Cálculo Ejemplos

$5 \csc (x) \sec (x)$

Paso 1

Obtén la primera derivada.

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Paso 1.1

Como $5$ es constante con respecto a $x$ , la derivada de $5 \csc (x) \sec (x)$ con respecto a $x$ es $5 \frac{d}{d x} [\csc (x) \sec (x)]$ .

$5 \frac{d}{d x} [\csc (x) \sec (x)]$

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

$5 (\csc (x) \frac{d}{d x} [\sec (x)] + \sec (x) \frac{d}{d x} [\csc (x)])$

Paso 1.3

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

$5 (\csc (x) (\sec (x) \tan (x)) + \sec (x) \frac{d}{d x} [\csc (x)])$

Paso 1.4

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

$5 (\csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot (x)))$

Paso 1.5

Simplifica.

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Paso 1.5.1

Aplica la propiedad distributiva.

$5 (\csc (x) \sec (x) \tan (x)) + 5 (\sec (x) (- \csc (x) \cot (x)))$

Paso 1.5.2

Multiplica $- 1$ por $5$ .

$5 \csc (x) \sec (x) \tan (x) - 5 \sec (x) \csc (x) \cot (x)$

Paso 1.5.3

Reordena los términos.

$f' (x) = 5 \csc (x) \sec (x) \tan (x) - 5 \cot (x) \csc (x) \sec (x)$

Paso 2

Obtener la segunda derivada.

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Paso 2.1

Según la regla de la suma, la derivada de $5 \csc (x) \sec (x) \tan (x) - 5 \cot (x) \csc (x) \sec (x)$ con respecto a $x$ es $\frac{d}{d x} [5 \csc (x) \sec (x) \tan (x)] + \frac{d}{d x} [- 5 \cot (x) \csc (x) \sec (x)]$ .

$\frac{d}{d x} [5 \csc (x) \sec (x) \tan (x)] + \frac{d}{d x} [- 5 \cot (x) \csc (x) \sec (x)]$

Paso 2.2

Evalúa $\frac{d}{d x} [5 \csc (x) \sec (x) \tan (x)]$ .

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Paso 2.2.1

Como $5$ es constante con respecto a $x$ , la derivada de $5 \csc (x) \sec (x) \tan (x)$ con respecto a $x$ es $5 \frac{d}{d x} [\csc (x) \sec (x) \tan (x)]$ .

$5 \frac{d}{d x} [\csc (x) \sec (x) \tan (x)] + \frac{d}{d x} [- 5 \cot (x) \csc (x) \sec (x)]$

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

$5 (\csc (x) \sec (x) \frac{d}{d x} [\tan (x)] + \tan (x) \frac{d}{d x} [\csc (x) \sec (x)]) + \frac{d}{d x} [- 5 \cot (x) \csc (x) \sec (x)]$

Paso 2.2.3

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

$5 (\csc (x) \sec (x) \sec^{2} (x) + \tan (x) \frac{d}{d x} [\csc (x) \sec (x)]) + \frac{d}{d x} [- 5 \cot (x) \csc (x) \sec (x)]$

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

$5 (\csc (x) \sec (x) \sec^{2} (x) + \tan (x) (\csc (x) \frac{d}{d x} [\sec (x)] + \sec (x) \frac{d}{d x} [\csc (x)])) + \frac{d}{d x} [- 5 \cot (x) \csc (x) \sec (x)]$

Paso 2.2.5

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

$5 (\csc (x) \sec (x) \sec^{2} (x) + \tan (x) (\csc (x) (\sec (x) \tan (x)) + \sec (x) \frac{d}{d x} [\csc (x)])) + \frac{d}{d x} [- 5 \cot (x) \csc (x) \sec (x)]$

Paso 2.2.6

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

$5 (\csc (x) \sec (x) \sec^{2} (x) + \tan (x) (\csc (x) (\sec (x) \tan (x)) + \sec (x) (- \csc (x) \cot (x)))) + \frac{d}{d x} [- 5 \cot (x) \csc (x) \sec (x)]$

Paso 2.2.7

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

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Paso 2.2.7.1

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

$5 (\csc (x) (\sec^{2} (x) \sec (x)) + \tan (x) (\csc (x) (\sec (x) \tan (x)) + \sec (x) (- \csc (x) \cot (x)))) + \frac{d}{d x} [- 5 \cot (x) \csc (x) \sec (x)]$

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 $1$ .

$5 (\csc (x) (\sec^{2} (x) \sec^{1} (x)) + \tan (x) (\csc (x) (\sec (x) \tan (x)) + \sec (x) (- \csc (x) \cot (x)))) + \frac{d}{d x} [- 5 \cot (x) \csc (x) \sec (x)]$

Paso 2.2.7.2.2

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

$5 (\csc (x) {\sec (x)}^{2 + 1} + \tan (x) (\csc (x) (\sec (x) \tan (x)) + \sec (x) (- \csc (x) \cot (x)))) + \frac{d}{d x} [- 5 \cot (x) \csc (x) \sec (x)]$

Paso 2.2.7.3

Suma $2$ y $1$ .

