Cálculo Ejemplos

$f (x) = 3 \tan (x) + 2 \cot (x)$

Paso 1

Obtén la primera derivada.

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Paso 1.1

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

$\frac{d}{d x} [3 \tan (x)] + \frac{d}{d x} [2 \cot (x)]$

Paso 1.2

Evalúa $\frac{d}{d x} [3 \tan (x)]$ .

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Paso 1.2.1

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

$3 \frac{d}{d x} [\tan (x)] + \frac{d}{d x} [2 \cot (x)]$

Paso 1.2.2

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

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

Paso 1.3

Evalúa $\frac{d}{d x} [2 \cot (x)]$ .

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Paso 1.3.1

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

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

Paso 1.3.2

La derivada de $\cot (x)$ con respecto a $x$ es $- \csc^{2} (x)$ .

$3 \sec^{2} (x) + 2 (- \csc^{2} (x))$

Paso 1.3.3

Multiplica $- 1$ por $2$ .

$f' (x) = 3 \sec^{2} (x) - 2 \csc^{2} (x)$

Paso 2

Obtener la segunda derivada.

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Paso 2.1

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

$\frac{d}{d x} [3 \sec^{2} (x)] + \frac{d}{d x} [- 2 \csc^{2} (x)]$

Paso 2.2

Evalúa $\frac{d}{d x} [3 \sec^{2} (x)]$ .

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Paso 2.2.1

Como $3$ es constante con respecto a $x$ , la derivada de $3 \sec^{2} (x)$ con respecto a $x$ es $3 \frac{d}{d x} [\sec^{2} (x)]$ .

$3 \frac{d}{d x} [\sec^{2} (x)] + \frac{d}{d x} [- 2 \csc^{2} (x)]$

Paso 2.2.2

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

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Paso 2.2.2.1

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

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

Paso 2.2.2.2

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 $n = 2$ .

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

Paso 2.2.2.3

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

$3 (2 \sec (x) \frac{d}{d x} [\sec (x)]) + \frac{d}{d x} [- 2 \csc^{2} (x)]$

Paso 2.2.3

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

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

Paso 2.2.4

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

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

Paso 2.2.5

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

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

Paso 2.2.6

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

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

Paso 2.2.7

Suma $1$ y $1$ .

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

Paso 2.2.8

Multiplica $2$ por $3$ .

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

Paso 2.3

Evalúa $\frac{d}{d x} [- 2 \csc^{2} (x)]$ .

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Paso 2.3.1

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

$6 \sec^{2} (x) \tan (x) - 2 \frac{d}{d x} [\csc^{2} (x)]$

Paso 2.3.2

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

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Paso 2.3.2.1

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

$6 \sec^{2} (x) \tan (x) - 2 (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [\csc (x)])$

Paso 2.3.2.2

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 $n = 2$ .

$6 \sec^{2} (x) \tan (x) - 2 (2 u_{2} \frac{d}{d x} [\csc (x)])$

Paso 2.3.2.3

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

$6 \sec^{2} (x) \tan (x) - 2 (2 \csc (x) \frac{d}{d x} [\csc (x)])$

Paso 2.3.3

La derivada de $\csc (x)$ con respecto a $x$ es $- \csc (x) \cot (x)$ .

$6 \sec^{2} (x) \tan (x) - 2 (2 \csc (x) (- \csc (x) \cot (x)))$

Paso 2.3.4

Multiplica $- 1$ por $2$ .

$6 \sec^{2} (x) \tan (x) - 2 (- 2 \csc (x) (\csc (x) \cot (x)))$

Paso 2.3.5

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

$6 \sec^{2} (x) \tan (x) - 2 (- 2 (\csc^{1} (x) \csc (x)) \cot (x))$

Paso 2.3.6

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

$6 \sec^{2} (x) \tan (x) - 2 (- 2 (\csc^{1} (x) \csc^{1} (x)) \cot (x))$

Paso 2.3.7

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

$6 \sec^{2} (x) \tan (x) - 2 (- 2 {\csc (x)}^{1 + 1} \cot (x))$

Paso 2.3.8

Suma $1$ y $1$ .

$6 \sec^{2} (x) \tan (x) - 2 (- 2 \csc^{2} (x) \cot (x))$

Paso 2.3.9

Multiplica $- 2$ por $- 2$ .

$f'' (x) = 6 \sec^{2} (x) \tan (x) + 4 \csc^{2} (x) \cot (x)$

Paso 3

Obtén la tercera derivada.

