln (x)} ((\frac{1}{2} + 2 x^{5}) \frac{1}{x} + \ln (x) (10 x^{4}))$

Paso 7.2

Aplica la propiedad distributiva.

$- 4 e^{\frac{1}{2} \ln (x) + 2 x^{5} \ln (x)} (\frac{1}{2} \cdot \frac{1}{x} + 2 x^{5} \frac{1}{x} + \ln (x) (10 x^{4}))$

Paso 7.3

Combina los términos.

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Paso 7.3.1

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

$- 4 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\frac{1}{2} \cdot \frac{1}{x} + 2 x^{5} \frac{1}{x} + \ln (x) (10 x^{4}))$

Paso 7.3.2

Multiplica $\frac{1}{2}$ por $\frac{1}{x}$ .

$- 4 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\frac{1}{2 x} + 2 x^{5} \frac{1}{x} + \ln (x) (10 x^{4}))$

Paso 7.3.3

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

$- 4 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\frac{1}{2 x} + x^{5} \frac{2}{x} + \ln (x) (10 x^{4}))$

Paso 7.3.4

Combina $x^{5}$ y $\frac{2}{x}$ .

$- 4 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\frac{1}{2 x} + \frac{x^{5} \cdot 2}{x} + \ln (x) (10 x^{4}))$

Paso 7.3.5

Mueve $2$ a la izquierda de $x^{5}$ .

$- 4 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\frac{1}{2 x} + \frac{2 \cdot x^{5}}{x} + \ln (x) (10 x^{4}))$

Paso 7.3.6

Cancela el factor común de $x^{5}$ y $x$ .

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Paso 7.3.6.1

Factoriza $x$ de $2 x^{5}$ .

$- 4 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\frac{1}{2 x} + \frac{x (2 x^{4})}{x} + \ln (x) (10 x^{4}))$

Paso 7.3.6.2

Cancela los factores comunes.

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Paso 7.3.6.2.1

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

$- 4 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\frac{1}{2 x} + \frac{x (2 x^{4})}{x^{1}} + \ln (x) (10 x^{4}))$

Paso 7.3.6.2.2

Factoriza $x$ de $x^{1}$ .

$- 4 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\frac{1}{2 x} + \frac{x (2 x^{4})}{x \cdot 1} + \ln (x) (10 x^{4}))$

Paso 7.3.6.2.3

Cancela el factor común.

$- 4 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\frac{1}{2 x} + \frac{x (2 x^{4})}{x \cdot 1} + \ln (x) (10 x^{4}))$

Paso 7.3.6.2.4

Reescribe la expresión.

$- 4 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\frac{1}{2 x} + \frac{2 x^{4}}{1} + \ln (x) (10 x^{4}))$

Paso 7.3.6.2.5

Divide $2 x^{4}$ por $1$ .

$- 4 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\frac{1}{2 x} + 2 x^{4} + \ln (x) (10 x^{4}))$

Paso 7.4

Reescribe con la propiedad conmutativa de la multiplicación.

$- 4 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\frac{1}{2 x} + 2 x^{4} + 10 \ln (x) x^{4})$

Paso 7.5

Aplica la propiedad distributiva.

$- 4 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} \frac{1}{2 x} - 4 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (2 x^{4}) - 4 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (10 \ln (x) x^{4})$

Paso 7.6

Simplifica.

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Paso 7.6.1

Cancela el factor común de $2$ .

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Paso 7.6.1.1

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

$2 (- 2 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)}) \frac{1}{2 x} - 4 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (2 x^{4}) - 4 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (10 \ln (x) x^{4})$

Paso 7.6.1.2

Factoriza $2$ de $2 x$ .

$2 (- 2 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)}) \frac{1}{2 (x)} - 4 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (2 x^{4}) - 4 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (10 \ln (x) x^{4})$

Paso 7.6.1.3

Cancela el factor común.

$2 (- 2 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)}) \frac{1}{2 x} - 4 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (2 x^{4}) - 4 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (10 \ln (x) x^{4})$

Paso 7.6.1.4

Reescribe la expresión.

