Cálculo Ejemplos

$4 x \sec (x)$

Paso 1

Obtén la primera derivada.

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Paso 1.1

Como $4$ es constante con respecto a $x$ , la derivada de $4 x \sec (x)$ con respecto a $x$ es $4 \frac{d}{d x} [x \sec (x)]$ .

$4 \frac{d}{d x} [x \sec (x)]$

Paso 1.2

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 $f (x) = x$ y $g (x) = \sec (x)$ .

$4 (x \frac{d}{d x} [\sec (x)] + \sec (x) \frac{d}{d x} [x])$

Paso 1.3

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

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

Paso 1.4

Diferencia con la regla de la potencia.

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Paso 1.4.1

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

$4 (x (\sec (x) \tan (x)) + \sec (x) \cdot 1)$

Paso 1.4.2

Multiplica $\sec (x)$ por $1$ .

$4 (x (\sec (x) \tan (x)) + \sec (x))$

Paso 1.5

Simplifica.

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Paso 1.5.1

Aplica la propiedad distributiva.

$4 (x (\sec (x) \tan (x))) + 4 \sec (x)$

Paso 1.5.2

Reordena los términos.

$f' (x) = 4 x \sec (x) \tan (x) + 4 \sec (x)$

Paso 2

Obtener la segunda derivada.

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Paso 2.1

Según la regla de la suma, la derivada de $4 x \sec (x) \tan (x) + 4 \sec (x)$ con respecto a $x$ es $\frac{d}{d x} [4 x \sec (x) \tan (x)] + \frac{d}{d x} [4 \sec (x)]$ .

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

Paso 2.2

Evalúa $\frac{d}{d x} [4 x \sec (x) \tan (x)]$ .

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Paso 2.2.1

Como $4$ es constante con respecto a $x$ , la derivada de $4 x \sec (x) \tan (x)$ con respecto a $x$ es $4 \frac{d}{d x} [x \sec (x) \tan (x)]$ .

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

Paso 2.2.2

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 $f (x) = x \sec (x)$ y $g (x) = \tan (x)$ .

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

Paso 2.2.3

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

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

Paso 2.2.4

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 $f (x) = x$ y $g (x) = \sec (x)$ .

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

Paso 2.2.5

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

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

Paso 2.2.6

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

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

Paso 2.2.7

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

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Paso 2.2.7.1

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

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

Paso 2.2.7.2

Multiplica $\sec^{2} (x)$ por $\sec (x)$ .

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Paso 2.2.7.2.1

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

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

Paso 2.2.7.2.2

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

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

Paso 2.2.7.3

Suma $2$ y $1$ .

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

Paso 2.2.8

Multiplica $\sec (x)$ por $1$ .

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

Paso 2.3

Evalúa $\frac{d}{d x} [4 \sec (x)]$ .

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Paso 2.3.1

Como $4$ es constante con respecto a $x$ , la derivada de $4 \sec (x)$ con respecto a $x$ es $4 \frac{d}{d x} [\sec (x)]$ .

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

Paso 2.3.2

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

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

Paso 2.4

Simplifica.

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Paso 2.4.1

Aplica la propiedad distributiva.

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

Paso 2.4.2

Aplica la propiedad distributiva.

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

Paso 2.4.3

Combina los términos.

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Paso 2.4.3.1

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

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

Paso 2.4.3.2

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

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

Paso 2.4.3.3

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

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

Paso 2.4.3.4

Suma $1$ y $1$ .

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

Paso 2.4.3.5

Reordena los factores de $4 \tan (x) \sec (x)$ .

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

Paso 2.4.3.6

Suma $4 \sec (x) \tan (x)$ y $4 \sec (x) \tan (x)$ .

$4 x \sec^{3} (x) + 4 x (\sec (x) \tan^{2} (x)) + 8 \sec (x) \tan (x)$

Paso 2.4.4

Reordena los términos.

$f'' (x) = 4 x \sec^{3} (x) + 4 x \tan^{2} (x) \sec (x) + 8 \sec (x) \tan (x)$

Paso 3

Obtén la tercera derivada.

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Paso 3.1

Según la regla de la suma, la derivada de $4 x \sec^{3} (x) + 4 x \tan^{2} (x) \sec (x) + 8 \sec (x) \tan (x)$ con respecto a $x$ es $\frac{d}{d x} [4 x \sec^{3} (x)] + \frac{d}{d x} [4 x \tan^{2} (x) \sec (x)] + \frac{d}{d x} [8 \sec (x) \tan (x)]$ .

$\frac{d}{d x} [4 x \sec^{3} (x)] + \frac{d}{d x} [4 x \tan^{2} (x) \sec (x)] + \frac{d}{d x} [8 \sec (x) \tan (x)]$

Paso 3.2

Evalúa $\frac{d}{d x} [4 x \sec^{3} (x)]$ .

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Paso 3.2.1

Como $4$ es constante con respecto a $x$ , la derivada de $4 x \sec^{3} (x)$ con respecto a $x$ es $4 \frac{d}{d x} [x \sec^{3} (x)]$ .

$4 \frac{d}{d x} [x \sec^{3} (x)] + \frac{d}{d x} [4 x \tan^{2} (x) \sec (x)] + \frac{d}{d x} [8 \sec (x) \tan (x)]$

Paso 3.2.2

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 $f (x) = x$ y $g (x) = \sec^{3} (x)$ .

$4 (x \frac{d}{d x} [\sec^{3} (x)] + \sec^{3} (x) \frac{d}{d x} [x]) + \frac{d}{d x} [4 x \tan^{2} (x) \sec (x)] + \frac{d}{d x} [8 \sec (x) \tan (x)]$

Paso 3.2.3

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

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Paso 3.2.3.1

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

$4 (x (\frac{d}{d u_{1}} [{u_{1}}^{3}] \frac{d}{d x} [\sec (x)]) + \sec^{3} (x) \frac{d}{d x} [x]) + \frac{d}{d x} [4 x \tan^{2} (x) \sec (x)] + \frac{d}{d x} [8 \sec (x) \tan (x)]$

Paso 3.2.3.2

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 $n = 3$ .

$4 (x (3 {u_{1}}^{2} \frac{d}{d x} [\sec (x)]) + \sec^{3} (x) \frac{d}{d x} [x]) + \frac{d}{d x} [4 x \tan^{2} (x) \sec (x)] + \frac{d}{d x} [8 \sec (x) \tan (x)]$

Paso 3.2.3.3

Reemplaza todos los casos de $u_{1}$ con $\sec (x)$ .

$4 (x (3 \sec^{2} (x) \frac{d}{d x} [\sec (x)]) + \sec^{3} (x) \frac{d}{d x} [x]) + \frac{d}{d x} [4 x \tan^{2} (x) \sec (x)] + \frac{d}{d x} [8 \sec (x) \tan (x)]$

Paso 3.2.4

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

$4 (x (3 \sec^{2} (x) (\sec (x) \tan (x))) + \sec^{3} (x) \frac{d}{d x} [x]) + \frac{d}{d x} [4 x \tan^{2} (x) \sec (x)] + \frac{d}{d x} [8 \sec (x) \tan (x)]$

Paso 3.2.5

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

$4 (x (3 \sec^{2} (x) (\sec (x) \tan (x))) + \sec^{3} (x) \cdot 1) + \frac{d}{d x} [4 x \tan^{2} (x) \sec (x)] + \frac{d}{d x} [8 \sec (x) \tan (x)]$

Paso 3.2.6

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

$4 (x (3 (\sec^{1} (x) \sec^{2} (x)) \tan (x)) + \sec^{3} (x) \cdot 1) + \frac{d}{d x} [4 x \tan^{2} (x) \sec (x)] + \frac{d}{d x} [8 \sec (x) \tan (x)]$

Paso 3.2.7

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

$4 (x (3 {\sec (x)}^{1 + 2} \tan (x)) + \sec^{3} (x) \cdot 1) + \frac{d}{d x} [4 x \tan^{2} (x) \sec (x)] + \frac{d}{d x} [8 \sec (x) \tan (x)]$

Paso 3.2.8

Suma $1$ y $2$ .