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Paso 3.1

Según la regla de la suma, la derivada de $6 \sec^{2} (x) \tan (x) + 4 \csc^{2} (x) \cot (x)$ con respecto a $x$ es $\frac{d}{d x} [6 \sec^{2} (x) \tan (x)] + \frac{d}{d x} [4 \csc^{2} (x) \cot (x)]$ .

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

Paso 3.2

Evalúa $\frac{d}{d x} [6 \sec^{2} (x) \tan (x)]$ .

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Paso 3.2.1

Como $6$ es constante con respecto a $x$ , la derivada de $6 \sec^{2} (x) \tan (x)$ con respecto a $x$ es $6 \frac{d}{d x} [\sec^{2} (x) \tan (x)]$ .

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

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

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

Paso 3.2.3

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

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

Paso 3.2.4

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

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Paso 3.2.4.1

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

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

Paso 3.2.4.2

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 $n = 2$ .

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

Paso 3.2.4.3

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

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

Paso 3.2.5

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

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

Paso 3.2.6

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

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Paso 3.2.6.1

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

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

Paso 3.2.6.2

Suma $2$ y $2$ .

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

Paso 3.2.7

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

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

Paso 3.2.8

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

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

Paso 3.2.9

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

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

Paso 3.2.10

Suma $1$ y $1$ .

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

Paso 3.2.11

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

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

Paso 3.2.12

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

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

Paso 3.2.13

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

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

Paso 3.2.14

Suma $1$ y $1$ .

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

Paso 3.3

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

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Paso 3.3.1

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

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

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

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

Paso 3.3.3

La derivada de $\cot (x)$ con respecto a $x$ es $- \csc^{2} (x)$ .

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

Paso 3.3.4

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

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Paso 3.3.4.1

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

$6 (\sec^{4} (x) + 2 \sec^{2} (x) \tan^{2} (x)) + 4 (\csc^{2} (x) (- \csc^{2} (x)) + \cot (x) (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [\csc (x)]))$

Paso 3.3.4.2

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 $n = 2$ .

$6 (\sec^{4} (x) + 2 \sec^{2} (x) \tan^{2} (x)) + 4 (\csc^{2} (x) (- \csc^{2} (x)) + \cot (x) (2 u_{2} \frac{d}{d x} [\csc (x)]))$

Paso 3.3.4.3

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

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

Paso 3.3.5

La derivada de $\csc (x)$ con respecto a $x$ es $- \csc (x) \cot (x)$ .

$6 (\sec^{4} (x) + 2 \sec^{2} (x) \tan^{2} (x)) + 4 (\csc^{2} (x) (- \csc^{2} (x)) + \cot (x) (2 \csc (x) (- \csc (x) \cot (x))))$

Paso 3.3.6

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

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Paso 3.3.6.1

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

$6 (\sec^{4} (x) + 2 \sec^{2} (x) \tan^{2} (x)) + 4 (\csc^{2} (x) \csc^{2} (x) \cdot - 1 + \cot (x) (2 \csc (x) (- \csc (x) \cot (x))))$

Paso 3.3.6.2

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

$6 (\sec^{4} (x) + 2 \sec^{2} (x) \tan^{2} (x)) + 4 ({\csc (x)}^{2 + 2} \cdot - 1 + \cot (x) (2 \csc (x) (- \csc (x) \cot (x))))$

Paso 3.3.6.3

Suma $2$ y $2$ .

$6 (\sec^{4} (x) + 2 \sec^{2} (x) \tan^{2} (x)) + 4 (\csc^{4} (x) \cdot - 1 + \cot (x) (2 \csc (x) (- \csc (x) \cot (x))))$

Paso 3.3.7

Mueve $- 1$ a la izquierda de $\csc^{4} (x)$ .

$6 (\sec^{4} (x) + 2 \sec^{2} (x) \tan^{2} (x)) + 4 (- 1 \cdot \csc^{4} (x) + \cot (x) (2 \csc (x) (- \csc (x) \cot (x))))$

Paso 3.3.8

Reescribe $- 1 \csc^{4} (x)$ como $- \csc^{4} (x)$ .

$6 (\sec^{4} (x) + 2 \sec^{2} (x) \tan^{2} (x)) + 4 (- \csc^{4} (x) + \cot (x) (2 \csc (x) (- \csc (x) \cot (x))))$

Paso 3.3.9

Multiplica $- 1$ por $2$ .