$- 2 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} \frac{1}{x} - 4 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (2 x^{4}) - 4 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (10 \ln (x) x^{4})$

Paso 7.6.2

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

$\frac{- 2}{x} e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} - 4 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (2 x^{4}) - 4 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (10 \ln (x) x^{4})$

Paso 7.6.3

Combina $\frac{- 2}{x}$ y $e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)}$ .

$\frac{- 2 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)}}{x} - 4 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (2 x^{4}) - 4 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (10 \ln (x) x^{4})$

Paso 7.6.4

Reescribe con la propiedad conmutativa de la multiplicación.

$\frac{- 2 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)}}{x} - 4 \cdot 2 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} x^{4} - 4 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (10 \ln (x) x^{4})$

Paso 7.6.5

Multiplica $10$ por $- 4$ .

$\frac{- 2 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)}}{x} - 4 \cdot 2 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} x^{4} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

Paso 7.7

Simplifica cada término.

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Paso 7.7.1

Simplifica el numerador.

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Paso 7.7.1.1

Para escribir $2 x^{5} \ln (x)$ como una fracción con un denominador común, multiplica por $\frac{2}{2}$ .

$\frac{- 2 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x) \cdot \frac{2}{2}}}{x} - 4 \cdot 2 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} x^{4} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

Paso 7.7.1.2

Combina $2 x^{5} \ln (x)$ y $\frac{2}{2}$ .

$\frac{- 2 e^{\frac{\ln (x)}{2} + \frac{2 x^{5} \ln (x) \cdot 2}{2}}}{x} - 4 \cdot 2 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} x^{4} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

Paso 7.7.1.3

Combina los numeradores sobre el denominador común.

$\frac{- 2 e^{\frac{\ln (x) + 2 x^{5} \ln (x) \cdot 2}{2}}}{x} - 4 \cdot 2 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} x^{4} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

Paso 7.7.1.4

Simplifica el numerador.

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Paso 7.7.1.4.1

Factoriza $\ln (x)$ de $\ln (x) + 2 x^{5} \ln (x) \cdot 2$ .

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Paso 7.7.1.4.1.1

Multiplica por $1$ .

$\frac{- 2 e^{\frac{\ln (x) \cdot 1 + 2 x^{5} \ln (x) \cdot 2}{2}}}{x} - 4 \cdot 2 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} x^{4} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

Paso 7.7.1.4.1.2

Factoriza $\ln (x)$ de $2 x^{5} \ln (x) \cdot 2$ .

$\frac{- 2 e^{\frac{\ln (x) \cdot 1 + \ln (x) (2 x^{5} \cdot 2)}{2}}}{x} - 4 \cdot 2 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} x^{4} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

Paso 7.7.1.4.1.3

Factoriza $\ln (x)$ de $\ln (x) \cdot 1 + \ln (x) (2 x^{5} \cdot 2)$ .

$\frac{- 2 e^{\frac{\ln (x) (1 + 2 x^{5} \cdot 2)}{2}}}{x} - 4 \cdot 2 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} x^{4} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

Paso 7.7.1.4.2

Multiplica $2$ por $2$ .

$\frac{- 2 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}}}{x} - 4 \cdot 2 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} x^{4} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

Paso 7.7.2

Mueve el negativo al frente de la fracción.

$- \frac{2 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}}}{x} - 4 \cdot 2 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} x^{4} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

Paso 7.7.3

Multiplica $- 4$ por $2$ .

$- \frac{2 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}}}{x} - 8 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} x^{4} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

Paso 7.8

Para escribir $- 8 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} x^{4}$ como una fracción con un denominador común, multiplica por $\frac{x}{x}$ .

$- \frac{2 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}}}{x} - 8 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} x^{4} \cdot \frac{x}{x} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

Paso 7.9

Combina $- 8 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} x^{4}$ y $\frac{x}{x}$ .

$- \frac{2 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}}}{x} + \frac{- 8 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} x^{4} x}{x} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

Paso 7.10

Combina los numeradores sobre el denominador común.