$5 (\csc (x) \sec^{3} (x) + \tan (x) (\csc (x) (\sec (x) \tan (x)) + \sec (x) (- \csc (x) \cot (x)))) + \frac{d}{d x} [- 5 \cot (x) \csc (x) \sec (x)]$

$5 (\csc (x) \sec^{3} (x) + \tan (x) (\csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot (x)))) + \frac{d}{d x} [- 5 \cot (x) \csc (x) \sec (x)]$

Paso 2.3

Evalúa $\frac{d}{d x} [- 5 \cot (x) \csc (x) \sec (x)]$ .

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Paso 2.3.1

Como $- 5$ es constante con respecto a $x$ , la derivada de $- 5 \cot (x) \csc (x) \sec (x)$ con respecto a $x$ es $- 5 \frac{d}{d x} [\cot (x) \csc (x) \sec (x)]$ .

$5 (\csc (x) \sec^{3} (x) + \tan (x) (\csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot (x)))) - 5 \frac{d}{d x} [\cot (x) \csc (x) \sec (x)]$

Paso 2.3.2

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

$5 (\csc (x) \sec^{3} (x) + \tan (x) (\csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot (x)))) - 5 (\cot (x) \csc (x) \frac{d}{d x} [\sec (x)] + \sec (x) \frac{d}{d x} [\cot (x) \csc (x)])$

Paso 2.3.3

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

$5 (\csc (x) \sec^{3} (x) + \tan (x) (\csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot (x)))) - 5 (\cot (x) \csc (x) (\sec (x) \tan (x)) + \sec (x) \frac{d}{d x} [\cot (x) \csc (x)])$

Paso 2.3.4

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 $f (x) = \cot (x)$ y $g (x) = \csc (x)$ .

$5 (\csc (x) \sec^{3} (x) + \tan (x) (\csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot (x)))) - 5 (\cot (x) \csc (x) (\sec (x) \tan (x)) + \sec (x) (\cot (x) \frac{d}{d x} [\csc (x)] + \csc (x) \frac{d}{d x} [\cot (x)]))$

Paso 2.3.5

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

$5 (\csc (x) \sec^{3} (x) + \tan (x) (\csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot (x)))) - 5 (\cot (x) \csc (x) (\sec (x) \tan (x)) + \sec (x) (\cot (x) (- \csc (x) \cot (x)) + \csc (x) \frac{d}{d x} [\cot (x)]))$

Paso 2.3.6

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

Paso 2.3.7

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

$5 (\csc (x) \sec^{3} (x) + \tan (x) (\csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot (x)))) - 5 (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) (\cot^{1} (x) \cot (x)) + \csc (x) (- \csc^{2} (x))))$

Paso 2.3.8

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

$5 (\csc (x) \sec^{3} (x) + \tan (x) (\csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot (x)))) - 5 (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) (\cot^{1} (x) \cot^{1} (x)) + \csc (x) (- \csc^{2} (x))))$

Paso 2.3.9

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

$5 (\csc (x) \sec^{3} (x) + \tan (x) (\csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot (x)))) - 5 (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) {\cot (x)}^{1 + 1} + \csc (x) (- \csc^{2} (x))))$

Paso 2.3.10

Suma $1$ y $1$ .

$5 (\csc (x) \sec^{3} (x) + \tan (x) (\csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot (x)))) - 5 (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot^{2} (x) + \csc (x) (- \csc^{2} (x))))$

Paso 2.3.11

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

$5 (\csc (x) \sec^{3} (x) + \tan (x) (\csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot (x)))) - 5 (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot^{2} (x) - (\csc^{1} (x) \csc^{2} (x))))$

Paso 2.3.12

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

$5 (\csc (x) \sec^{3} (x) + \tan (x) (\csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot (x)))) - 5 (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot^{2} (x) - {\csc (x)}^{1 + 2}))$

Paso 2.3.13

Suma $1$ y $2$ .

$5 (\csc (x) \sec^{3} (x) + \tan (x) (\csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot (x)))) - 5 (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot^{2} (x) - \csc^{3} (x)))$

Paso 2.4

Simplifica.

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Paso 2.4.1

Aplica la propiedad distributiva.

$5 (\csc (x) \sec^{3} (x) + \tan (x) (\csc (x) \sec (x) \tan (x)) + \tan (x) (\sec (x) (- \csc (x) \cot (x)))) - 5 (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot^{2} (x) - \csc^{3} (x)))$

Paso 2.4.2

Aplica la propiedad distributiva.

$5 (\csc (x) \sec^{3} (x)) + 5 (\tan (x) (\csc (x) \sec (x) \tan (x))) + 5 (\tan (x) (\sec (x) (- \csc (x) \cot (x)))) - 5 (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot^{2} (x) - \csc^{3} (x)))$

Paso 2.4.3

Aplica la propiedad distributiva.

Paso 2.4.4

Aplica la propiedad distributiva.

