Cálculo Ejemplos

$f (x) = \ln (\sec^{4} (x) \cdot \tan^{2} (x))$

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 $g (x) = \sec^{4} (x) \tan^{2} (x)$ .

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Paso 1.1

Para aplicar la regla de la cadena, establece $u_{1}$ como $\sec^{4} (x) \tan^{2} (x)$ .

$\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d x} [\sec^{4} (x) \tan^{2} (x)]$

Paso 1.2

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

$\frac{1}{\sec^{4} (x) \tan^{2} (x)} \frac{d}{d x} [\sec^{4} (x) \tan^{2} (x)]$

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 $f (x) = \sec^{4} (x)$ y $g (x) = \tan^{2} (x)$ .

$\frac{1}{\sec^{4} (x) \tan^{2} (x)} (\sec^{4} (x) \frac{d}{d x} [\tan^{2} (x)] + \tan^{2} (x) \frac{d}{d x} [\sec^{4} (x)])$

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 $f (x) = x^{2}$ y $g (x) = \tan (x)$ .

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Paso 3.1

Para aplicar la regla de la cadena, establece $u_{2}$ como $\tan (x)$ .

$\frac{1}{\sec^{4} (x) \tan^{2} (x)} (\sec^{4} (x) (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [\tan (x)]) + \tan^{2} (x) \frac{d}{d x} [\sec^{4} (x)])$

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 $n = 2$ .

$\frac{1}{\sec^{4} (x) \tan^{2} (x)} (\sec^{4} (x) (2 u_{2} \frac{d}{d x} [\tan (x)]) + \tan^{2} (x) \frac{d}{d x} [\sec^{4} (x)])$

Paso 3.3

Reemplaza todos los casos de $u_{2}$ con $\tan (x)$ .

$\frac{1}{\sec^{4} (x) \tan^{2} (x)} (\sec^{4} (x) (2 \tan (x) \frac{d}{d x} [\tan (x)]) + \tan^{2} (x) \frac{d}{d x} [\sec^{4} (x)])$

Paso 4

Mueve $2$ a la izquierda de $\sec^{4} (x)$ .

$\frac{1}{\sec^{4} (x) \tan^{2} (x)} (2 \cdot \sec^{4} (x) \tan (x) \frac{d}{d x} [\tan (x)] + \tan^{2} (x) \frac{d}{d x} [\sec^{4} (x)])$

Paso 5

La derivada de $\tan (x)$ con respecto a $x$ es $\sec^{2} (x)$ .

$\frac{1}{\sec^{4} (x) \tan^{2} (x)} (2 \sec^{4} (x) \tan (x) \sec^{2} (x) + \tan^{2} (x) \frac{d}{d x} [\sec^{4} (x)])$

Paso 6

Multiplica $\sec^{4} (x)$ por $\sec^{2} (x)$ sumando los exponentes.

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Paso 6.1

Mueve $\sec^{2} (x)$ .

$\frac{1}{\sec^{4} (x) \tan^{2} (x)} (2 (\sec^{2} (x) \sec^{4} (x)) \tan (x) + \tan^{2} (x) \frac{d}{d x} [\sec^{4} (x)])$

Paso 6.2

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

$\frac{1}{\sec^{4} (x) \tan^{2} (x)} (2 {\sec (x)}^{2 + 4} \tan (x) + \tan^{2} (x) \frac{d}{d x} [\sec^{4} (x)])$

Paso 6.3

Suma $2$ y $4$ .

$\frac{1}{\sec^{4} (x) \tan^{2} (x)} (2 \sec^{6} (x) \tan (x) + \tan^{2} (x) \frac{d}{d x} [\sec^{4} (x)])$

Paso 7

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

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Paso 7.1

Para aplicar la regla de la cadena, establece $u_{3}$ como $\sec (x)$ .

$\frac{1}{\sec^{4} (x) \tan^{2} (x)} (2 \sec^{6} (x) \tan (x) + \tan^{2} (x) (\frac{d}{d u_{3}} [{u_{3}}^{4}] \frac{d}{d x} [\sec (x)]))$

Paso 7.2

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 $n = 4$ .

$\frac{1}{\sec^{4} (x) \tan^{2} (x)} (2 \sec^{6} (x) \tan (x) + \tan^{2} (x) (4 {u_{3}}^{3} \frac{d}{d x} [\sec (x)]))$

Paso 7.3

Reemplaza todos los casos de $u_{3}$ con $\sec (x)$ .

$\frac{1}{\sec^{4} (x) \tan^{2} (x)} (2 \sec^{6} (x) \tan (x) + \tan^{2} (x) (4 \sec^{3} (x) \frac{d}{d x} [\sec (x)]))$

Paso 8

Mueve $4$ a la izquierda de $\tan^{2} (x)$ .

$\frac{1}{\sec^{4} (x) \tan^{2} (x)} (2 \sec^{6} (x) \tan (x) + 4 \cdot \tan^{2} (x) \sec^{3} (x) \frac{d}{d x} [\sec (x)])$

Paso 9

La derivada de $\sec (x)$ con respecto a $x$ es $\sec (x) \tan (x)$ .