$\frac{- 2 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} - 8 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} x^{4} x}{x} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

Paso 7.11

Simplifica el numerador.

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Paso 7.11.1

Factoriza $2$ de $- 2 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} - 8 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} x^{4} x$ .

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Paso 7.11.1.1

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

$\frac{2 (- e^{\frac{\ln (x) (1 + 4 x^{5})}{2}}) - 8 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} x^{4} x}{x} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

Paso 7.11.1.2

Factoriza $2$ de $- 8 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} x^{4} x$ .

$\frac{2 (- e^{\frac{\ln (x) (1 + 4 x^{5})}{2}}) + 2 (- 4 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} x^{4} x)}{x} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

Paso 7.11.1.3

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

$\frac{2 (- e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} - 4 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} x^{4} x)}{x} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

Paso 7.11.2

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

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Paso 7.11.2.1

Mueve $x$ .

$\frac{2 (- e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} - 4 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (x \cdot x^{4}))}{x} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

Paso 7.11.2.2

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

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Paso 7.11.2.2.1

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

$\frac{2 (- e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} - 4 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (x^{1} x^{4}))}{x} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

Paso 7.11.2.2.2

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

$\frac{2 (- e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} - 4 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} x^{1 + 4})}{x} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

Paso 7.11.2.3

Suma $1$ y $4$ .

$\frac{2 (- e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} - 4 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} x^{5})}{x} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

Paso 7.11.3

Simplifica cada término.

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Paso 7.11.3.1

Para escribir $2 x^{5} \ln (x)$ como una fracción con un denominador común, multiplica por $\frac{2}{2}$ .

$\frac{2 (- e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} - 4 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x) \cdot \frac{2}{2}} x^{5})}{x} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

Paso 7.11.3.2

Combina $2 x^{5} \ln (x)$ y $\frac{2}{2}$ .

$\frac{2 (- e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} - 4 e^{\frac{\ln (x)}{2} + \frac{2 x^{5} \ln (x) \cdot 2}{2}} x^{5})}{x} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

Paso 7.11.3.3

Combina los numeradores sobre el denominador común.

$\frac{2 (- e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} - 4 e^{\frac{\ln (x) + 2 x^{5} \ln (x) \cdot 2}{2}} x^{5})}{x} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

Paso 7.11.3.4

Simplifica el numerador.

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Paso 7.11.3.4.1

Factoriza $\ln (x)$ de $\ln (x) + 2 x^{5} \ln (x) \cdot 2$ .

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Paso 7.11.3.4.1.1

Multiplica por $1$ .

$\frac{2 (- e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} - 4 e^{\frac{\ln (x) \cdot 1 + 2 x^{5} \ln (x) \cdot 2}{2}} x^{5})}{x} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

Paso 7.11.3.4.1.2

Factoriza $\ln (x)$ de $2 x^{5} \ln (x) \cdot 2$ .

$\frac{2 (- e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} - 4 e^{\frac{\ln (x) \cdot 1 + \ln (x) (2 x^{5} \cdot 2)}{2}} x^{5})}{x} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

Paso 7.11.3.4.1.3

Factoriza $\ln (x)$ de $\ln (x) \cdot 1 + \ln (x) (2 x^{5} \cdot 2)$ .

$\frac{2 (- e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} - 4 e^{\frac{\ln (x) (1 + 2 x^{5} \cdot 2)}{2}} x^{5})}{x} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

Paso 7.11.3.4.2

Multiplica $2$ por $2$ .

$\frac{2 (- e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} - 4 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} x^{5})}{x} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

Paso 7.11.4

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

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Paso 7.11.4.1

Reordena la expresión.

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Paso 7.11.4.1.1

Reordena $1$ y $4 x^{5}$ .

$\frac{2 (- e^{\frac{\ln (x) (4 x^{5} + 1)}{2}} - 4 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} x^{5})}{x} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

Paso 7.11.4.1.2

Reordena $1$ y $4 x^{5}$ .