$4 (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x) \cdot 1) + \frac{d}{d x} [4 x \tan^{2} (x) \sec (x)] + \frac{d}{d x} [8 \sec (x) \tan (x)]$

Paso 3.2.9

Multiplica $\sec^{3} (x)$ por $1$ .

$4 (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x)) + \frac{d}{d x} [4 x \tan^{2} (x) \sec (x)] + \frac{d}{d x} [8 \sec (x) \tan (x)]$

Paso 3.3

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

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Paso 3.3.1

Como $4$ es constante con respecto a $x$ , la derivada de $4 x \tan^{2} (x) \sec (x)$ con respecto a $x$ es $4 \frac{d}{d x} [x \tan^{2} (x) \sec (x)]$ .

$4 (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x)) + 4 \frac{d}{d x} [x \tan^{2} (x) \sec (x)] + \frac{d}{d x} [8 \sec (x) \tan (x)]$

Paso 3.3.2

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

$4 (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x)) + 4 (x \tan^{2} (x) \frac{d}{d x} [\sec (x)] + \sec (x) \frac{d}{d x} [x \tan^{2} (x)]) + \frac{d}{d x} [8 \sec (x) \tan (x)]$

Paso 3.3.3

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

$4 (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x)) + 4 (x \tan^{2} (x) (\sec (x) \tan (x)) + \sec (x) \frac{d}{d x} [x \tan^{2} (x)]) + \frac{d}{d x} [8 \sec (x) \tan (x)]$

Paso 3.3.4

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

$4 (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x)) + 4 (x \tan^{2} (x) (\sec (x) \tan (x)) + \sec (x) (x \frac{d}{d x} [\tan^{2} (x)] + \tan^{2} (x) \frac{d}{d x} [x])) + \frac{d}{d x} [8 \sec (x) \tan (x)]$

Paso 3.3.5

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 3.3.5.1

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

$4 (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x)) + 4 (x \tan^{2} (x) (\sec (x) \tan (x)) + \sec (x) (x (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [\tan (x)]) + \tan^{2} (x) \frac{d}{d x} [x])) + \frac{d}{d x} [8 \sec (x) \tan (x)]$

Paso 3.3.5.2

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 $n = 2$ .

$4 (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x)) + 4 (x \tan^{2} (x) (\sec (x) \tan (x)) + \sec (x) (x (2 u_{2} \frac{d}{d x} [\tan (x)]) + \tan^{2} (x) \frac{d}{d x} [x])) + \frac{d}{d x} [8 \sec (x) \tan (x)]$

Paso 3.3.5.3

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

$4 (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x)) + 4 (x \tan^{2} (x) (\sec (x) \tan (x)) + \sec (x) (x (2 \tan (x) \frac{d}{d x} [\tan (x)]) + \tan^{2} (x) \frac{d}{d x} [x])) + \frac{d}{d x} [8 \sec (x) \tan (x)]$

Paso 3.3.6

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

$4 (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x)) + 4 (x \tan^{2} (x) (\sec (x) \tan (x)) + \sec (x) (x (2 \tan (x) \sec^{2} (x)) + \tan^{2} (x) \frac{d}{d x} [x])) + \frac{d}{d x} [8 \sec (x) \tan (x)]$

Paso 3.3.7

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

$4 (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x)) + 4 (x \tan^{2} (x) (\sec (x) \tan (x)) + \sec (x) (x (2 \tan (x) \sec^{2} (x)) + \tan^{2} (x) \cdot 1)) + \frac{d}{d x} [8 \sec (x) \tan (x)]$

Paso 3.3.8

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

$4 (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x)) + 4 (x (\tan^{1} (x) \tan^{2} (x)) \sec (x) + \sec (x) (x (2 \tan (x) \sec^{2} (x)) + \tan^{2} (x) \cdot 1)) + \frac{d}{d x} [8 \sec (x) \tan (x)]$

Paso 3.3.9

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

$4 (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x)) + 4 (x {\tan (x)}^{1 + 2} \sec (x) + \sec (x) (x (2 \tan (x) \sec^{2} (x)) + \tan^{2} (x) \cdot 1)) + \frac{d}{d x} [8 \sec (x) \tan (x)]$

Paso 3.3.10

Suma $1$ y $2$ .

$4 (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x)) + 4 (x \tan^{3} (x) \sec (x) + \sec (x) (x (2 \tan (x) \sec^{2} (x)) + \tan^{2} (x) \cdot 1)) + \frac{d}{d x} [8 \sec (x) \tan (x)]$

Paso 3.3.11

Multiplica $\tan^{2} (x)$ por $1$ .

$4 (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x)) + 4 (x \tan^{3} (x) \sec (x) + \sec (x) (x (2 \tan (x) \sec^{2} (x)) + \tan^{2} (x))) + \frac{d}{d x} [8 \sec (x) \tan (x)]$

Paso 3.4

Evalúa $\frac{d}{d x} [8 \sec (x) \tan (x)]$ .

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Paso 3.4.1

Como $8$ es constante con respecto a $x$ , la derivada de $8 \sec (x) \tan (x)$ con respecto a $x$ es $8 \frac{d}{d x} [\sec (x) \tan (x)]$ .

$4 (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x)) + 4 (x \tan^{3} (x) \sec (x) + \sec (x) (x (2 \tan (x) \sec^{2} (x)) + \tan^{2} (x))) + 8 \frac{d}{d x} [\sec (x) \tan (x)]$

Paso 3.4.2

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 $f (x) = \sec (x)$ y $g (x) = \tan (x)$ .

$4 (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x)) + 4 (x \tan^{3} (x) \sec (x) + \sec (x) (x (2 \tan (x) \sec^{2} (x)) + \tan^{2} (x))) + 8 (\sec (x) \frac{d}{d x} [\tan (x)] + \tan (x) \frac{d}{d x} [\sec (x)])$

Paso 3.4.3

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

$4 (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x)) + 4 (x \tan^{3} (x) \sec (x) + \sec (x) (x (2 \tan (x) \sec^{2} (x)) + \tan^{2} (x))) + 8 (\sec (x) \sec^{2} (x) + \tan (x) \frac{d}{d x} [\sec (x)])$

Paso 3.4.4

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

$4 (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x)) + 4 (x \tan^{3} (x) \sec (x) + \sec (x) (x (2 \tan (x) \sec^{2} (x)) + \tan^{2} (x))) + 8 (\sec (x) \sec^{2} (x) + \tan (x) (\sec (x) \tan (x)))$

Paso 3.4.5

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

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Paso 3.4.5.1

Multiplica $\sec (x)$ por $\sec^{2} (x)$ .

