Cálculo Ejemplos

$f (x) = \ln (x^{4} (\cos (x) + \sin (x)))$

Paso 1

Diferencia con la regla de la cadena, que establece que $\frac{d}{d x} [f (g (x))]$ es $f' (g (x)) g' (x)$ donde $f (x) = \ln (x)$ y $g (x) = x^{4} (\cos (x) + \sin (x))$ .

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Paso 1.1

Para aplicar la regla de la cadena, establece $u$ como $x^{4} (\cos (x) + \sin (x))$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [x^{4} (\cos (x) + \sin (x))]$

Paso 1.2

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

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

Paso 1.3

Reemplaza todos los casos de $u$ con $x^{4} (\cos (x) + \sin (x))$ .

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

Paso 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 $f (x) = x^{4}$ y $g (x) = \cos (x) + \sin (x)$ .

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

Paso 3

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

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

Paso 4

La derivada de $\cos (x)$ con respecto a $x$ es $- \sin (x)$ .

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

Paso 5

La derivada de $\sin (x)$ con respecto a $x$ es $\cos (x)$ .

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

Paso 6

Diferencia con la regla de la potencia.

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Paso 6.1

Diferencia con la regla de la potencia, que establece que $\frac{d}{d x} [x^{n}]$ es $n x^{n - 1}$ donde $n = 4$ .

$\frac{1}{x^{4} (\cos (x) + \sin (x))} (x^{4} (- \sin (x) + \cos (x)) + (\cos (x) + \sin (x)) (4 x^{3}))$

Paso 6.2

Mueve $4$ a la izquierda de $\cos (x) + \sin (x)$ .

$\frac{1}{x^{4} (\cos (x) + \sin (x))} (x^{4} (- \sin (x) + \cos (x)) + 4 \cdot (\cos (x) + \sin (x)) x^{3})$

Paso 7

Simplifica.

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Paso 7.1

Aplica la propiedad distributiva.

$\frac{1}{x^{4} \cos (x) + x^{4} \sin (x)} (x^{4} (- \sin (x) + \cos (x)) + 4 (\cos (x) + \sin (x)) x^{3})$

Paso 7.2

Aplica la propiedad distributiva.

$\frac{1}{x^{4} \cos (x) + x^{4} \sin (x)} (x^{4} (- \sin (x)) + x^{4} \cos (x) + 4 (\cos (x) + \sin (x)) x^{3})$

Paso 7.3

Aplica la propiedad distributiva.

$\frac{1}{x^{4} \cos (x) + x^{4} \sin (x)} (x^{4} (- \sin (x)) + x^{4} \cos (x) + (4 \cos (x) + 4 \sin (x)) x^{3})$

Paso 7.4

Aplica la propiedad distributiva.

$\frac{1}{x^{4} \cos (x) + x^{4} \sin (x)} (x^{4} (- \sin (x)) + x^{4} \cos (x) + 4 \cos (x) x^{3} + 4 \sin (x) x^{3})$

Paso 7.5

Reordena los factores de $\frac{1}{x^{4} \cos (x) + x^{4} \sin (x)} (x^{4} (- \sin (x)) + x^{4} \cos (x) + 4 \cos (x) x^{3} + 4 \sin (x) x^{3})$ .

$(x^{4} (- \sin (x)) + x^{4} \cos (x) + 4 \cos (x) x^{3} + 4 \sin (x) x^{3}) \frac{1}{x^{4} \cos (x) + x^{4} \sin (x)}$

Paso 7.6

Factoriza $x^{4}$ de $x^{4} \cos (x) + x^{4} \sin (x)$ .

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Paso 7.6.1

Factoriza $x^{4}$ de $x^{4} \cos (x)$ .

$(x^{4} (- \sin (x)) + x^{4} \cos (x) + 4 \cos (x) x^{3} + 4 \sin (x) x^{3}) \frac{1}{x^{4} (\cos (x)) + x^{4} \sin (x)}$

Paso 7.6.2

Factoriza $x^{4}$ de $x^{4} \sin (x)$ .

$(x^{4} (- \sin (x)) + x^{4} \cos (x) + 4 \cos (x) x^{3} + 4 \sin (x) x^{3}) \frac{1}{x^{4} (\cos (x)) + x^{4} (\sin (x))}$

Paso 7.6.3

Factoriza $x^{4}$ de $x^{4} (\cos (x)) + x^{4} (\sin (x))$ .

$(x^{4} (- \sin (x)) + x^{4} \cos (x) + 4 \cos (x) x^{3} + 4 \sin (x) x^{3}) \frac{1}{x^{4} (\cos (x) + \sin (x))}$

Paso 7.7

Reescribe con la propiedad conmutativa de la multiplicación.

$(- x^{4} \sin (x) + x^{4} \cos (x) + 4 \cos (x) x^{3} + 4 \sin (x) x^{3}) \frac{1}{x^{4} (\cos (x) + \sin (x))}$

Paso 7.8

Aplica la propiedad distributiva.