$\frac{2 (- e^{\frac{\ln (x) (4 x^{5} + 1)}{2}} - 4 e^{\frac{\ln (x) (4 x^{5} + 1)}{2}} x^{5})}{x} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

Paso 7.11.4.1.3

Mueve $e^{\frac{\ln (x) (4 x^{5} + 1)}{2}}$ .

$\frac{2 (- e^{\frac{\ln (x) (4 x^{5} + 1)}{2}} - 4 x^{5} e^{\frac{\ln (x) (4 x^{5} + 1)}{2}})}{x} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

Paso 7.11.4.2

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

$\frac{2 (e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (- e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (1 + 4 x^{5})}{2}}) - 4 x^{5} e^{\frac{\ln (x) (4 x^{5} + 1)}{2}})}{x} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

Paso 7.11.4.3

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

$\frac{2 (e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (- e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (1 + 4 x^{5})}{2}}) + e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (- 4 x^{5} e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (1 + 4 x^{5})}{2}}))}{x} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

Paso 7.11.4.4

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

$\frac{2 (e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (- e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (1 + 4 x^{5})}{2}} - 4 x^{5} e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (1 + 4 x^{5})}{2}}))}{x} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

Paso 7.11.5

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

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Paso 7.11.5.1

Reordena la expresión.

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Paso 7.11.5.1.1

Reordena $1$ y $4 x^{5}$ .

$\frac{2 (e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (- e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (4 x^{5} + 1)}{2}} - 4 x^{5} e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (1 + 4 x^{5})}{2}}))}{x} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

Paso 7.11.5.1.2

Reordena $1$ y $4 x^{5}$ .

$\frac{2 (e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (- e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (4 x^{5} + 1)}{2}} - 4 x^{5} e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (4 x^{5} + 1)}{2}}))}{x} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

Paso 7.11.5.2

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

$\frac{2 (e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (4 x^{5} + 1)}{2}} \cdot - 1 - 4 x^{5} e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (4 x^{5} + 1)}{2}}))}{x} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

Paso 7.11.5.3

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

$\frac{2 (e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (4 x^{5} + 1)}{2}} \cdot - 1 + e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (4 x^{5} + 1)}{2}} (- 4 x^{5})))}{x} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

Paso 7.11.5.4

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

$\frac{2 (e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (4 x^{5} + 1)}{2}} (- 1 - 4 x^{5})))}{x} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

$\frac{2 (e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (4 x^{5} + 1)}{2}} (- 1 - 4 x^{5}))}{x} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

Paso 7.11.6

Multiplica $e^{\frac{\ln (x) (1 + 4 x^{5})}{2}}$ por $e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (4 x^{5} + 1)}{2}}$ sumando los exponentes.

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Paso 7.11.6.1

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

$\frac{2 (e^{\frac{\ln (x) (1 + 4 x^{5})}{2} + \frac{\ln (x) (4 x^{5} + 1) - \ln (x) (4 x^{5} + 1)}{2}} (- 1 - 4 x^{5}))}{x} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

Paso 7.11.6.2

Resta $\ln (x) (4 x^{5} + 1)$ de $\ln (x) (4 x^{5} + 1)$ .

$\frac{2 (e^{\frac{\ln (x) (1 + 4 x^{5})}{2} + \frac{0}{2}} (- 1 - 4 x^{5}))}{x} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

Paso 7.11.6.3

Combina los términos opuestos en $\frac{\ln (x) (1 + 4 x^{5})}{2} + \frac{0}{2}$ .

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Paso 7.11.6.3.1

Divide $0$ por $2$ .

$\frac{2 (e^{\frac{\ln (x) (1 + 4 x^{5})}{2} + 0} (- 1 - 4 x^{5}))}{x} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

Paso 7.11.6.3.2

Suma $\frac{\ln (x) (1 + 4 x^{5})}{2}$ y $0$ .

$\frac{2 (e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (- 1 - 4 x^{5}))}{x} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

$\frac{2 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (- 1 - 4 x^{5})}{x} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$

Paso 7.12

Para escribir $- 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$ como una fracción con un denominador común, multiplica por $\frac{x}{x}$ .