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Paso 3.4.5.1.1

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

$4 (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x)) + 4 (x \tan^{3} (x) \sec (x) + \sec (x) (x (2 \tan (x) \sec^{2} (x)) + \tan^{2} (x))) + 8 (\sec^{1} (x) \sec^{2} (x) + \tan (x) (\sec (x) \tan (x)))$

Paso 3.4.5.1.2

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

$4 (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x)) + 4 (x \tan^{3} (x) \sec (x) + \sec (x) (x (2 \tan (x) \sec^{2} (x)) + \tan^{2} (x))) + 8 ({\sec (x)}^{1 + 2} + \tan (x) (\sec (x) \tan (x)))$

Paso 3.4.5.2

Suma $1$ y $2$ .

$4 (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x)) + 4 (x \tan^{3} (x) \sec (x) + \sec (x) (x (2 \tan (x) \sec^{2} (x)) + \tan^{2} (x))) + 8 (\sec^{3} (x) + \tan (x) (\sec (x) \tan (x)))$

Paso 3.4.6

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

$4 (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x)) + 4 (x \tan^{3} (x) \sec (x) + \sec (x) (x (2 \tan (x) \sec^{2} (x)) + \tan^{2} (x))) + 8 (\sec^{3} (x) + \tan^{1} (x) \tan (x) \sec (x))$

Paso 3.4.7

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

$4 (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x)) + 4 (x \tan^{3} (x) \sec (x) + \sec (x) (x (2 \tan (x) \sec^{2} (x)) + \tan^{2} (x))) + 8 (\sec^{3} (x) + \tan^{1} (x) \tan^{1} (x) \sec (x))$

Paso 3.4.8

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

$4 (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x)) + 4 (x \tan^{3} (x) \sec (x) + \sec (x) (x (2 \tan (x) \sec^{2} (x)) + \tan^{2} (x))) + 8 (\sec^{3} (x) + {\tan (x)}^{1 + 1} \sec (x))$

Paso 3.4.9

Suma $1$ y $1$ .

$4 (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x)) + 4 (x \tan^{3} (x) \sec (x) + \sec (x) (x (2 \tan (x) \sec^{2} (x)) + \tan^{2} (x))) + 8 (\sec^{3} (x) + \tan^{2} (x) \sec (x))$

Paso 3.5

Simplifica.

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Paso 3.5.1

Aplica la propiedad distributiva.

$4 (x (3 \sec^{3} (x) \tan (x))) + 4 \sec^{3} (x) + 4 (x \tan^{3} (x) \sec (x) + \sec (x) (x (2 \tan (x) \sec^{2} (x)) + \tan^{2} (x))) + 8 (\sec^{3} (x) + \tan^{2} (x) \sec (x))$

Paso 3.5.2

Aplica la propiedad distributiva.

$4 (x (3 \sec^{3} (x) \tan (x))) + 4 \sec^{3} (x) + 4 (x \tan^{3} (x) \sec (x) + \sec (x) (x (2 \tan (x) \sec^{2} (x))) + \sec (x) \tan^{2} (x)) + 8 (\sec^{3} (x) + \tan^{2} (x) \sec (x))$

Paso 3.5.3

Aplica la propiedad distributiva.

$4 (x (3 \sec^{3} (x) \tan (x))) + 4 \sec^{3} (x) + 4 (x \tan^{3} (x) \sec (x)) + 4 (\sec (x) (x (2 \tan (x) \sec^{2} (x)))) + 4 (\sec (x) \tan^{2} (x)) + 8 (\sec^{3} (x) + \tan^{2} (x) \sec (x))$

Paso 3.5.4

Aplica la propiedad distributiva.

$4 (x (3 \sec^{3} (x) \tan (x))) + 4 \sec^{3} (x) + 4 (x \tan^{3} (x) \sec (x)) + 4 (\sec (x) (x (2 \tan (x) \sec^{2} (x)))) + 4 (\sec (x) \tan^{2} (x)) + 8 \sec^{3} (x) + 8 (\tan^{2} (x) \sec (x))$

Paso 3.5.5

Combina los términos.

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Paso 3.5.5.1

Multiplica $3$ por $4$ .

$12 (x (\sec^{3} (x) \tan (x))) + 4 \sec^{3} (x) + 4 (x \tan^{3} (x) \sec (x)) + 4 (\sec (x) (x (2 \tan (x) \sec^{2} (x)))) + 4 (\sec (x) \tan^{2} (x)) + 8 \sec^{3} (x) + 8 (\tan^{2} (x) \sec (x))$

Paso 3.5.5.2

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

$12 x (\sec^{3} (x) \tan (x)) + 4 \sec^{3} (x) + 4 x \tan^{3} (x) \sec (x) + 4 (x (2 \tan (x) (\sec^{1} (x) \sec^{2} (x)))) + 4 (\sec (x) \tan^{2} (x)) + 8 \sec^{3} (x) + 8 (\tan^{2} (x) \sec (x))$

Paso 3.5.5.3

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

$12 x (\sec^{3} (x) \tan (x)) + 4 \sec^{3} (x) + 4 x \tan^{3} (x) \sec (x) + 4 (x (2 \tan (x) {\sec (x)}^{1 + 2})) + 4 (\sec (x) \tan^{2} (x)) + 8 \sec^{3} (x) + 8 (\tan^{2} (x) \sec (x))$

Paso 3.5.5.4

Suma $1$ y $2$ .

$12 x (\sec^{3} (x) \tan (x)) + 4 \sec^{3} (x) + 4 x \tan^{3} (x) \sec (x) + 4 (x (2 \tan (x) \sec^{3} (x))) + 4 (\sec (x) \tan^{2} (x)) + 8 \sec^{3} (x) + 8 (\tan^{2} (x) \sec (x))$

Paso 3.5.5.5

Multiplica $2$ por $4$ .

$12 x (\sec^{3} (x) \tan (x)) + 4 \sec^{3} (x) + 4 x \tan^{3} (x) \sec (x) + 8 (x (\tan (x) \sec^{3} (x))) + 4 (\sec (x) \tan^{2} (x)) + 8 \sec^{3} (x) + 8 (\tan^{2} (x) \sec (x))$

Paso 3.5.5.6

Reordena los factores de $8 x (\tan (x) \sec^{3} (x))$ .

$12 x (\sec^{3} (x) \tan (x)) + 4 \sec^{3} (x) + 4 x \tan^{3} (x) \sec (x) + 8 x \sec^{3} (x) \tan (x) + 4 \sec (x) \tan^{2} (x) + 8 \sec^{3} (x) + 8 (\tan^{2} (x) \sec (x))$

Paso 3.5.5.7

Suma $12 x \sec^{3} (x) \tan (x)$ y $8 x \sec^{3} (x) \tan (x)$ .

$20 x \sec^{3} (x) \tan (x) + 4 \sec^{3} (x) + 4 x \tan^{3} (x) \sec (x) + 4 \sec (x) \tan^{2} (x) + 8 \sec^{3} (x) + 8 (\tan^{2} (x) \sec (x))$

Paso 3.5.5.8

Suma $4 \sec^{3} (x)$ y $8 \sec^{3} (x)$ .

$20 x \sec^{3} (x) \tan (x) + 4 x \tan^{3} (x) \sec (x) + 4 \sec (x) \tan^{2} (x) + 12 \sec^{3} (x) + 8 \tan^{2} (x) \sec (x)$

Paso 3.5.5.9

Reordena los factores de $4 \sec (x) \tan^{2} (x)$ .