$6 (\sec^{4} (x) + 2 \sec^{2} (x) \tan^{2} (x)) + 4 (- \csc^{4} (x) + \cot (x) (- 2 \csc (x) (\csc (x) \cot (x))))$

Paso 3.3.10

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

$6 (\sec^{4} (x) + 2 \sec^{2} (x) \tan^{2} (x)) + 4 (- \csc^{4} (x) + \cot (x) (- 2 (\csc^{1} (x) \csc (x)) \cot (x)))$

Paso 3.3.11

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

$6 (\sec^{4} (x) + 2 \sec^{2} (x) \tan^{2} (x)) + 4 (- \csc^{4} (x) + \cot (x) (- 2 (\csc^{1} (x) \csc^{1} (x)) \cot (x)))$

Paso 3.3.12

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

$6 (\sec^{4} (x) + 2 \sec^{2} (x) \tan^{2} (x)) + 4 (- \csc^{4} (x) + \cot (x) (- 2 {\csc (x)}^{1 + 1} \cot (x)))$

Paso 3.3.13

Suma $1$ y $1$ .

$6 (\sec^{4} (x) + 2 \sec^{2} (x) \tan^{2} (x)) + 4 (- \csc^{4} (x) + \cot (x) (- 2 \csc^{2} (x) \cot (x)))$

Paso 3.3.14

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

$6 (\sec^{4} (x) + 2 \sec^{2} (x) \tan^{2} (x)) + 4 (- \csc^{4} (x) - 2 \csc^{2} (x) (\cot^{1} (x) \cot (x)))$

Paso 3.3.15

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

$6 (\sec^{4} (x) + 2 \sec^{2} (x) \tan^{2} (x)) + 4 (- \csc^{4} (x) - 2 \csc^{2} (x) (\cot^{1} (x) \cot^{1} (x)))$

Paso 3.3.16

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

$6 (\sec^{4} (x) + 2 \sec^{2} (x) \tan^{2} (x)) + 4 (- \csc^{4} (x) - 2 \csc^{2} (x) {\cot (x)}^{1 + 1})$

Paso 3.3.17

Suma $1$ y $1$ .

$6 (\sec^{4} (x) + 2 \sec^{2} (x) \tan^{2} (x)) + 4 (- \csc^{4} (x) - 2 \csc^{2} (x) \cot^{2} (x))$

Paso 3.4

Simplifica.

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Paso 3.4.1

Aplica la propiedad distributiva.

$6 \sec^{4} (x) + 6 (2 \sec^{2} (x) \tan^{2} (x)) + 4 (- \csc^{4} (x) - 2 \csc^{2} (x) \cot^{2} (x))$

Paso 3.4.2

Aplica la propiedad distributiva.

$6 \sec^{4} (x) + 6 (2 \sec^{2} (x) \tan^{2} (x)) + 4 (- \csc^{4} (x)) + 4 (- 2 \csc^{2} (x) \cot^{2} (x))$

Paso 3.4.3

Combina los términos.

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Paso 3.4.3.1

Multiplica $2$ por $6$ .

$6 \sec^{4} (x) + 12 (\sec^{2} (x) \tan^{2} (x)) + 4 (- \csc^{4} (x)) + 4 (- 2 \csc^{2} (x) \cot^{2} (x))$

Paso 3.4.3.2

Multiplica $- 1$ por $4$ .

$6 \sec^{4} (x) + 12 \sec^{2} (x) \tan^{2} (x) - 4 \csc^{4} (x) + 4 (- 2 \csc^{2} (x) \cot^{2} (x))$

Paso 3.4.3.3

Multiplica $- 2$ por $4$ .

$6 \sec^{4} (x) + 12 \sec^{2} (x) \tan^{2} (x) - 4 \csc^{4} (x) - 8 \csc^{2} (x) \cot^{2} (x)$

Paso 3.4.4

Reordena los términos.

$f''' (x) = 12 \sec^{2} (x) \tan^{2} (x) - 8 \cot^{2} (x) \csc^{2} (x) + 6 \sec^{4} (x) - 4 \csc^{4} (x)$

Paso 4

Obtén la cuarta derivada.

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Paso 4.1

Según la regla de la suma, la derivada de $12 \sec^{2} (x) \tan^{2} (x) - 8 \cot^{2} (x) \csc^{2} (x) + 6 \sec^{4} (x) - 4 \csc^{4} (x)$ con respecto a $x$ es $\frac{d}{d x} [12 \sec^{2} (x) \tan^{2} (x)] + \frac{d}{d x} [- 8 \cot^{2} (x) \csc^{2} (x)] + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$ .