$\frac{2 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (- 1 - 4 x^{5})}{x} - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4}) \cdot \frac{x}{x}$

Paso 7.13

Combina $- 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4})$ y $\frac{x}{x}$ .

$\frac{2 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (- 1 - 4 x^{5})}{x} + \frac{- 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4}) x}{x}$

Paso 7.14

Combina los numeradores sobre el denominador común.

$\frac{2 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (- 1 - 4 x^{5}) - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4}) x}{x}$

Paso 7.15

Simplifica el numerador.

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Paso 7.15.1

Factoriza $2$ de $2 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (- 1 - 4 x^{5}) - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4}) x$ .

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Paso 7.15.1.1

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

$\frac{2 (e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (- 1 - 4 x^{5})) - 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4}) x}{x}$

Paso 7.15.1.2

Factoriza $2$ de $- 40 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4}) x$ .

$\frac{2 (e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (- 1 - 4 x^{5})) + 2 (- 20 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4}) x)}{x}$

Paso 7.15.1.3

Factoriza $2$ de $2 (e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (- 1 - 4 x^{5})) + 2 (- 20 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4}) x)$ .

$\frac{2 (e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (- 1 - 4 x^{5}) - 20 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4}) x)}{x}$

Paso 7.15.2

Aplica la propiedad distributiva.

$\frac{2 (e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} \cdot - 1 + e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (- 4 x^{5}) - 20 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4}) x)}{x}$

Paso 7.15.3

Mueve $- 1$ a la izquierda de $e^{\frac{\ln (x) (1 + 4 x^{5})}{2}}$ .

$\frac{2 (- 1 \cdot e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} + e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (- 4 x^{5}) - 20 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4}) x)}{x}$

Paso 7.15.4

Reescribe con la propiedad conmutativa de la multiplicación.

$\frac{2 (- 1 \cdot e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} - 4 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} x^{5} - 20 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4}) x)}{x}$

Paso 7.15.5

Reescribe $- 1 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}}$ como $- e^{\frac{\ln (x) (1 + 4 x^{5})}{2}}$ .

$\frac{2 (- e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} - 4 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} x^{5} - 20 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{4}) x)}{x}$

Paso 7.15.6

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

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Paso 7.15.6.1

Mueve $x$ .

$\frac{2 (- e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} - 4 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} x^{5} - 20 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) (x \cdot x^{4})))}{x}$

Paso 7.15.6.2

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

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Paso 7.15.6.2.1

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

$\frac{2 (- e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} - 4 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} x^{5} - 20 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) (x^{1} x^{4})))}{x}$

Paso 7.15.6.2.2

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

$\frac{2 (- e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} - 4 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} x^{5} - 20 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{1 + 4}))}{x}$

Paso 7.15.6.3

Suma $1$ y $4$ .

$\frac{2 (- e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} - 4 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} x^{5} - 20 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x)} (\ln (x) x^{5}))}{x}$

Paso 7.15.7

Simplifica cada término.

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Paso 7.15.7.1

Para escribir $2 x^{5} \ln (x)$ como una fracción con un denominador común, multiplica por $\frac{2}{2}$ .

$\frac{2 (- e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} - 4 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} x^{5} - 20 e^{\frac{\ln (x)}{2} + 2 x^{5} \ln (x) \cdot \frac{2}{2}} (\ln (x) x^{5}))}{x}$

Paso 7.15.7.2

Combina $2 x^{5} \ln (x)$ y $\frac{2}{2}$ .

$\frac{2 (- e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} - 4 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} x^{5} - 20 e^{\frac{\ln (x)}{2} + \frac{2 x^{5} \ln (x) \cdot 2}{2}} (\ln (x) x^{5}))}{x}$

Paso 7.15.7.3

Combina los numeradores sobre el denominador común.

$\frac{2 (- e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} - 4 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} x^{5} - 20 e^{\frac{\ln (x) + 2 x^{5} \ln (x) \cdot 2}{2}} (\ln (x) x^{5}))}{x}$

Paso 7.15.7.4

Simplifica el numerador.