$20 x \sec^{3} (x) \tan (x) + 4 x \tan^{3} (x) \sec (x) + 4 \tan^{2} (x) \sec (x) + 12 \sec^{3} (x) + 8 \tan^{2} (x) \sec (x)$

Paso 3.5.5.10

Suma $4 \tan^{2} (x) \sec (x)$ y $8 \tan^{2} (x) \sec (x)$ .

$f''' (x) = 20 x \sec^{3} (x) \tan (x) + 4 x \tan^{3} (x) \sec (x) + 12 \tan^{2} (x) \sec (x) + 12 \sec^{3} (x)$

Paso 4

Obtén la cuarta derivada.

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Paso 4.1

Según la regla de la suma, la derivada de $20 x \sec^{3} (x) \tan (x) + 4 x \tan^{3} (x) \sec (x) + 12 \tan^{2} (x) \sec (x) + 12 \sec^{3} (x)$ con respecto a $x$ es $\frac{d}{d x} [20 x \sec^{3} (x) \tan (x)] + \frac{d}{d x} [4 x \tan^{3} (x) \sec (x)] + \frac{d}{d x} [12 \tan^{2} (x) \sec (x)] + \frac{d}{d x} [12 \sec^{3} (x)]$ .

$\frac{d}{d x} [20 x \sec^{3} (x) \tan (x)] + \frac{d}{d x} [4 x \tan^{3} (x) \sec (x)] + \frac{d}{d x} [12 \tan^{2} (x) \sec (x)] + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.2

Evalúa $\frac{d}{d x} [20 x \sec^{3} (x) \tan (x)]$ .

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Paso 4.2.1

Como $20$ es constante con respecto a $x$ , la derivada de $20 x \sec^{3} (x) \tan (x)$ con respecto a $x$ es $20 \frac{d}{d x} [x \sec^{3} (x) \tan (x)]$ .

$20 \frac{d}{d x} [x \sec^{3} (x) \tan (x)] + \frac{d}{d x} [4 x \tan^{3} (x) \sec (x)] + \frac{d}{d x} [12 \tan^{2} (x) \sec (x)] + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.2.2

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 $f (x) = x \sec^{3} (x)$ y $g (x) = \tan (x)$ .

$20 (x \sec^{3} (x) \frac{d}{d x} [\tan (x)] + \tan (x) \frac{d}{d x} [x \sec^{3} (x)]) + \frac{d}{d x} [4 x \tan^{3} (x) \sec (x)] + \frac{d}{d x} [12 \tan^{2} (x) \sec (x)] + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.2.3

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

$20 (x \sec^{3} (x) \sec^{2} (x) + \tan (x) \frac{d}{d x} [x \sec^{3} (x)]) + \frac{d}{d x} [4 x \tan^{3} (x) \sec (x)] + \frac{d}{d x} [12 \tan^{2} (x) \sec (x)] + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.2.4

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 $f (x) = x$ y $g (x) = \sec^{3} (x)$ .

$20 (x \sec^{3} (x) \sec^{2} (x) + \tan (x) (x \frac{d}{d x} [\sec^{3} (x)] + \sec^{3} (x) \frac{d}{d x} [x])) + \frac{d}{d x} [4 x \tan^{3} (x) \sec (x)] + \frac{d}{d x} [12 \tan^{2} (x) \sec (x)] + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.2.5

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 $g (x) = \sec (x)$ .

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Paso 4.2.5.1

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

$20 (x \sec^{3} (x) \sec^{2} (x) + \tan (x) (x (\frac{d}{d u_{1}} [{u_{1}}^{3}] \frac{d}{d x} [\sec (x)]) + \sec^{3} (x) \frac{d}{d x} [x])) + \frac{d}{d x} [4 x \tan^{3} (x) \sec (x)] + \frac{d}{d x} [12 \tan^{2} (x) \sec (x)] + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.2.5.2

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 $n = 3$ .

$20 (x \sec^{3} (x) \sec^{2} (x) + \tan (x) (x (3 {u_{1}}^{2} \frac{d}{d x} [\sec (x)]) + \sec^{3} (x) \frac{d}{d x} [x])) + \frac{d}{d x} [4 x \tan^{3} (x) \sec (x)] + \frac{d}{d x} [12 \tan^{2} (x) \sec (x)] + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.2.5.3

Reemplaza todos los casos de $u_{1}$ con $\sec (x)$ .

$20 (x \sec^{3} (x) \sec^{2} (x) + \tan (x) (x (3 \sec^{2} (x) \frac{d}{d x} [\sec (x)]) + \sec^{3} (x) \frac{d}{d x} [x])) + \frac{d}{d x} [4 x \tan^{3} (x) \sec (x)] + \frac{d}{d x} [12 \tan^{2} (x) \sec (x)] + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.2.6

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

$20 (x \sec^{3} (x) \sec^{2} (x) + \tan (x) (x (3 \sec^{2} (x) (\sec (x) \tan (x))) + \sec^{3} (x) \frac{d}{d x} [x])) + \frac{d}{d x} [4 x \tan^{3} (x) \sec (x)] + \frac{d}{d x} [12 \tan^{2} (x) \sec (x)] + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.2.7

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

$20 (x \sec^{3} (x) \sec^{2} (x) + \tan (x) (x (3 \sec^{2} (x) (\sec (x) \tan (x))) + \sec^{3} (x) \cdot 1)) + \frac{d}{d x} [4 x \tan^{3} (x) \sec (x)] + \frac{d}{d x} [12 \tan^{2} (x) \sec (x)] + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.2.8

Multiplica $\sec^{3} (x)$ por $\sec^{2} (x)$ sumando los exponentes.

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Paso 4.2.8.1

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

$20 (x (\sec^{2} (x) \sec^{3} (x)) + \tan (x) (x (3 \sec^{2} (x) (\sec (x) \tan (x))) + \sec^{3} (x) \cdot 1)) + \frac{d}{d x} [4 x \tan^{3} (x) \sec (x)] + \frac{d}{d x} [12 \tan^{2} (x) \sec (x)] + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.2.8.2

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

$20 (x {\sec (x)}^{2 + 3} + \tan (x) (x (3 \sec^{2} (x) (\sec (x) \tan (x))) + \sec^{3} (x) \cdot 1)) + \frac{d}{d x} [4 x \tan^{3} (x) \sec (x)] + \frac{d}{d x} [12 \tan^{2} (x) \sec (x)] + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.2.8.3

Suma $2$ y $3$ .

$20 (x \sec^{5} (x) + \tan (x) (x (3 \sec^{2} (x) (\sec (x) \tan (x))) + \sec^{3} (x) \cdot 1)) + \frac{d}{d x} [4 x \tan^{3} (x) \sec (x)] + \frac{d}{d x} [12 \tan^{2} (x) \sec (x)] + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.2.9

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

$20 (x \sec^{5} (x) + \tan (x) (x (3 (\sec^{1} (x) \sec^{2} (x)) \tan (x)) + \sec^{3} (x) \cdot 1)) + \frac{d}{d x} [4 x \tan^{3} (x) \sec (x)] + \frac{d}{d x} [12 \tan^{2} (x) \sec (x)] + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.2.10

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

$20 (x \sec^{5} (x) + \tan (x) (x (3 {\sec (x)}^{1 + 2} \tan (x)) + \sec^{3} (x) \cdot 1)) + \frac{d}{d x} [4 x \tan^{3} (x) \sec (x)] + \frac{d}{d x} [12 \tan^{2} (x) \sec (x)] + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.2.11

Suma $1$ y $2$ .