$\frac{d}{d x} [12 \sec^{2} (x) \tan^{2} (x)] + \frac{d}{d x} [- 8 \cot^{2} (x) \csc^{2} (x)] + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.2

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

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Paso 4.2.1

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

$12 \frac{d}{d x} [\sec^{2} (x) \tan^{2} (x)] + \frac{d}{d x} [- 8 \cot^{2} (x) \csc^{2} (x)] + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.2.2

$12 (\sec^{2} (x) \frac{d}{d x} [\tan^{2} (x)] + \tan^{2} (x) \frac{d}{d x} [\sec^{2} (x)]) + \frac{d}{d x} [- 8 \cot^{2} (x) \csc^{2} (x)] + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.2.3

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

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Paso 4.2.3.1

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

$12 (\sec^{2} (x) (\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [\tan (x)]) + \tan^{2} (x) \frac{d}{d x} [\sec^{2} (x)]) + \frac{d}{d x} [- 8 \cot^{2} (x) \csc^{2} (x)] + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.2.3.2

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 $n = 2$ .

$12 (\sec^{2} (x) (2 u_{1} \frac{d}{d x} [\tan (x)]) + \tan^{2} (x) \frac{d}{d x} [\sec^{2} (x)]) + \frac{d}{d x} [- 8 \cot^{2} (x) \csc^{2} (x)] + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.2.3.3

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

$12 (\sec^{2} (x) (2 \tan (x) \frac{d}{d x} [\tan (x)]) + \tan^{2} (x) \frac{d}{d x} [\sec^{2} (x)]) + \frac{d}{d x} [- 8 \cot^{2} (x) \csc^{2} (x)] + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.2.4

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

$12 (\sec^{2} (x) (2 \tan (x) \sec^{2} (x)) + \tan^{2} (x) \frac{d}{d x} [\sec^{2} (x)]) + \frac{d}{d x} [- 8 \cot^{2} (x) \csc^{2} (x)] + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

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

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Paso 4.2.5.1

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

$12 (\sec^{2} (x) (2 \tan (x) \sec^{2} (x)) + \tan^{2} (x) (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [\sec (x)])) + \frac{d}{d x} [- 8 \cot^{2} (x) \csc^{2} (x)] + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.2.5.2

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 $n = 2$ .

$12 (\sec^{2} (x) (2 \tan (x) \sec^{2} (x)) + \tan^{2} (x) (2 u_{2} \frac{d}{d x} [\sec (x)])) + \frac{d}{d x} [- 8 \cot^{2} (x) \csc^{2} (x)] + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.2.5.3

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

$12 (\sec^{2} (x) (2 \tan (x) \sec^{2} (x)) + \tan^{2} (x) (2 \sec (x) \frac{d}{d x} [\sec (x)])) + \frac{d}{d x} [- 8 \cot^{2} (x) \csc^{2} (x)] + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.2.6

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

$12 (\sec^{2} (x) (2 \tan (x) \sec^{2} (x)) + \tan^{2} (x) (2 \sec (x) (\sec (x) \tan (x)))) + \frac{d}{d x} [- 8 \cot^{2} (x) \csc^{2} (x)] + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.2.7

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

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Paso 4.2.7.1

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

$12 (\sec^{2} (x) \sec^{2} (x) (2 \tan (x)) + \tan^{2} (x) (2 \sec (x) (\sec (x) \tan (x)))) + \frac{d}{d x} [- 8 \cot^{2} (x) \csc^{2} (x)] + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.2.7.2

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

$12 ({\sec (x)}^{2 + 2} (2 \tan (x)) + \tan^{2} (x) (2 \sec (x) (\sec (x) \tan (x)))) + \frac{d}{d x} [- 8 \cot^{2} (x) \csc^{2} (x)] + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.2.7.3

Suma $2$ y $2$ .