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Paso 7.15.7.4.1

Factoriza $\ln (x)$ de $\ln (x) + 2 x^{5} \ln (x) \cdot 2$ .

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Paso 7.15.7.4.1.1

Multiplica por $1$ .

$\frac{2 (- e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} - 4 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} x^{5} - 20 e^{\frac{\ln (x) \cdot 1 + 2 x^{5} \ln (x) \cdot 2}{2}} (\ln (x) x^{5}))}{x}$

Paso 7.15.7.4.1.2

Factoriza $\ln (x)$ de $2 x^{5} \ln (x) \cdot 2$ .

$\frac{2 (- e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} - 4 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} x^{5} - 20 e^{\frac{\ln (x) \cdot 1 + \ln (x) (2 x^{5} \cdot 2)}{2}} (\ln (x) x^{5}))}{x}$

Paso 7.15.7.4.1.3

Factoriza $\ln (x)$ de $\ln (x) \cdot 1 + \ln (x) (2 x^{5} \cdot 2)$ .

$\frac{2 (- e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} - 4 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} x^{5} - 20 e^{\frac{\ln (x) (1 + 2 x^{5} \cdot 2)}{2}} (\ln (x) x^{5}))}{x}$

Paso 7.15.7.4.2

Multiplica $2$ por $2$ .

$\frac{2 (- e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} - 4 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} x^{5} - 20 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (\ln (x) x^{5}))}{x}$

Paso 7.15.8

Factoriza $e^{\frac{\ln (x) (1 + 4 x^{5})}{2}}$ de $- e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} - 4 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} x^{5} - 20 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (\ln (x) x^{5})$ .

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Paso 7.15.8.1

Reordena la expresión.

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Paso 7.15.8.1.1

Reordena $1$ y $4 x^{5}$ .

$\frac{2 (- e^{\frac{\ln (x) (4 x^{5} + 1)}{2}} - 4 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} x^{5} - 20 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (\ln (x) x^{5}))}{x}$

Paso 7.15.8.1.2

Reordena $1$ y $4 x^{5}$ .

$\frac{2 (- e^{\frac{\ln (x) (4 x^{5} + 1)}{2}} - 4 e^{\frac{\ln (x) (4 x^{5} + 1)}{2}} x^{5} - 20 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (\ln (x) x^{5}))}{x}$

Paso 7.15.8.1.3

Mueve $e^{\frac{\ln (x) (4 x^{5} + 1)}{2}}$ .

$\frac{2 (- e^{\frac{\ln (x) (4 x^{5} + 1)}{2}} - 4 x^{5} e^{\frac{\ln (x) (4 x^{5} + 1)}{2}} - 20 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (\ln (x) x^{5}))}{x}$

Paso 7.15.8.1.4

Reordena $1$ y $4 x^{5}$ .

$\frac{2 (- e^{\frac{\ln (x) (4 x^{5} + 1)}{2}} - 4 x^{5} e^{\frac{\ln (x) (4 x^{5} + 1)}{2}} - 20 e^{\frac{\ln (x) (4 x^{5} + 1)}{2}} (\ln (x) x^{5}))}{x}$

Paso 7.15.8.1.5

Mueve $\ln (x)$ .

$\frac{2 (- e^{\frac{\ln (x) (4 x^{5} + 1)}{2}} - 4 x^{5} e^{\frac{\ln (x) (4 x^{5} + 1)}{2}} - 20 e^{\frac{\ln (x) (4 x^{5} + 1)}{2}} x^{5} \ln (x))}{x}$

Paso 7.15.8.1.6

Mueve $e^{\frac{\ln (x) (4 x^{5} + 1)}{2}}$ .