$20 (x \sec^{5} (x) + \tan (x) (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x) \cdot 1)) + \frac{d}{d x} [4 x \tan^{3} (x) \sec (x)] + \frac{d}{d x} [12 \tan^{2} (x) \sec (x)] + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.2.12

Multiplica $\sec^{3} (x)$ por $1$ .

$20 (x \sec^{5} (x) + \tan (x) (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x))) + \frac{d}{d x} [4 x \tan^{3} (x) \sec (x)] + \frac{d}{d x} [12 \tan^{2} (x) \sec (x)] + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.3

Evalúa $\frac{d}{d x} [4 x \tan^{3} (x) \sec (x)]$ .

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Paso 4.3.1

Como $4$ es constante con respecto a $x$ , la derivada de $4 x \tan^{3} (x) \sec (x)$ con respecto a $x$ es $4 \frac{d}{d x} [x \tan^{3} (x) \sec (x)]$ .

$20 (x \sec^{5} (x) + \tan (x) (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x))) + 4 \frac{d}{d x} [x \tan^{3} (x) \sec (x)] + \frac{d}{d x} [12 \tan^{2} (x) \sec (x)] + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.3.2

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 $f (x) = x \tan^{3} (x)$ y $g (x) = \sec (x)$ .

$20 (x \sec^{5} (x) + \tan (x) (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x))) + 4 (x \tan^{3} (x) \frac{d}{d x} [\sec (x)] + \sec (x) \frac{d}{d x} [x \tan^{3} (x)]) + \frac{d}{d x} [12 \tan^{2} (x) \sec (x)] + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.3.3

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

$20 (x \sec^{5} (x) + \tan (x) (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x))) + 4 (x \tan^{3} (x) (\sec (x) \tan (x)) + \sec (x) \frac{d}{d x} [x \tan^{3} (x)]) + \frac{d}{d x} [12 \tan^{2} (x) \sec (x)] + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.3.4

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 $f (x) = x$ y $g (x) = \tan^{3} (x)$ .

$20 (x \sec^{5} (x) + \tan (x) (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x))) + 4 (x \tan^{3} (x) (\sec (x) \tan (x)) + \sec (x) (x \frac{d}{d x} [\tan^{3} (x)] + \tan^{3} (x) \frac{d}{d x} [x])) + \frac{d}{d x} [12 \tan^{2} (x) \sec (x)] + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.3.5

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 $g (x) = \tan (x)$ .

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Paso 4.3.5.1

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

$20 (x \sec^{5} (x) + \tan (x) (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x))) + 4 (x \tan^{3} (x) (\sec (x) \tan (x)) + \sec (x) (x (\frac{d}{d u_{2}} [{u_{2}}^{3}] \frac{d}{d x} [\tan (x)]) + \tan^{3} (x) \frac{d}{d x} [x])) + \frac{d}{d x} [12 \tan^{2} (x) \sec (x)] + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.3.5.2

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 $n = 3$ .

$20 (x \sec^{5} (x) + \tan (x) (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x))) + 4 (x \tan^{3} (x) (\sec (x) \tan (x)) + \sec (x) (x (3 {u_{2}}^{2} \frac{d}{d x} [\tan (x)]) + \tan^{3} (x) \frac{d}{d x} [x])) + \frac{d}{d x} [12 \tan^{2} (x) \sec (x)] + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.3.5.3

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

$20 (x \sec^{5} (x) + \tan (x) (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x))) + 4 (x \tan^{3} (x) (\sec (x) \tan (x)) + \sec (x) (x (3 \tan^{2} (x) \frac{d}{d x} [\tan (x)]) + \tan^{3} (x) \frac{d}{d x} [x])) + \frac{d}{d x} [12 \tan^{2} (x) \sec (x)] + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.3.6

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

$20 (x \sec^{5} (x) + \tan (x) (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x))) + 4 (x \tan^{3} (x) (\sec (x) \tan (x)) + \sec (x) (x (3 \tan^{2} (x) \sec^{2} (x)) + \tan^{3} (x) \frac{d}{d x} [x])) + \frac{d}{d x} [12 \tan^{2} (x) \sec (x)] + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.3.7

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

$20 (x \sec^{5} (x) + \tan (x) (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x))) + 4 (x \tan^{3} (x) (\sec (x) \tan (x)) + \sec (x) (x (3 \tan^{2} (x) \sec^{2} (x)) + \tan^{3} (x) \cdot 1)) + \frac{d}{d x} [12 \tan^{2} (x) \sec (x)] + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.3.8

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

$20 (x \sec^{5} (x) + \tan (x) (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x))) + 4 (x (\tan^{1} (x) \tan^{3} (x)) \sec (x) + \sec (x) (x (3 \tan^{2} (x) \sec^{2} (x)) + \tan^{3} (x) \cdot 1)) + \frac{d}{d x} [12 \tan^{2} (x) \sec (x)] + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.3.9

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

$20 (x \sec^{5} (x) + \tan (x) (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x))) + 4 (x {\tan (x)}^{1 + 3} \sec (x) + \sec (x) (x (3 \tan^{2} (x) \sec^{2} (x)) + \tan^{3} (x) \cdot 1)) + \frac{d}{d x} [12 \tan^{2} (x) \sec (x)] + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.3.10

Suma $1$ y $3$ .

$20 (x \sec^{5} (x) + \tan (x) (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x))) + 4 (x \tan^{4} (x) \sec (x) + \sec (x) (x (3 \tan^{2} (x) \sec^{2} (x)) + \tan^{3} (x) \cdot 1)) + \frac{d}{d x} [12 \tan^{2} (x) \sec (x)] + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.3.11

Multiplica $\tan^{3} (x)$ por $1$ .

$20 (x \sec^{5} (x) + \tan (x) (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x))) + 4 (x \tan^{4} (x) \sec (x) + \sec (x) (x (3 \tan^{2} (x) \sec^{2} (x)) + \tan^{3} (x))) + \frac{d}{d x} [12 \tan^{2} (x) \sec (x)] + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.4

Evalúa $\frac{d}{d x} [12 \tan^{2} (x) \sec (x)]$ .

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Paso 4.4.1

Como $12$ es constante con respecto a $x$ , la derivada de $12 \tan^{2} (x) \sec (x)$ con respecto a $x$ es $12 \frac{d}{d x} [\tan^{2} (x) \sec (x)]$ .