$12 (\sec^{4} (x) (2 \tan (x)) + \tan^{2} (x) (2 \sec (x) (\sec (x) \tan (x)))) + \frac{d}{d x} [- 8 \cot^{2} (x) \csc^{2} (x)] + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.2.8

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

$12 (2 \cdot \sec^{4} (x) \tan (x) + \tan^{2} (x) (2 \sec (x) (\sec (x) \tan (x)))) + \frac{d}{d x} [- 8 \cot^{2} (x) \csc^{2} (x)] + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.2.9

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

$12 (2 \sec^{4} (x) \tan (x) + \tan^{2} (x) (2 (\sec^{1} (x) \sec (x)) \tan (x))) + \frac{d}{d x} [- 8 \cot^{2} (x) \csc^{2} (x)] + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.2.10

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

$12 (2 \sec^{4} (x) \tan (x) + \tan^{2} (x) (2 (\sec^{1} (x) \sec^{1} (x)) \tan (x))) + \frac{d}{d x} [- 8 \cot^{2} (x) \csc^{2} (x)] + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.2.11

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

$12 (2 \sec^{4} (x) \tan (x) + \tan^{2} (x) (2 {\sec (x)}^{1 + 1} \tan (x))) + \frac{d}{d x} [- 8 \cot^{2} (x) \csc^{2} (x)] + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.2.12

Suma $1$ y $1$ .

$12 (2 \sec^{4} (x) \tan (x) + \tan^{2} (x) (2 \sec^{2} (x) \tan (x))) + \frac{d}{d x} [- 8 \cot^{2} (x) \csc^{2} (x)] + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.2.13

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

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Paso 4.2.13.1

Mueve $\tan (x)$ .

$12 (2 \sec^{4} (x) \tan (x) + \tan (x) \tan^{2} (x) (2 \sec^{2} (x))) + \frac{d}{d x} [- 8 \cot^{2} (x) \csc^{2} (x)] + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.2.13.2

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

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Paso 4.2.13.2.1

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

$12 (2 \sec^{4} (x) \tan (x) + \tan^{1} (x) \tan^{2} (x) (2 \sec^{2} (x))) + \frac{d}{d x} [- 8 \cot^{2} (x) \csc^{2} (x)] + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.2.13.2.2

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

$12 (2 \sec^{4} (x) \tan (x) + {\tan (x)}^{1 + 2} (2 \sec^{2} (x))) + \frac{d}{d x} [- 8 \cot^{2} (x) \csc^{2} (x)] + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.2.13.3

Suma $1$ y $2$ .

$12 (2 \sec^{4} (x) \tan (x) + \tan^{3} (x) (2 \sec^{2} (x))) + \frac{d}{d x} [- 8 \cot^{2} (x) \csc^{2} (x)] + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.2.14

Mueve $2$ a la izquierda de $\tan^{3} (x)$ .

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) + \frac{d}{d x} [- 8 \cot^{2} (x) \csc^{2} (x)] + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.3

Evalúa $\frac{d}{d x} [- 8 \cot^{2} (x) \csc^{2} (x)]$ .

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Paso 4.3.1

Como $- 8$ es constante con respecto a $x$ , la derivada de $- 8 \cot^{2} (x) \csc^{2} (x)$ con respecto a $x$ es $- 8 \frac{d}{d x} [\cot^{2} (x) \csc^{2} (x)]$ .

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) - 8 \frac{d}{d x} [\cot^{2} (x) \csc^{2} (x)] + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

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

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{2} (x) \frac{d}{d x} [\csc^{2} (x)] + \csc^{2} (x) \frac{d}{d x} [\cot^{2} (x)]) + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.3.3

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

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Paso 4.3.3.1

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

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{2} (x) (\frac{d}{d u_{3}} [{u_{3}}^{2}] \frac{d}{d x} [\csc (x)]) + \csc^{2} (x) \frac{d}{d x} [\cot^{2} (x)]) + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.3.3.2

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 $n = 2$ .

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{2} (x) (2 u_{3} \frac{d}{d x} [\csc (x)]) + \csc^{2} (x) \frac{d}{d x} [\cot^{2} (x)]) + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.3.3.3

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

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{2} (x) (2 \csc (x) \frac{d}{d x} [\csc (x)]) + \csc^{2} (x) \frac{d}{d x} [\cot^{2} (x)]) + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.3.4

La derivada de $\csc (x)$ con respecto a $x$ es $- \csc (x) \cot (x)$ .

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{2} (x) (2 \csc (x) (- \csc (x) \cot (x))) + \csc^{2} (x) \frac{d}{d x} [\cot^{2} (x)]) + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

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

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Paso 4.3.5.1

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

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{2} (x) (2 \csc (x) (- \csc (x) \cot (x))) + \csc^{2} (x) (\frac{d}{d u_{4}} [{u_{4}}^{2}] \frac{d}{d x} [\cot (x)])) + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.3.5.2

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 $n = 2$ .