$\frac{2 (- e^{\frac{\ln (x) (4 x^{5} + 1)}{2}} - 4 x^{5} e^{\frac{\ln (x) (4 x^{5} + 1)}{2}} - 20 x^{5} e^{\frac{\ln (x) (4 x^{5} + 1)}{2}} \ln (x))}{x}$

Paso 7.15.8.2

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

$\frac{2 (e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (- e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (1 + 4 x^{5})}{2}}) - 4 x^{5} e^{\frac{\ln (x) (4 x^{5} + 1)}{2}} - 20 x^{5} e^{\frac{\ln (x) (4 x^{5} + 1)}{2}} \ln (x))}{x}$

Paso 7.15.8.3

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

$\frac{2 (e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (- e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (1 + 4 x^{5})}{2}}) + e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (- 4 x^{5} e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (1 + 4 x^{5})}{2}}) - 20 x^{5} e^{\frac{\ln (x) (4 x^{5} + 1)}{2}} \ln (x))}{x}$

Paso 7.15.8.4

Factoriza $e^{\frac{\ln (x) (1 + 4 x^{5})}{2}}$ de $- 20 x^{5} e^{\frac{\ln (x) (4 x^{5} + 1)}{2}} \ln (x)$ .

$\frac{2 (e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (- e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (1 + 4 x^{5})}{2}}) + e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (- 4 x^{5} e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (1 + 4 x^{5})}{2}}) + e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (- 20 x^{5} e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (1 + 4 x^{5})}{2}} \ln (x)))}{x}$

Paso 7.15.8.5

$\frac{2 (e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (- e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (1 + 4 x^{5})}{2}} - 4 x^{5} e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (1 + 4 x^{5})}{2}}) + e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (- 20 x^{5} e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (1 + 4 x^{5})}{2}} \ln (x)))}{x}$

Paso 7.15.8.6

Factoriza $e^{\frac{\ln (x) (1 + 4 x^{5})}{2}}$ de $e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (- e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (1 + 4 x^{5})}{2}} - 4 x^{5} e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (1 + 4 x^{5})}{2}}) + e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (- 20 x^{5} e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (1 + 4 x^{5})}{2}} \ln (x))$ .

$\frac{2 (e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (- e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (1 + 4 x^{5})}{2}} - 4 x^{5} e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (1 + 4 x^{5})}{2}} - 20 x^{5} e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (1 + 4 x^{5})}{2}} \ln (x)))}{x}$

Paso 7.15.9

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Paso 7.15.9.1

Reordena la expresión.

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Paso 7.15.9.1.1

Reordena $1$ y $4 x^{5}$ .

$\frac{2 (e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (- e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (4 x^{5} + 1)}{2}} - 4 x^{5} e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (1 + 4 x^{5})}{2}} - 20 x^{5} e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (1 + 4 x^{5})}{2}} \ln (x)))}{x}$

Paso 7.15.9.1.2

Reordena $1$ y $4 x^{5}$ .

$\frac{2 (e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (- e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (4 x^{5} + 1)}{2}} - 4 x^{5} e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (4 x^{5} + 1)}{2}} - 20 x^{5} e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (1 + 4 x^{5})}{2}} \ln (x)))}{x}$

Paso 7.15.9.1.3

Reordena $1$ y $4 x^{5}$ .

Paso 7.15.9.2

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

$\frac{2 (e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (4 x^{5} + 1)}{2}} \cdot - 1 - 4 x^{5} e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (4 x^{5} + 1)}{2}} - 20 x^{5} e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (4 x^{5} + 1)}{2}} \ln (x)))}{x}$

Paso 7.15.9.3

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

$\frac{2 (e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (4 x^{5} + 1)}{2}} \cdot - 1 + e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (4 x^{5} + 1)}{2}} (- 4 x^{5}) - 20 x^{5} e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (4 x^{5} + 1)}{2}} \ln (x)))}{x}$

Paso 7.15.9.4

Factoriza $e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (1 + 4 x^{5})}{2}}$ de $- 20 x^{5} e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (4 x^{5} + 1)}{2}} \ln (x)$ .

$\frac{2 (e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (4 x^{5} + 1)}{2}} \cdot - 1 + e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (4 x^{5} + 1)}{2}} (- 4 x^{5}) + e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (4 x^{5} + 1)}{2}} (- 20 x^{5} \ln (x))))}{x}$

Paso 7.15.9.5

$\frac{2 (e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (4 x^{5} + 1)}{2}} \cdot (- 1 - 4 x^{5}) + e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (4 x^{5} + 1)}{2}} (- 20 x^{5} \ln (x))))}{x}$

Paso 7.15.9.6

Factoriza $e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (4 x^{5} + 1)}{2}}$ de $e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (4 x^{5} + 1)}{2}} \cdot (- 1 - 4 x^{5}) + e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (4 x^{5} + 1)}{2}} (- 20 x^{5} \ln (x))$ .