$20 (x \sec^{5} (x) + \tan (x) (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x))) + 4 (x \tan^{4} (x) \sec (x) + \sec (x) (x (3 \tan^{2} (x) \sec^{2} (x)) + \tan^{3} (x))) + 12 \frac{d}{d x} [\tan^{2} (x) \sec (x)] + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.4.2

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

$20 (x \sec^{5} (x) + \tan (x) (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x))) + 4 (x \tan^{4} (x) \sec (x) + \sec (x) (x (3 \tan^{2} (x) \sec^{2} (x)) + \tan^{3} (x))) + 12 (\tan^{2} (x) \frac{d}{d x} [\sec (x)] + \sec (x) \frac{d}{d x} [\tan^{2} (x)]) + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.4.3

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

$20 (x \sec^{5} (x) + \tan (x) (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x))) + 4 (x \tan^{4} (x) \sec (x) + \sec (x) (x (3 \tan^{2} (x) \sec^{2} (x)) + \tan^{3} (x))) + 12 (\tan^{2} (x) (\sec (x) \tan (x)) + \sec (x) \frac{d}{d x} [\tan^{2} (x)]) + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.4.4

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

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Paso 4.4.4.1

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

$20 (x \sec^{5} (x) + \tan (x) (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x))) + 4 (x \tan^{4} (x) \sec (x) + \sec (x) (x (3 \tan^{2} (x) \sec^{2} (x)) + \tan^{3} (x))) + 12 (\tan^{2} (x) (\sec (x) \tan (x)) + \sec (x) (\frac{d}{d u_{3}} [{u_{3}}^{2}] \frac{d}{d x} [\tan (x)])) + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.4.4.2

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 $n = 2$ .

$20 (x \sec^{5} (x) + \tan (x) (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x))) + 4 (x \tan^{4} (x) \sec (x) + \sec (x) (x (3 \tan^{2} (x) \sec^{2} (x)) + \tan^{3} (x))) + 12 (\tan^{2} (x) (\sec (x) \tan (x)) + \sec (x) (2 u_{3} \frac{d}{d x} [\tan (x)])) + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.4.4.3

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

$20 (x \sec^{5} (x) + \tan (x) (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x))) + 4 (x \tan^{4} (x) \sec (x) + \sec (x) (x (3 \tan^{2} (x) \sec^{2} (x)) + \tan^{3} (x))) + 12 (\tan^{2} (x) (\sec (x) \tan (x)) + \sec (x) (2 \tan (x) \frac{d}{d x} [\tan (x)])) + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.4.5

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

$20 (x \sec^{5} (x) + \tan (x) (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x))) + 4 (x \tan^{4} (x) \sec (x) + \sec (x) (x (3 \tan^{2} (x) \sec^{2} (x)) + \tan^{3} (x))) + 12 (\tan^{2} (x) (\sec (x) \tan (x)) + \sec (x) (2 \tan (x) \sec^{2} (x))) + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.4.6

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

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Paso 4.4.6.1

Mueve $\tan (x)$ .

$20 (x \sec^{5} (x) + \tan (x) (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x))) + 4 (x \tan^{4} (x) \sec (x) + \sec (x) (x (3 \tan^{2} (x) \sec^{2} (x)) + \tan^{3} (x))) + 12 (\tan (x) \tan^{2} (x) \sec (x) + \sec (x) (2 \tan (x) \sec^{2} (x))) + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.4.6.2

Multiplica $\tan (x)$ por $\tan^{2} (x)$ .

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Paso 4.4.6.2.1

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

$20 (x \sec^{5} (x) + \tan (x) (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x))) + 4 (x \tan^{4} (x) \sec (x) + \sec (x) (x (3 \tan^{2} (x) \sec^{2} (x)) + \tan^{3} (x))) + 12 (\tan^{1} (x) \tan^{2} (x) \sec (x) + \sec (x) (2 \tan (x) \sec^{2} (x))) + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.4.6.2.2

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

$20 (x \sec^{5} (x) + \tan (x) (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x))) + 4 (x \tan^{4} (x) \sec (x) + \sec (x) (x (3 \tan^{2} (x) \sec^{2} (x)) + \tan^{3} (x))) + 12 ({\tan (x)}^{1 + 2} \sec (x) + \sec (x) (2 \tan (x) \sec^{2} (x))) + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.4.6.3

Suma $1$ y $2$ .

$20 (x \sec^{5} (x) + \tan (x) (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x))) + 4 (x \tan^{4} (x) \sec (x) + \sec (x) (x (3 \tan^{2} (x) \sec^{2} (x)) + \tan^{3} (x))) + 12 (\tan^{3} (x) \sec (x) + \sec (x) (2 \tan (x) \sec^{2} (x))) + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.4.7

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

$20 (x \sec^{5} (x) + \tan (x) (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x))) + 4 (x \tan^{4} (x) \sec (x) + \sec (x) (x (3 \tan^{2} (x) \sec^{2} (x)) + \tan^{3} (x))) + 12 (\tan^{3} (x) \sec (x) + 2 \tan (x) (\sec^{1} (x) \sec^{2} (x))) + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.4.8

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

$20 (x \sec^{5} (x) + \tan (x) (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x))) + 4 (x \tan^{4} (x) \sec (x) + \sec (x) (x (3 \tan^{2} (x) \sec^{2} (x)) + \tan^{3} (x))) + 12 (\tan^{3} (x) \sec (x) + 2 \tan (x) {\sec (x)}^{1 + 2}) + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.4.9

Suma $1$ y $2$ .

$20 (x \sec^{5} (x) + \tan (x) (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x))) + 4 (x \tan^{4} (x) \sec (x) + \sec (x) (x (3 \tan^{2} (x) \sec^{2} (x)) + \tan^{3} (x))) + 12 (\tan^{3} (x) \sec (x) + 2 \tan (x) \sec^{3} (x)) + \frac{d}{d x} [12 \sec^{3} (x)]$

Paso 4.5

Evalúa $\frac{d}{d x} [12 \sec^{3} (x)]$ .

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Paso 4.5.1

Como $12$ es constante con respecto a $x$ , la derivada de $12 \sec^{3} (x)$ con respecto a $x$ es $12 \frac{d}{d x} [\sec^{3} (x)]$ .

$20 (x \sec^{5} (x) + \tan (x) (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x))) + 4 (x \tan^{4} (x) \sec (x) + \sec (x) (x (3 \tan^{2} (x) \sec^{2} (x)) + \tan^{3} (x))) + 12 (\tan^{3} (x) \sec (x) + 2 \tan (x) \sec^{3} (x)) + 12 \frac{d}{d x} [\sec^{3} (x)]$

Paso 4.5.2

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

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Paso 4.5.2.1

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

$20 (x \sec^{5} (x) + \tan (x) (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x))) + 4 (x \tan^{4} (x) \sec (x) + \sec (x) (x (3 \tan^{2} (x) \sec^{2} (x)) + \tan^{3} (x))) + 12 (\tan^{3} (x) \sec (x) + 2 \tan (x) \sec^{3} (x)) + 12 (\frac{d}{d u_{4}} [{u_{4}}^{3}] \frac{d}{d x} [\sec (x)])$

Paso 4.5.2.2

Diferencia con la regla de la potencia, que establece que $\frac{d}{d u_{4}} [{u_{4}}^{n}]$ es $n {u_{4}}^{n - 1}$ donde $n = 3$ .

$20 (x \sec^{5} (x) + \tan (x) (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x))) + 4 (x \tan^{4} (x) \sec (x) + \sec (x) (x (3 \tan^{2} (x) \sec^{2} (x)) + \tan^{3} (x))) + 12 (\tan^{3} (x) \sec (x) + 2 \tan (x) \sec^{3} (x)) + 12 (3 {u_{4}}^{2} \frac{d}{d x} [\sec (x)])$

Paso 4.5.2.3

Reemplaza todos los casos de $u_{4}$ con $\sec (x)$ .