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{2} (x) (2 \csc (x) (- \csc (x) \cot (x))) + \csc^{2} (x) (2 u_{4} \frac{d}{d x} [\cot (x)])) + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.3.5.3

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

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{2} (x) (2 \csc (x) (- \csc (x) \cot (x))) + \csc^{2} (x) (2 \cot (x) \frac{d}{d x} [\cot (x)])) + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.3.6

La derivada de $\cot (x)$ con respecto a $x$ es $- \csc^{2} (x)$ .

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{2} (x) (2 \csc (x) (- \csc (x) \cot (x))) + \csc^{2} (x) (2 \cot (x) (- \csc^{2} (x)))) + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.3.7

Multiplica $- 1$ por $2$ .

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{2} (x) (- 2 \csc (x) (\csc (x) \cot (x))) + \csc^{2} (x) (2 \cot (x) (- \csc^{2} (x)))) + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.3.8

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

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{2} (x) (- 2 (\csc^{1} (x) \csc (x)) \cot (x)) + \csc^{2} (x) (2 \cot (x) (- \csc^{2} (x)))) + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.3.9

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

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{2} (x) (- 2 (\csc^{1} (x) \csc^{1} (x)) \cot (x)) + \csc^{2} (x) (2 \cot (x) (- \csc^{2} (x)))) + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.3.10

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

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{2} (x) (- 2 {\csc (x)}^{1 + 1} \cot (x)) + \csc^{2} (x) (2 \cot (x) (- \csc^{2} (x)))) + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.3.11

Suma $1$ y $1$ .

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{2} (x) (- 2 \csc^{2} (x) \cot (x)) + \csc^{2} (x) (2 \cot (x) (- \csc^{2} (x)))) + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.3.12

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

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Paso 4.3.12.1

Mueve $\cot (x)$ .

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot (x) \cot^{2} (x) (- 2 \csc^{2} (x)) + \csc^{2} (x) (2 \cot (x) (- \csc^{2} (x)))) + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.3.12.2

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

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Paso 4.3.12.2.1

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

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{1} (x) \cot^{2} (x) (- 2 \csc^{2} (x)) + \csc^{2} (x) (2 \cot (x) (- \csc^{2} (x)))) + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.3.12.2.2

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

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) - 8 ({\cot (x)}^{1 + 2} (- 2 \csc^{2} (x)) + \csc^{2} (x) (2 \cot (x) (- \csc^{2} (x)))) + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.3.12.3

Suma $1$ y $2$ .

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{3} (x) (- 2 \csc^{2} (x)) + \csc^{2} (x) (2 \cot (x) (- \csc^{2} (x)))) + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.3.13

Multiplica $- 1$ por $2$ .

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{3} (x) (- 2 \csc^{2} (x)) + \csc^{2} (x) (- 2 \cot (x) \csc^{2} (x))) + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.3.14

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

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Paso 4.3.14.1

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

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{3} (x) (- 2 \csc^{2} (x)) + \csc^{2} (x) \csc^{2} (x) (- 2 \cot (x))) + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.3.14.2

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

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{3} (x) (- 2 \csc^{2} (x)) + {\csc (x)}^{2 + 2} (- 2 \cot (x))) + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.3.14.3

Suma $2$ y $2$ .

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{3} (x) (- 2 \csc^{2} (x)) + \csc^{4} (x) (- 2 \cot (x))) + \frac{d}{d x} [6 \sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.4

Evalúa $\frac{d}{d x} [6 \sec^{4} (x)]$ .

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Paso 4.4.1

Como $6$ es constante con respecto a $x$ , la derivada de $6 \sec^{4} (x)$ con respecto a $x$ es $6 \frac{d}{d x} [\sec^{4} (x)]$ .

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{3} (x) (- 2 \csc^{2} (x)) + \csc^{4} (x) (- 2 \cot (x))) + 6 \frac{d}{d x} [\sec^{4} (x)] + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.4.2

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

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Paso 4.4.2.1

Para aplicar la regla de la cadena, establece $u_{5}$ como $\sec (x)$ .

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{3} (x) (- 2 \csc^{2} (x)) + \csc^{4} (x) (- 2 \cot (x))) + 6 (\frac{d}{d u_{5}} [{u_{5}}^{4}] \frac{d}{d x} [\sec (x)]) + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.4.2.2

Diferencia con la regla de la potencia, que establece que $\frac{d}{d u_{5}} [{u_{5}}^{n}]$ es $n {u_{5}}^{n - 1}$ donde $n = 4$ .

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{3} (x) (- 2 \csc^{2} (x)) + \csc^{4} (x) (- 2 \cot (x))) + 6 (4 {u_{5}}^{3} \frac{d}{d x} [\sec (x)]) + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.4.2.3

Reemplaza todos los casos de $u_{5}$ con $\sec (x)$ .