$\frac{2 (e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (4 x^{5} + 1)}{2}} (- 1 - 4 x^{5} - 20 x^{5} \ln (x))))}{x}$

$\frac{2 (e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (4 x^{5} + 1)}{2}} (- 1 - 4 x^{5} - 20 x^{5} \ln (x)))}{x}$

Paso 7.15.10

Multiplica $e^{\frac{\ln (x) (1 + 4 x^{5})}{2}}$ por $e^{\frac{\ln (x) (4 x^{5} + 1) - \ln (x) (4 x^{5} + 1)}{2}}$ sumando los exponentes.

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Paso 7.15.10.1

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

$\frac{2 (e^{\frac{\ln (x) (1 + 4 x^{5})}{2} + \frac{\ln (x) (4 x^{5} + 1) - \ln (x) (4 x^{5} + 1)}{2}} (- 1 - 4 x^{5} - 20 x^{5} \ln (x)))}{x}$

Paso 7.15.10.2

Resta $\ln (x) (4 x^{5} + 1)$ de $\ln (x) (4 x^{5} + 1)$ .

$\frac{2 (e^{\frac{\ln (x) (1 + 4 x^{5})}{2} + \frac{0}{2}} (- 1 - 4 x^{5} - 20 x^{5} \ln (x)))}{x}$

Paso 7.15.10.3

Combina los términos opuestos en $\frac{\ln (x) (1 + 4 x^{5})}{2} + \frac{0}{2}$ .

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Paso 7.15.10.3.1

Divide $0$ por $2$ .

$\frac{2 (e^{\frac{\ln (x) (1 + 4 x^{5})}{2} + 0} (- 1 - 4 x^{5} - 20 x^{5} \ln (x)))}{x}$

Paso 7.15.10.3.2

Suma $\frac{\ln (x) (1 + 4 x^{5})}{2}$ y $0$ .

$\frac{2 (e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (- 1 - 4 x^{5} - 20 x^{5} \ln (x)))}{x}$

$\frac{2 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (- 1 - 4 x^{5} - 20 x^{5} \ln (x))}{x}$

Paso 7.16

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

$\frac{2 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (- 1 (1) - 4 x^{5} - 20 x^{5} \ln (x))}{x}$

Paso 7.17

Factoriza $- 1$ de $- 4 x^{5}$ .

$\frac{2 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (- 1 (1) - (4 x^{5}) - 20 x^{5} \ln (x))}{x}$

Paso 7.18

Factoriza $- 1$ de $- 1 (1) - (4 x^{5})$ .

$\frac{2 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (- 1 (1 + 4 x^{5}) - 20 x^{5} \ln (x))}{x}$

Paso 7.19

Factoriza $- 1$ de $- 20 x^{5} \ln (x)$ .

$\frac{2 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (- 1 (1 + 4 x^{5}) - (20 x^{5} \ln (x)))}{x}$

Paso 7.20

Factoriza $- 1$ de $- 1 (1 + 4 x^{5}) - (20 x^{5} \ln (x))$ .

$\frac{2 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (- 1 (1 + 4 x^{5} + 20 x^{5} \ln (x)))}{x}$

Paso 7.21

Mueve el negativo al frente de la fracción.

$- \frac{(2 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}}) (1 + 4 x^{5} + 20 x^{5} \ln (x))}{x}$

Paso 7.22

Reordena los factores en $- \frac{(2 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}}) (1 + 4 x^{5} + 20 x^{5} \ln (x))}{x}$ .

$- \frac{2 e^{\frac{\ln (x) (1 + 4 x^{5})}{2}} (1 + 4 x^{5} + 20 x^{5} \ln (x))}{x}$