$20 (x \sec^{5} (x) + \tan (x) (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x))) + 4 (x \tan^{4} (x) \sec (x) + \sec (x) (x (3 \tan^{2} (x) \sec^{2} (x)) + \tan^{3} (x))) + 12 (\tan^{3} (x) \sec (x) + 2 \tan (x) \sec^{3} (x)) + 12 (3 \sec^{2} (x) \frac{d}{d x} [\sec (x)])$

Paso 4.5.3

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

$20 (x \sec^{5} (x) + \tan (x) (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x))) + 4 (x \tan^{4} (x) \sec (x) + \sec (x) (x (3 \tan^{2} (x) \sec^{2} (x)) + \tan^{3} (x))) + 12 (\tan^{3} (x) \sec (x) + 2 \tan (x) \sec^{3} (x)) + 12 (3 \sec^{2} (x) (\sec (x) \tan (x)))$

Paso 4.5.4

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

$20 (x \sec^{5} (x) + \tan (x) (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x))) + 4 (x \tan^{4} (x) \sec (x) + \sec (x) (x (3 \tan^{2} (x) \sec^{2} (x)) + \tan^{3} (x))) + 12 (\tan^{3} (x) \sec (x) + 2 \tan (x) \sec^{3} (x)) + 12 (3 (\sec^{1} (x) \sec^{2} (x)) \tan (x))$

Paso 4.5.5

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

$20 (x \sec^{5} (x) + \tan (x) (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x))) + 4 (x \tan^{4} (x) \sec (x) + \sec (x) (x (3 \tan^{2} (x) \sec^{2} (x)) + \tan^{3} (x))) + 12 (\tan^{3} (x) \sec (x) + 2 \tan (x) \sec^{3} (x)) + 12 (3 {\sec (x)}^{1 + 2} \tan (x))$

Paso 4.5.6

Suma $1$ y $2$ .

$20 (x \sec^{5} (x) + \tan (x) (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x))) + 4 (x \tan^{4} (x) \sec (x) + \sec (x) (x (3 \tan^{2} (x) \sec^{2} (x)) + \tan^{3} (x))) + 12 (\tan^{3} (x) \sec (x) + 2 \tan (x) \sec^{3} (x)) + 12 (3 \sec^{3} (x) \tan (x))$

Paso 4.5.7

Multiplica $3$ por $12$ .

$20 (x \sec^{5} (x) + \tan (x) (x (3 \sec^{3} (x) \tan (x)) + \sec^{3} (x))) + 4 (x \tan^{4} (x) \sec (x) + \sec (x) (x (3 \tan^{2} (x) \sec^{2} (x)) + \tan^{3} (x))) + 12 (\tan^{3} (x) \sec (x) + 2 \tan (x) \sec^{3} (x)) + 36 \sec^{3} (x) \tan (x)$

Paso 4.6

Simplifica.

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Paso 4.6.1

Aplica la propiedad distributiva.

$20 (x \sec^{5} (x) + \tan (x) (x (3 \sec^{3} (x) \tan (x))) + \tan (x) \sec^{3} (x)) + 4 (x \tan^{4} (x) \sec (x) + \sec (x) (x (3 \tan^{2} (x) \sec^{2} (x)) + \tan^{3} (x))) + 12 (\tan^{3} (x) \sec (x) + 2 \tan (x) \sec^{3} (x)) + 36 \sec^{3} (x) \tan (x)$

Paso 4.6.2

Aplica la propiedad distributiva.

$20 (x \sec^{5} (x)) + 20 (\tan (x) (x (3 \sec^{3} (x) \tan (x)))) + 20 (\tan (x) \sec^{3} (x)) + 4 (x \tan^{4} (x) \sec (x) + \sec (x) (x (3 \tan^{2} (x) \sec^{2} (x)) + \tan^{3} (x))) + 12 (\tan^{3} (x) \sec (x) + 2 \tan (x) \sec^{3} (x)) + 36 \sec^{3} (x) \tan (x)$

Paso 4.6.3

Aplica la propiedad distributiva.

$20 (x \sec^{5} (x)) + 20 (\tan (x) (x (3 \sec^{3} (x) \tan (x)))) + 20 (\tan (x) \sec^{3} (x)) + 4 (x \tan^{4} (x) \sec (x) + \sec (x) (x (3 \tan^{2} (x) \sec^{2} (x))) + \sec (x) \tan^{3} (x)) + 12 (\tan^{3} (x) \sec (x) + 2 \tan (x) \sec^{3} (x)) + 36 \sec^{3} (x) \tan (x)$

Paso 4.6.4

Aplica la propiedad distributiva.

$20 (x \sec^{5} (x)) + 20 (\tan (x) (x (3 \sec^{3} (x) \tan (x)))) + 20 (\tan (x) \sec^{3} (x)) + 4 (x \tan^{4} (x) \sec (x)) + 4 (\sec (x) (x (3 \tan^{2} (x) \sec^{2} (x)))) + 4 (\sec (x) \tan^{3} (x)) + 12 (\tan^{3} (x) \sec (x) + 2 \tan (x) \sec^{3} (x)) + 36 \sec^{3} (x) \tan (x)$

Paso 4.6.5

Aplica la propiedad distributiva.

$20 (x \sec^{5} (x)) + 20 (\tan (x) (x (3 \sec^{3} (x) \tan (x)))) + 20 (\tan (x) \sec^{3} (x)) + 4 (x \tan^{4} (x) \sec (x)) + 4 (\sec (x) (x (3 \tan^{2} (x) \sec^{2} (x)))) + 4 (\sec (x) \tan^{3} (x)) + 12 (\tan^{3} (x) \sec (x)) + 12 (2 \tan (x) \sec^{3} (x)) + 36 \sec^{3} (x) \tan (x)$

Paso 4.6.6

Combina los términos.

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Paso 4.6.6.1

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

$20 x \sec^{5} (x) + 20 (x (3 \sec^{3} (x) (\tan^{1} (x) \tan (x)))) + 20 (\tan (x) \sec^{3} (x)) + 4 (x \tan^{4} (x) \sec (x)) + 4 (\sec (x) (x (3 \tan^{2} (x) \sec^{2} (x)))) + 4 (\sec (x) \tan^{3} (x)) + 12 (\tan^{3} (x) \sec (x)) + 12 (2 \tan (x) \sec^{3} (x)) + 36 \sec^{3} (x) \tan (x)$

Paso 4.6.6.2

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

$20 x \sec^{5} (x) + 20 (x (3 \sec^{3} (x) (\tan^{1} (x) \tan^{1} (x)))) + 20 (\tan (x) \sec^{3} (x)) + 4 (x \tan^{4} (x) \sec (x)) + 4 (\sec (x) (x (3 \tan^{2} (x) \sec^{2} (x)))) + 4 (\sec (x) \tan^{3} (x)) + 12 (\tan^{3} (x) \sec (x)) + 12 (2 \tan (x) \sec^{3} (x)) + 36 \sec^{3} (x) \tan (x)$

Paso 4.6.6.3

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

$20 x \sec^{5} (x) + 20 (x (3 \sec^{3} (x) {\tan (x)}^{1 + 1})) + 20 (\tan (x) \sec^{3} (x)) + 4 (x \tan^{4} (x) \sec (x)) + 4 (\sec (x) (x (3 \tan^{2} (x) \sec^{2} (x)))) + 4 (\sec (x) \tan^{3} (x)) + 12 (\tan^{3} (x) \sec (x)) + 12 (2 \tan (x) \sec^{3} (x)) + 36 \sec^{3} (x) \tan (x)$

Paso 4.6.6.4

Suma $1$ y $1$ .