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{3} (x) (- 2 \csc^{2} (x)) + \csc^{4} (x) (- 2 \cot (x))) + 6 (4 \sec^{3} (x) \frac{d}{d x} [\sec (x)]) + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.4.3

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

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{3} (x) (- 2 \csc^{2} (x)) + \csc^{4} (x) (- 2 \cot (x))) + 6 (4 \sec^{3} (x) (\sec (x) \tan (x))) + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.4.4

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

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{3} (x) (- 2 \csc^{2} (x)) + \csc^{4} (x) (- 2 \cot (x))) + 6 (4 (\sec^{1} (x) \sec^{3} (x)) \tan (x)) + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.4.5

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

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{3} (x) (- 2 \csc^{2} (x)) + \csc^{4} (x) (- 2 \cot (x))) + 6 (4 {\sec (x)}^{1 + 3} \tan (x)) + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.4.6

Suma $1$ y $3$ .

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{3} (x) (- 2 \csc^{2} (x)) + \csc^{4} (x) (- 2 \cot (x))) + 6 (4 \sec^{4} (x) \tan (x)) + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.4.7

Multiplica $4$ por $6$ .

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{3} (x) (- 2 \csc^{2} (x)) + \csc^{4} (x) (- 2 \cot (x))) + 24 \sec^{4} (x) \tan (x) + \frac{d}{d x} [- 4 \csc^{4} (x)]$

Paso 4.5

Evalúa $\frac{d}{d x} [- 4 \csc^{4} (x)]$ .

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Paso 4.5.1

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

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{3} (x) (- 2 \csc^{2} (x)) + \csc^{4} (x) (- 2 \cot (x))) + 24 \sec^{4} (x) \tan (x) - 4 \frac{d}{d x} [\csc^{4} (x)]$

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 $f (x) = x^{4}$ y $g (x) = \csc (x)$ .

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Paso 4.5.2.1

Para aplicar la regla de la cadena, establece $u_{6}$ como $\csc (x)$ .

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{3} (x) (- 2 \csc^{2} (x)) + \csc^{4} (x) (- 2 \cot (x))) + 24 \sec^{4} (x) \tan (x) - 4 (\frac{d}{d u_{6}} [{u_{6}}^{4}] \frac{d}{d x} [\csc (x)])$

Paso 4.5.2.2

Diferencia con la regla de la potencia, que establece que $\frac{d}{d u_{6}} [{u_{6}}^{n}]$ es $n {u_{6}}^{n - 1}$ donde $n = 4$ .

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{3} (x) (- 2 \csc^{2} (x)) + \csc^{4} (x) (- 2 \cot (x))) + 24 \sec^{4} (x) \tan (x) - 4 (4 {u_{6}}^{3} \frac{d}{d x} [\csc (x)])$

Paso 4.5.2.3

Reemplaza todos los casos de $u_{6}$ con $\csc (x)$ .

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{3} (x) (- 2 \csc^{2} (x)) + \csc^{4} (x) (- 2 \cot (x))) + 24 \sec^{4} (x) \tan (x) - 4 (4 \csc^{3} (x) \frac{d}{d x} [\csc (x)])$

Paso 4.5.3

La derivada de $\csc (x)$ con respecto a $x$ es $- \csc (x) \cot (x)$ .

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{3} (x) (- 2 \csc^{2} (x)) + \csc^{4} (x) (- 2 \cot (x))) + 24 \sec^{4} (x) \tan (x) - 4 (4 \csc^{3} (x) (- \csc (x) \cot (x)))$

Paso 4.5.4

Multiplica $- 1$ por $4$ .

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{3} (x) (- 2 \csc^{2} (x)) + \csc^{4} (x) (- 2 \cot (x))) + 24 \sec^{4} (x) \tan (x) - 4 (- 4 \csc^{3} (x) (\csc (x) \cot (x)))$

Paso 4.5.5

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

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{3} (x) (- 2 \csc^{2} (x)) + \csc^{4} (x) (- 2 \cot (x))) + 24 \sec^{4} (x) \tan (x) - 4 (- 4 (\csc^{1} (x) \csc^{3} (x)) \cot (x))$

Paso 4.5.6

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

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{3} (x) (- 2 \csc^{2} (x)) + \csc^{4} (x) (- 2 \cot (x))) + 24 \sec^{4} (x) \tan (x) - 4 (- 4 {\csc (x)}^{1 + 3} \cot (x))$

Paso 4.5.7

Suma $1$ y $3$ .