$5 (\csc (x) \sec^{3} (x)) + 5 (\tan (x) (\csc (x) \sec (x) \tan (x))) + 5 (\tan (x) (\sec (x) (- \csc (x) \cot (x)))) - 5 (\cot (x) \csc (x) \sec (x) \tan (x)) - 5 (\sec (x) (- \csc (x) \cot^{2} (x))) - 5 (\sec (x) (- \csc^{3} (x)))$

Paso 2.4.5

Combina los términos.

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Paso 2.4.5.1

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

$5 \csc (x) \sec^{3} (x) + 5 (\tan^{1} (x) \tan (x) (\csc (x) \sec (x))) + 5 (\tan (x) (\sec (x) (- \csc (x) \cot (x)))) - 5 (\cot (x) \csc (x) \sec (x) \tan (x)) - 5 (\sec (x) (- \csc (x) \cot^{2} (x))) - 5 (\sec (x) (- \csc^{3} (x)))$

Paso 2.4.5.2

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

$5 \csc (x) \sec^{3} (x) + 5 (\tan^{1} (x) \tan^{1} (x) (\csc (x) \sec (x))) + 5 (\tan (x) (\sec (x) (- \csc (x) \cot (x)))) - 5 (\cot (x) \csc (x) \sec (x) \tan (x)) - 5 (\sec (x) (- \csc (x) \cot^{2} (x))) - 5 (\sec (x) (- \csc^{3} (x)))$

Paso 2.4.5.3

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

$5 \csc (x) \sec^{3} (x) + 5 ({\tan (x)}^{1 + 1} (\csc (x) \sec (x))) + 5 (\tan (x) (\sec (x) (- \csc (x) \cot (x)))) - 5 (\cot (x) \csc (x) \sec (x) \tan (x)) - 5 (\sec (x) (- \csc (x) \cot^{2} (x))) - 5 (\sec (x) (- \csc^{3} (x)))$

Paso 2.4.5.4

Suma $1$ y $1$ .

$5 \csc (x) \sec^{3} (x) + 5 (\tan^{2} (x) (\csc (x) \sec (x))) + 5 (\tan (x) (\sec (x) (- \csc (x) \cot (x)))) - 5 (\cot (x) \csc (x) \sec (x) \tan (x)) - 5 (\sec (x) (- \csc (x) \cot^{2} (x))) - 5 (\sec (x) (- \csc^{3} (x)))$

Paso 2.4.5.5

Multiplica $- 1$ por $5$ .

$5 \csc (x) \sec^{3} (x) + 5 \tan^{2} (x) \csc (x) \sec (x) - 5 (\tan (x) \sec (x) (\csc (x) \cot (x))) - 5 (\cot (x) \csc (x) \sec (x) \tan (x)) - 5 (\sec (x) (- \csc (x) \cot^{2} (x))) - 5 (\sec (x) (- \csc^{3} (x)))$

Paso 2.4.5.6

Multiplica $- 1$ por $- 5$ .

$5 \csc (x) \sec^{3} (x) + 5 \tan^{2} (x) \csc (x) \sec (x) - 5 \tan (x) \sec (x) \csc (x) \cot (x) - 5 \cot (x) \csc (x) \sec (x) \tan (x) + 5 (\sec (x) (\csc (x) \cot^{2} (x))) - 5 (\sec (x) (- \csc^{3} (x)))$

Paso 2.4.5.7

Multiplica $- 1$ por $- 5$ .

$5 \csc (x) \sec^{3} (x) + 5 \tan^{2} (x) \csc (x) \sec (x) - 5 \tan (x) \sec (x) \csc (x) \cot (x) - 5 \cot (x) \csc (x) \sec (x) \tan (x) + 5 \sec (x) \csc (x) \cot^{2} (x) + 5 (\sec (x) (\csc^{3} (x)))$

Paso 2.4.5.8

Reordena los factores de $- 5 \tan (x) \sec (x) \csc (x) \cot (x)$ .

$5 \csc (x) \sec^{3} (x) + 5 \tan^{2} (x) \csc (x) \sec (x) - 5 \cot (x) \csc (x) \sec (x) \tan (x) - 5 \cot (x) \csc (x) \sec (x) \tan (x) + 5 \sec (x) \csc (x) \cot^{2} (x) + 5 \sec (x) \csc^{3} (x)$

Paso 2.4.5.9

Resta $5 \cot (x) \csc (x) \sec (x) \tan (x)$ de $- 5 \cot (x) \csc (x) \sec (x) \tan (x)$ .

$5 \csc (x) \sec^{3} (x) + 5 \tan^{2} (x) \csc (x) \sec (x) - 10 \cot (x) \csc (x) \sec (x) \tan (x) + 5 \sec (x) \csc (x) \cot^{2} (x) + 5 \sec (x) \csc^{3} (x)$

Paso 2.4.6

Reordena los términos.

$f'' (x) = 5 \sec^{3} (x) \csc (x) + 5 \tan^{2} (x) \csc (x) \sec (x) - 10 \cot (x) \csc (x) \sec (x) \tan (x) + 5 \cot^{2} (x) \csc (x) \sec (x) + 5 \csc^{3} (x) \sec (x)$