$20 x \sec^{5} (x) + 20 (x (3 \sec^{3} (x) \tan^{2} (x))) + 20 (\tan (x) \sec^{3} (x)) + 4 (x \tan^{4} (x) \sec (x)) + 4 (\sec (x) (x (3 \tan^{2} (x) \sec^{2} (x)))) + 4 (\sec (x) \tan^{3} (x)) + 12 (\tan^{3} (x) \sec (x)) + 12 (2 \tan (x) \sec^{3} (x)) + 36 \sec^{3} (x) \tan (x)$

Paso 4.6.6.5

Multiplica $3$ por $20$ .

$20 x \sec^{5} (x) + 60 (x (\sec^{3} (x) \tan^{2} (x))) + 20 (\tan (x) \sec^{3} (x)) + 4 (x \tan^{4} (x) \sec (x)) + 4 (\sec (x) (x (3 \tan^{2} (x) \sec^{2} (x)))) + 4 (\sec (x) \tan^{3} (x)) + 12 (\tan^{3} (x) \sec (x)) + 12 (2 \tan (x) \sec^{3} (x)) + 36 \sec^{3} (x) \tan (x)$

Paso 4.6.6.6

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

$20 x \sec^{5} (x) + 60 x (\sec^{3} (x) \tan^{2} (x)) + 20 \tan (x) \sec^{3} (x) + 4 x \tan^{4} (x) \sec (x) + 4 (x (3 \tan^{2} (x) (\sec^{1} (x) \sec^{2} (x)))) + 4 (\sec (x) \tan^{3} (x)) + 12 (\tan^{3} (x) \sec (x)) + 12 (2 \tan (x) \sec^{3} (x)) + 36 \sec^{3} (x) \tan (x)$

Paso 4.6.6.7

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

$20 x \sec^{5} (x) + 60 x (\sec^{3} (x) \tan^{2} (x)) + 20 \tan (x) \sec^{3} (x) + 4 x \tan^{4} (x) \sec (x) + 4 (x (3 \tan^{2} (x) {\sec (x)}^{1 + 2})) + 4 (\sec (x) \tan^{3} (x)) + 12 (\tan^{3} (x) \sec (x)) + 12 (2 \tan (x) \sec^{3} (x)) + 36 \sec^{3} (x) \tan (x)$

Paso 4.6.6.8

Suma $1$ y $2$ .

$20 x \sec^{5} (x) + 60 x (\sec^{3} (x) \tan^{2} (x)) + 20 \tan (x) \sec^{3} (x) + 4 x \tan^{4} (x) \sec (x) + 4 (x (3 \tan^{2} (x) \sec^{3} (x))) + 4 (\sec (x) \tan^{3} (x)) + 12 (\tan^{3} (x) \sec (x)) + 12 (2 \tan (x) \sec^{3} (x)) + 36 \sec^{3} (x) \tan (x)$

Paso 4.6.6.9

Multiplica $3$ por $4$ .

$20 x \sec^{5} (x) + 60 x (\sec^{3} (x) \tan^{2} (x)) + 20 \tan (x) \sec^{3} (x) + 4 x \tan^{4} (x) \sec (x) + 12 (x (\tan^{2} (x) \sec^{3} (x))) + 4 (\sec (x) \tan^{3} (x)) + 12 (\tan^{3} (x) \sec (x)) + 12 (2 \tan (x) \sec^{3} (x)) + 36 \sec^{3} (x) \tan (x)$

Paso 4.6.6.10

Reordena los factores de $12 x (\tan^{2} (x) \sec^{3} (x))$ .

$20 x \sec^{5} (x) + 60 x (\sec^{3} (x) \tan^{2} (x)) + 20 \tan (x) \sec^{3} (x) + 4 x \tan^{4} (x) \sec (x) + 12 x \sec^{3} (x) \tan^{2} (x) + 4 \sec (x) \tan^{3} (x) + 12 (\tan^{3} (x) \sec (x)) + 12 (2 \tan (x) \sec^{3} (x)) + 36 \sec^{3} (x) \tan (x)$

Paso 4.6.6.11

Suma $60 x \sec^{3} (x) \tan^{2} (x)$ y $12 x \sec^{3} (x) \tan^{2} (x)$ .

$20 x \sec^{5} (x) + 72 x \sec^{3} (x) \tan^{2} (x) + 20 \tan (x) \sec^{3} (x) + 4 x \tan^{4} (x) \sec (x) + 4 \sec (x) \tan^{3} (x) + 12 (\tan^{3} (x) \sec (x)) + 12 (2 \tan (x) \sec^{3} (x)) + 36 \sec^{3} (x) \tan (x)$

Paso 4.6.6.12

Multiplica $2$ por $12$ .

$20 x \sec^{5} (x) + 72 x \sec^{3} (x) \tan^{2} (x) + 20 \tan (x) \sec^{3} (x) + 4 x \tan^{4} (x) \sec (x) + 4 \sec (x) \tan^{3} (x) + 12 \tan^{3} (x) \sec (x) + 24 (\tan (x) \sec^{3} (x)) + 36 \sec^{3} (x) \tan (x)$

Paso 4.6.6.13

Reordena los factores de $4 \sec (x) \tan^{3} (x)$ .

$20 x \sec^{5} (x) + 72 x \sec^{3} (x) \tan^{2} (x) + 20 \tan (x) \sec^{3} (x) + 4 x \tan^{4} (x) \sec (x) + 4 \tan^{3} (x) \sec (x) + 12 \tan^{3} (x) \sec (x) + 24 \tan (x) \sec^{3} (x) + 36 \sec^{3} (x) \tan (x)$

Paso 4.6.6.14

Suma $4 \tan^{3} (x) \sec (x)$ y $12 \tan^{3} (x) \sec (x)$ .

$20 x \sec^{5} (x) + 72 x \sec^{3} (x) \tan^{2} (x) + 20 \tan (x) \sec^{3} (x) + 4 x \tan^{4} (x) \sec (x) + 16 \tan^{3} (x) \sec (x) + 24 \tan (x) \sec^{3} (x) + 36 \sec^{3} (x) \tan (x)$

Paso 4.6.6.15

Suma $20 \tan (x) \sec^{3} (x)$ y $24 \tan (x) \sec^{3} (x)$ .

$20 x \sec^{5} (x) + 72 x \sec^{3} (x) \tan^{2} (x) + 44 \tan (x) \sec^{3} (x) + 4 x \tan^{4} (x) \sec (x) + 16 \tan^{3} (x) \sec (x) + 36 \sec^{3} (x) \tan (x)$

Paso 4.6.6.16

Reordena los factores de $44 \tan (x) \sec^{3} (x)$ .

$20 x \sec^{5} (x) + 72 x \sec^{3} (x) \tan^{2} (x) + 44 \sec^{3} (x) \tan (x) + 4 x \tan^{4} (x) \sec (x) + 16 \tan^{3} (x) \sec (x) + 36 \sec^{3} (x) \tan (x)$

Paso 4.6.6.17

Suma $44 \sec^{3} (x) \tan (x)$ y $36 \sec^{3} (x) \tan (x)$ .

$f^{4} (x) = 20 x \sec^{5} (x) + 72 x \sec^{3} (x) \tan^{2} (x) + 80 \sec^{3} (x) \tan (x) + 4 x \tan^{4} (x) \sec (x) + 16 \tan^{3} (x) \sec (x)$