$- x^{4} \sin (x) \frac{1}{x^{4} (\cos (x) + \sin (x))} + x^{4} \cos (x) \frac{1}{x^{4} (\cos (x) + \sin (x))} + 4 \cos (x) x^{3} \frac{1}{x^{4} (\cos (x) + \sin (x))} + 4 \sin (x) x^{3} \frac{1}{x^{4} (\cos (x) + \sin (x))}$

Paso 7.9

Simplifica.

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Paso 7.9.1

Cancela el factor común de $x^{4}$ .

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Paso 7.9.1.1

Factoriza $x^{4}$ de $- x^{4} \sin (x)$ .

$x^{4} (- \sin (x)) \frac{1}{x^{4} (\cos (x) + \sin (x))} + x^{4} \cos (x) \frac{1}{x^{4} (\cos (x) + \sin (x))} + 4 \cos (x) x^{3} \frac{1}{x^{4} (\cos (x) + \sin (x))} + 4 \sin (x) x^{3} \frac{1}{x^{4} (\cos (x) + \sin (x))}$

Paso 7.9.1.2

Cancela el factor común.

Paso 7.9.1.3

Reescribe la expresión.

$- \sin (x) \frac{1}{\cos (x) + \sin (x)} + x^{4} \cos (x) \frac{1}{x^{4} (\cos (x) + \sin (x))} + 4 \cos (x) x^{3} \frac{1}{x^{4} (\cos (x) + \sin (x))} + 4 \sin (x) x^{3} \frac{1}{x^{4} (\cos (x) + \sin (x))}$

Paso 7.9.2

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

$- \frac{\sin (x)}{\cos (x) + \sin (x)} + x^{4} \cos (x) \frac{1}{x^{4} (\cos (x) + \sin (x))} + 4 \cos (x) x^{3} \frac{1}{x^{4} (\cos (x) + \sin (x))} + 4 \sin (x) x^{3} \frac{1}{x^{4} (\cos (x) + \sin (x))}$

Paso 7.9.3

Cancela el factor común de $x^{4}$ .

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Paso 7.9.3.1

Factoriza $x^{4}$ de $x^{4} \cos (x)$ .

$- \frac{\sin (x)}{\cos (x) + \sin (x)} + x^{4} (\cos (x)) \frac{1}{x^{4} (\cos (x) + \sin (x))} + 4 \cos (x) x^{3} \frac{1}{x^{4} (\cos (x) + \sin (x))} + 4 \sin (x) x^{3} \frac{1}{x^{4} (\cos (x) + \sin (x))}$

Paso 7.9.3.2

Cancela el factor común.

Paso 7.9.3.3

Reescribe la expresión.

$- \frac{\sin (x)}{\cos (x) + \sin (x)} + \cos (x) \frac{1}{\cos (x) + \sin (x)} + 4 \cos (x) x^{3} \frac{1}{x^{4} (\cos (x) + \sin (x))} + 4 \sin (x) x^{3} \frac{1}{x^{4} (\cos (x) + \sin (x))}$

Paso 7.9.4

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

$- \frac{\sin (x)}{\cos (x) + \sin (x)} + \frac{\cos (x)}{\cos (x) + \sin (x)} + 4 \cos (x) x^{3} \frac{1}{x^{4} (\cos (x) + \sin (x))} + 4 \sin (x) x^{3} \frac{1}{x^{4} (\cos (x) + \sin (x))}$

Paso 7.9.5

Cancela el factor común de $x^{3}$ .

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Paso 7.9.5.1

Factoriza $x^{3}$ de $4 \cos (x) x^{3}$ .

$- \frac{\sin (x)}{\cos (x) + \sin (x)} + \frac{\cos (x)}{\cos (x) + \sin (x)} + x^{3} (4 \cos (x)) \frac{1}{x^{4} (\cos (x) + \sin (x))} + 4 \sin (x) x^{3} \frac{1}{x^{4} (\cos (x) + \sin (x))}$

Paso 7.9.5.2

Factoriza $x^{3}$ de $x^{4} (\cos (x) + \sin (x))$ .

$- \frac{\sin (x)}{\cos (x) + \sin (x)} + \frac{\cos (x)}{\cos (x) + \sin (x)} + x^{3} (4 \cos (x)) \frac{1}{x^{3} (x (\cos (x) + \sin (x)))} + 4 \sin (x) x^{3} \frac{1}{x^{4} (\cos (x) + \sin (x))}$

Paso 7.9.5.3

Cancela el factor común.

$- \frac{\sin (x)}{\cos (x) + \sin (x)} + \frac{\cos (x)}{\cos (x) + \sin (x)} + x^{3} (4 \cos (x)) \frac{1}{x^{3} (x (\cos (x) + \sin (x)))} + 4 \sin (x) x^{3} \frac{1}{x^{4} (\cos (x) + \sin (x))}$

Paso 7.9.5.4

Reescribe la expresión.