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{3} (x) (- 2 \csc^{2} (x)) + \csc^{4} (x) (- 2 \cot (x))) + 24 \sec^{4} (x) \tan (x) - 4 (- 4 \csc^{4} (x) \cot (x))$

Paso 4.5.8

Multiplica $- 4$ por $- 4$ .

$12 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{3} (x) (- 2 \csc^{2} (x)) + \csc^{4} (x) (- 2 \cot (x))) + 24 \sec^{4} (x) \tan (x) + 16 \csc^{4} (x) \cot (x)$

Paso 4.6

Simplifica.

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Paso 4.6.1

Aplica la propiedad distributiva.

$12 (2 \sec^{4} (x) \tan (x)) + 12 (2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{3} (x) (- 2 \csc^{2} (x)) + \csc^{4} (x) (- 2 \cot (x))) + 24 \sec^{4} (x) \tan (x) + 16 \csc^{4} (x) \cot (x)$

Paso 4.6.2

Aplica la propiedad distributiva.

$12 (2 \sec^{4} (x) \tan (x)) + 12 (2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{3} (x) (- 2 \csc^{2} (x))) - 8 (\csc^{4} (x) (- 2 \cot (x))) + 24 \sec^{4} (x) \tan (x) + 16 \csc^{4} (x) \cot (x)$

Paso 4.6.3

Combina los términos.

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Paso 4.6.3.1

Multiplica $2$ por $12$ .

$24 (\sec^{4} (x) \tan (x)) + 12 (2 \tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{3} (x) (- 2 \csc^{2} (x))) - 8 (\csc^{4} (x) (- 2 \cot (x))) + 24 \sec^{4} (x) \tan (x) + 16 \csc^{4} (x) \cot (x)$

Paso 4.6.3.2

Multiplica $2$ por $12$ .

$24 \sec^{4} (x) \tan (x) + 24 (\tan^{3} (x) \sec^{2} (x)) - 8 (\cot^{3} (x) (- 2 \csc^{2} (x))) - 8 (\csc^{4} (x) (- 2 \cot (x))) + 24 \sec^{4} (x) \tan (x) + 16 \csc^{4} (x) \cot (x)$

Paso 4.6.3.3

Multiplica $- 2$ por $- 8$ .

$24 \sec^{4} (x) \tan (x) + 24 \tan^{3} (x) \sec^{2} (x) + 16 (\cot^{3} (x) (\csc^{2} (x))) - 8 (\csc^{4} (x) (- 2 \cot (x))) + 24 \sec^{4} (x) \tan (x) + 16 \csc^{4} (x) \cot (x)$

Paso 4.6.3.4

Multiplica $- 2$ por $- 8$ .

$24 \sec^{4} (x) \tan (x) + 24 \tan^{3} (x) \sec^{2} (x) + 16 \cot^{3} (x) \csc^{2} (x) + 16 (\csc^{4} (x) (\cot (x))) + 24 \sec^{4} (x) \tan (x) + 16 \csc^{4} (x) \cot (x)$

Paso 4.6.3.5

Suma $24 \sec^{4} (x) \tan (x)$ y $24 \sec^{4} (x) \tan (x)$ .

$48 \sec^{4} (x) \tan (x) + 24 \tan^{3} (x) \sec^{2} (x) + 16 \cot^{3} (x) \csc^{2} (x) + 16 \csc^{4} (x) \cot (x) + 16 \csc^{4} (x) \cot (x)$

Paso 4.6.3.6

Suma $16 \csc^{4} (x) \cot (x)$ y $16 \csc^{4} (x) \cot (x)$ .

$f^{4} (x) = 48 \sec^{4} (x) \tan (x) + 24 \tan^{3} (x) \sec^{2} (x) + 16 \cot^{3} (x) \csc^{2} (x) + 32 \csc^{4} (x) \cot (x)$

Paso 5

La cuarta derivada de $f (x)$ con respecto a $x$ es $48 \sec^{4} (x) \tan (x) + 24 \tan^{3} (x) \sec^{2} (x) + 16 \cot^{3} (x) \csc^{2} (x) + 32 \csc^{4} (x) \cot (x)$ .

$48 \sec^{4} (x) \tan (x) + 24 \tan^{3} (x) \sec^{2} (x) + 16 \cot^{3} (x) \csc^{2} (x) + 32 \csc^{4} (x) \cot (x)$