$\frac{1}{\sec^{4} (x) \tan^{2} (x)} (2 \sec^{6} (x) \tan (x) + 4 \tan^{2} (x) \sec^{3} (x) (\sec (x) \tan (x)))$

Paso 10

Eleva $\tan (x)$ a la potencia de $1$ .

$\frac{1}{\sec^{4} (x) \tan^{2} (x)} (2 \sec^{6} (x) \tan (x) + 4 (\tan^{1} (x) \tan^{2} (x)) \sec^{3} (x) \sec (x))$

Paso 11

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

$\frac{1}{\sec^{4} (x) \tan^{2} (x)} (2 \sec^{6} (x) \tan (x) + 4 {\tan (x)}^{1 + 2} \sec^{3} (x) \sec (x))$

Paso 12

Suma $1$ y $2$ .

$\frac{1}{\sec^{4} (x) \tan^{2} (x)} (2 \sec^{6} (x) \tan (x) + 4 \tan^{3} (x) \sec^{3} (x) \sec (x))$

Paso 13

Eleva $\sec (x)$ a la potencia de $1$ .

$\frac{1}{\sec^{4} (x) \tan^{2} (x)} (2 \sec^{6} (x) \tan (x) + 4 \tan^{3} (x) (\sec^{1} (x) \sec^{3} (x)))$

Paso 14

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

$\frac{1}{\sec^{4} (x) \tan^{2} (x)} (2 \sec^{6} (x) \tan (x) + 4 \tan^{3} (x) {\sec (x)}^{1 + 3})$

Paso 15

Suma $1$ y $3$ .

$\frac{1}{\sec^{4} (x) \tan^{2} (x)} (2 \sec^{6} (x) \tan (x) + 4 \tan^{3} (x) \sec^{4} (x))$

Paso 16

Simplifica.

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Paso 16.1

Aplica la propiedad distributiva.

$\frac{1}{\sec^{4} (x) \tan^{2} (x)} (2 \sec^{6} (x) \tan (x)) + \frac{1}{\sec^{4} (x) \tan^{2} (x)} (4 \tan^{3} (x) \sec^{4} (x))$

Paso 16.2

Combina los términos.

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Paso 16.2.1

Combina $2$ y $\frac{1}{\sec^{4} (x) \tan^{2} (x)}$ .

$\frac{2}{\sec^{4} (x) \tan^{2} (x)} (\sec^{6} (x) \tan (x)) + \frac{1}{\sec^{4} (x) \tan^{2} (x)} (4 \tan^{3} (x) \sec^{4} (x))$

Paso 16.2.2

Combina $\sec^{6} (x)$ y $\frac{2}{\sec^{4} (x) \tan^{2} (x)}$ .

$\frac{\sec^{6} (x) \cdot 2}{\sec^{4} (x) \tan^{2} (x)} \tan (x) + \frac{1}{\sec^{4} (x) \tan^{2} (x)} (4 \tan^{3} (x) \sec^{4} (x))$

Paso 16.2.3

Combina $\frac{\sec^{6} (x) \cdot 2}{\sec^{4} (x) \tan^{2} (x)}$ y $\tan (x)$ .

$\frac{\sec^{6} (x) \cdot 2 \tan (x)}{\sec^{4} (x) \tan^{2} (x)} + \frac{1}{\sec^{4} (x) \tan^{2} (x)} (4 \tan^{3} (x) \sec^{4} (x))$

Paso 16.2.4

Mueve $2$ a la izquierda de $\sec^{6} (x)$ .

$\frac{2 \cdot \sec^{6} (x) \tan (x)}{\sec^{4} (x) \tan^{2} (x)} + \frac{1}{\sec^{4} (x) \tan^{2} (x)} (4 \tan^{3} (x) \sec^{4} (x))$

Paso 16.2.5

Cancela el factor común de $\sec^{6} (x)$ y $\sec^{4} (x)$ .

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Paso 16.2.5.1

Factoriza $\sec^{4} (x)$ de $2 \sec^{6} (x) \tan (x)$ .

$\frac{\sec^{4} (x) (2 \sec^{2} (x) \tan (x))}{\sec^{4} (x) \tan^{2} (x)} + \frac{1}{\sec^{4} (x) \tan^{2} (x)} (4 \tan^{3} (x) \sec^{4} (x))$

Paso 16.2.5.2

Cancela los factores comunes.

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Paso 16.2.5.2.1

Factoriza $\sec^{4} (x)$ de $\sec^{4} (x) \tan^{2} (x)$ .

$\frac{\sec^{4} (x) (2 \sec^{2} (x) \tan (x))}{\sec^{4} (x) (\tan^{2} (x))} + \frac{1}{\sec^{4} (x) \tan^{2} (x)} (4 \tan^{3} (x) \sec^{4} (x))$

Paso 16.2.5.2.2

Cancela el factor común.

$\frac{\sec^{4} (x) (2 \sec^{2} (x) \tan (x))}{\sec^{4} (x) \tan^{2} (x)} + \frac{1}{\sec^{4} (x) \tan^{2} (x)} (4 \tan^{3} (x) \sec^{4} (x))$

Paso 16.2.5.2.3

Reescribe la expresión.