$- \frac{\sin (x)}{\cos (x) + \sin (x)} + \frac{\cos (x)}{\cos (x) + \sin (x)} + 4 \cos (x) \frac{1}{x (\cos (x) + \sin (x))} + 4 \sin (x) x^{3} \frac{1}{x^{4} (\cos (x) + \sin (x))}$

Paso 7.9.6

Combina $\frac{1}{x (\cos (x) + \sin (x))}$ y $4$ .

$- \frac{\sin (x)}{\cos (x) + \sin (x)} + \frac{\cos (x)}{\cos (x) + \sin (x)} + \frac{4}{x (\cos (x) + \sin (x))} \cos (x) + 4 \sin (x) x^{3} \frac{1}{x^{4} (\cos (x) + \sin (x))}$

Paso 7.9.7

Combina $\frac{4}{x (\cos (x) + \sin (x))}$ y $\cos (x)$ .

$- \frac{\sin (x)}{\cos (x) + \sin (x)} + \frac{\cos (x)}{\cos (x) + \sin (x)} + \frac{4 \cos (x)}{x (\cos (x) + \sin (x))} + 4 \sin (x) x^{3} \frac{1}{x^{4} (\cos (x) + \sin (x))}$

Paso 7.9.8

Cancela el factor común de $x^{3}$ .

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Paso 7.9.8.1

Factoriza $x^{3}$ de $4 \sin (x) x^{3}$ .

$- \frac{\sin (x)}{\cos (x) + \sin (x)} + \frac{\cos (x)}{\cos (x) + \sin (x)} + \frac{4 \cos (x)}{x (\cos (x) + \sin (x))} + x^{3} (4 \sin (x)) \frac{1}{x^{4} (\cos (x) + \sin (x))}$

Paso 7.9.8.2

Factoriza $x^{3}$ de $x^{4} (\cos (x) + \sin (x))$ .

$- \frac{\sin (x)}{\cos (x) + \sin (x)} + \frac{\cos (x)}{\cos (x) + \sin (x)} + \frac{4 \cos (x)}{x (\cos (x) + \sin (x))} + x^{3} (4 \sin (x)) \frac{1}{x^{3} (x (\cos (x) + \sin (x)))}$

Paso 7.9.8.3

Cancela el factor común.

$- \frac{\sin (x)}{\cos (x) + \sin (x)} + \frac{\cos (x)}{\cos (x) + \sin (x)} + \frac{4 \cos (x)}{x (\cos (x) + \sin (x))} + x^{3} (4 \sin (x)) \frac{1}{x^{3} (x (\cos (x) + \sin (x)))}$

Paso 7.9.8.4

Reescribe la expresión.

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

Paso 7.9.9

Combina $\frac{1}{x (\cos (x) + \sin (x))}$ y $4$ .

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

Paso 7.9.10

Combina $\frac{4}{x (\cos (x) + \sin (x))}$ y $\sin (x)$ .

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

Paso 7.10

Combina los numeradores sobre el denominador común.

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

Paso 7.11

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

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

Paso 7.12

Escribe cada expresión con un denominador común de $(\cos (x) + \sin (x)) x$ , mediante la multiplicación de cada uno por un factor adecuado de $1$ .

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Paso 7.12.1

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

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

Paso 7.12.2

Reordena los factores de $(\cos (x) + \sin (x)) x$ .

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

Paso 7.13

Combina los numeradores sobre el denominador común.

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

Paso 7.14

Aplica la propiedad distributiva.

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

Paso 7.15

Combina los numeradores sobre el denominador común.

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

Paso 7.16

Factoriza $- 1$ de $- \sin (x) x$ .

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

Paso 7.17

Factoriza $- 1$ de $\cos (x) x$ .

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

Paso 7.18

Factoriza $- 1$ de $- (\sin (x) x) - (- \cos (x) x)$ .

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

Paso 7.19

Factoriza $- 1$ de $4 \cos (x)$ .

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

Paso 7.20

Factoriza $- 1$ de $- (\sin (x) x - \cos (x) x) - (- 4 \cos (x))$ .

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

Paso 7.21

Factoriza $- 1$ de $4 \sin (x)$ .

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

Paso 7.22

Factoriza $- 1$ de $- (\sin (x) x - \cos (x) x - 4 \cos (x)) - (- 4 \sin (x))$ .

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

Paso 7.23

Reescribe $- (\sin (x) x - \cos (x) x - 4 \cos (x) - 4 \sin (x))$ como $- 1 (\sin (x) x - \cos (x) x - 4 \cos (x) - 4 \sin (x))$ .

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

Paso 7.24

Mueve el negativo al frente de la fracción.

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

Paso 7.25

Reordena los factores en $- \frac{\sin (x) x - \cos (x) x - 4 \cos (x) - 4 \sin (x)}{x (\cos (x) + \sin (x))}$ .

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