Cálculo Ejemplos

$f (x) = \frac{\sec (x)}{2 - \cos (x)}$

Paso 1

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

$\frac{(2 - \cos (x)) \frac{d}{d x} [\sec (x)] - \sec (x) \frac{d}{d x} [2 - \cos (x)]}{{(2 - \cos (x))}^{2}}$

Paso 2

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

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

Paso 3

Diferencia.

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Paso 3.1

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

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

Paso 3.2

Como $2$ es constante con respecto a $x$ , la derivada de $2$ con respecto a $x$ es $0$ .

$\frac{(2 - \cos (x)) \sec (x) \tan (x) - \sec (x) (0 + \frac{d}{d x} [- \cos (x)])}{{(2 - \cos (x))}^{2}}$

Paso 3.3

Suma $0$ y $\frac{d}{d x} [- \cos (x)]$ .

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

Paso 3.4

Como $- 1$ es constante con respecto a $x$ , la derivada de $- \cos (x)$ con respecto a $x$ es $- \frac{d}{d x} [\cos (x)]$ .

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

Paso 3.5

Multiplica.

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Paso 3.5.1

Multiplica $- 1$ por $- 1$ .

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

Paso 3.5.2

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

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

Paso 4

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

$\frac{(2 - \cos (x)) \sec (x) \tan (x) + \sec (x) (- \sin (x))}{{(2 - \cos (x))}^{2}}$

Paso 5

Simplifica.

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Paso 5.1

Aplica la propiedad distributiva.

$\frac{(2 \sec (x) - \cos (x) \sec (x)) \tan (x) + \sec (x) (- \sin (x))}{{(2 - \cos (x))}^{2}}$

Paso 5.2

Aplica la propiedad distributiva.

$\frac{2 \sec (x) \tan (x) - \cos (x) \sec (x) \tan (x) + \sec (x) (- \sin (x))}{{(2 - \cos (x))}^{2}}$

Paso 5.3

Simplifica el numerador.

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Paso 5.3.1

Simplifica cada término.

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Paso 5.3.1.1

Reescribe $\sec (x)$ en términos de senos y cosenos.

$\frac{2 \frac{1}{\cos (x)} \tan (x) - \cos (x) \sec (x) \tan (x) + \sec (x) (- \sin (x))}{{(2 - \cos (x))}^{2}}$

Paso 5.3.1.2

Combina $2$ y $\frac{1}{\cos (x)}$ .

$\frac{\frac{2}{\cos (x)} \tan (x) - \cos (x) \sec (x) \tan (x) + \sec (x) (- \sin (x))}{{(2 - \cos (x))}^{2}}$

Paso 5.3.1.3

Reescribe $\tan (x)$ en términos de senos y cosenos.

$\frac{\frac{2}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)} - \cos (x) \sec (x) \tan (x) + \sec (x) (- \sin (x))}{{(2 - \cos (x))}^{2}}$

Paso 5.3.1.4

Multiplica $\frac{2}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$ .

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Paso 5.3.1.4.1

Multiplica $\frac{2}{\cos (x)}$ por $\frac{\sin (x)}{\cos (x)}$ .

$\frac{\frac{2 \sin (x)}{\cos (x) \cos (x)} - \cos (x) \sec (x) \tan (x) + \sec (x) (- \sin (x))}{{(2 - \cos (x))}^{2}}$

Paso 5.3.1.4.2

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

$\frac{\frac{2 \sin (x)}{\cos^{1} (x) \cos (x)} - \cos (x) \sec (x) \tan (x) + \sec (x) (- \sin (x))}{{(2 - \cos (x))}^{2}}$

Paso 5.3.1.4.3

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

$\frac{\frac{2 \sin (x)}{\cos^{1} (x) \cos^{1} (x)} - \cos (x) \sec (x) \tan (x) + \sec (x) (- \sin (x))}{{(2 - \cos (x))}^{2}}$

Paso 5.3.1.4.4

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

$\frac{\frac{2 \sin (x)}{{\cos (x)}^{1 + 1}} - \cos (x) \sec (x) \tan (x) + \sec (x) (- \sin (x))}{{(2 - \cos (x))}^{2}}$

Paso 5.3.1.4.5

Suma $1$ y $1$ .