$\frac{2 \sec^{2} (x) \tan (x)}{\tan^{2} (x)} + \frac{1}{\sec^{4} (x) \tan^{2} (x)} (4 \tan^{3} (x) \sec^{4} (x))$

Paso 16.2.6

Cancela el factor común de $\tan (x)$ y $\tan^{2} (x)$ .

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Paso 16.2.6.1

Factoriza $\tan (x)$ de $2 \sec^{2} (x) \tan (x)$ .

$\frac{\tan (x) (2 \sec^{2} (x))}{\tan^{2} (x)} + \frac{1}{\sec^{4} (x) \tan^{2} (x)} (4 \tan^{3} (x) \sec^{4} (x))$

Paso 16.2.6.2

Cancela los factores comunes.

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Paso 16.2.6.2.1

Factoriza $\tan (x)$ de $\tan^{2} (x)$ .

$\frac{\tan (x) (2 \sec^{2} (x))}{\tan (x) \tan (x)} + \frac{1}{\sec^{4} (x) \tan^{2} (x)} (4 \tan^{3} (x) \sec^{4} (x))$

Paso 16.2.6.2.2

Cancela el factor común.

$\frac{\tan (x) (2 \sec^{2} (x))}{\tan (x) \tan (x)} + \frac{1}{\sec^{4} (x) \tan^{2} (x)} (4 \tan^{3} (x) \sec^{4} (x))$

Paso 16.2.6.2.3

Reescribe la expresión.

$\frac{2 \sec^{2} (x)}{\tan (x)} + \frac{1}{\sec^{4} (x) \tan^{2} (x)} (4 \tan^{3} (x) \sec^{4} (x))$

Paso 16.2.7

Combina $4$ y $\frac{1}{\sec^{4} (x) \tan^{2} (x)}$ .

$\frac{2 \sec^{2} (x)}{\tan (x)} + \frac{4}{\sec^{4} (x) \tan^{2} (x)} (\tan^{3} (x) \sec^{4} (x))$

Paso 16.2.8

Combina $\tan^{3} (x)$ y $\frac{4}{\sec^{4} (x) \tan^{2} (x)}$ .

$\frac{2 \sec^{2} (x)}{\tan (x)} + \frac{\tan^{3} (x) \cdot 4}{\sec^{4} (x) \tan^{2} (x)} \sec^{4} (x)$

Paso 16.2.9

Combina $\frac{\tan^{3} (x) \cdot 4}{\sec^{4} (x) \tan^{2} (x)}$ y $\sec^{4} (x)$ .

$\frac{2 \sec^{2} (x)}{\tan (x)} + \frac{\tan^{3} (x) \cdot 4 \sec^{4} (x)}{\sec^{4} (x) \tan^{2} (x)}$

Paso 16.2.10

Mueve $4$ a la izquierda de $\tan^{3} (x)$ .

$\frac{2 \sec^{2} (x)}{\tan (x)} + \frac{4 \cdot \tan^{3} (x) \sec^{4} (x)}{\sec^{4} (x) \tan^{2} (x)}$

Paso 16.2.11

Cancela el factor común de $\tan^{3} (x)$ y $\tan^{2} (x)$ .

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Paso 16.2.11.1

Factoriza $\tan^{2} (x)$ de $4 \tan^{3} (x) \sec^{4} (x)$ .

$\frac{2 \sec^{2} (x)}{\tan (x)} + \frac{\tan^{2} (x) (4 \tan (x) \sec^{4} (x))}{\sec^{4} (x) \tan^{2} (x)}$

Paso 16.2.11.2

Cancela los factores comunes.

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Paso 16.2.11.2.1

Factoriza $\tan^{2} (x)$ de $\sec^{4} (x) \tan^{2} (x)$ .

$\frac{2 \sec^{2} (x)}{\tan (x)} + \frac{\tan^{2} (x) (4 \tan (x) \sec^{4} (x))}{\tan^{2} (x) \sec^{4} (x)}$

Paso 16.2.11.2.2

Cancela el factor común.

$\frac{2 \sec^{2} (x)}{\tan (x)} + \frac{\tan^{2} (x) (4 \tan (x) \sec^{4} (x))}{\tan^{2} (x) \sec^{4} (x)}$

Paso 16.2.11.2.3

Reescribe la expresión.

$\frac{2 \sec^{2} (x)}{\tan (x)} + \frac{4 \tan (x) \sec^{4} (x)}{\sec^{4} (x)}$

Paso 16.2.12

Cancela el factor común de $\sec^{4} (x)$ .

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Paso 16.2.12.1

Cancela el factor común.

$\frac{2 \sec^{2} (x)}{\tan (x)} + \frac{4 \tan (x) \sec^{4} (x)}{\sec^{4} (x)}$

Paso 16.2.12.2

Divide $4 \tan (x)$ por $1$ .

$\frac{2 \sec^{2} (x)}{\tan (x)} + 4 \tan (x)$