$\frac{\frac{2 \sin (x)}{\cos^{2} (x)} - \cos (x) \sec (x) \tan (x) + \sec (x) (- \sin (x))}{{(2 - \cos (x))}^{2}}$

Paso 5.3.1.5

Reescribe en términos de senos y cosenos, luego, cancela los factores comunes.

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Paso 5.3.1.5.1

Agrega paréntesis.

$\frac{\frac{2 \sin (x)}{\cos^{2} (x)} - (\cos (x) \sec (x)) \tan (x) + \sec (x) (- \sin (x))}{{(2 - \cos (x))}^{2}}$

Paso 5.3.1.5.2

Reordena $\cos (x)$ y $\sec (x)$ .

$\frac{\frac{2 \sin (x)}{\cos^{2} (x)} - (\sec (x) \cos (x)) \tan (x) + \sec (x) (- \sin (x))}{{(2 - \cos (x))}^{2}}$

Paso 5.3.1.5.3

Reescribe $- \cos (x) \sec (x)$ en términos de senos y cosenos.

$\frac{\frac{2 \sin (x)}{\cos^{2} (x)} - (\frac{1}{\cos (x)} \cos (x)) \tan (x) + \sec (x) (- \sin (x))}{{(2 - \cos (x))}^{2}}$

Paso 5.3.1.5.4

Cancela los factores comunes.

$\frac{\frac{2 \sin (x)}{\cos^{2} (x)} - 1 \cdot 1 \tan (x) + \sec (x) (- \sin (x))}{{(2 - \cos (x))}^{2}}$

Paso 5.3.1.6

Multiplica $- 1$ por $1$ .

$\frac{\frac{2 \sin (x)}{\cos^{2} (x)} - 1 \tan (x) + \sec (x) (- \sin (x))}{{(2 - \cos (x))}^{2}}$

Paso 5.3.1.7

Reescribe $\tan (x)$ en términos de senos y cosenos.

$\frac{\frac{2 \sin (x)}{\cos^{2} (x)} - 1 \frac{\sin (x)}{\cos (x)} + \sec (x) (- \sin (x))}{{(2 - \cos (x))}^{2}}$

Paso 5.3.1.8

Reescribe $- 1 \frac{\sin (x)}{\cos (x)}$ como $- \frac{\sin (x)}{\cos (x)}$ .

$\frac{\frac{2 \sin (x)}{\cos^{2} (x)} - \frac{\sin (x)}{\cos (x)} + \sec (x) (- \sin (x))}{{(2 - \cos (x))}^{2}}$

Paso 5.3.1.9

Reescribe con la propiedad conmutativa de la multiplicación.

$\frac{\frac{2 \sin (x)}{\cos^{2} (x)} - \frac{\sin (x)}{\cos (x)} - \sec (x) \sin (x)}{{(2 - \cos (x))}^{2}}$

Paso 5.3.1.10

Reescribe $\sec (x)$ en términos de senos y cosenos.

$\frac{\frac{2 \sin (x)}{\cos^{2} (x)} - \frac{\sin (x)}{\cos (x)} - \frac{1}{\cos (x)} \sin (x)}{{(2 - \cos (x))}^{2}}$

Paso 5.3.1.11

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

$\frac{\frac{2 \sin (x)}{\cos^{2} (x)} - \frac{\sin (x)}{\cos (x)} - \frac{\sin (x)}{\cos (x)}}{{(2 - \cos (x))}^{2}}$

Paso 5.3.2

Combina los numeradores sobre el denominador común.

$\frac{\frac{2 \sin (x)}{\cos^{2} (x)} + \frac{- \sin (x) - \sin (x)}{\cos (x)}}{{(2 - \cos (x))}^{2}}$

Paso 5.3.3

Resta $\sin (x)$ de $- \sin (x)$ .

$\frac{\frac{2 \sin (x)}{\cos^{2} (x)} + \frac{- 2 \sin (x)}{\cos (x)}}{{(2 - \cos (x))}^{2}}$

Paso 5.3.4

Mueve el negativo al frente de la fracción.

$\frac{\frac{2 \sin (x)}{\cos^{2} (x)} - \frac{2 \sin (x)}{\cos (x)}}{{(2 - \cos (x))}^{2}}$

Paso 5.3.5

Simplifica cada término.

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Paso 5.3.5.1

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

$\frac{\frac{2 \sin (x)}{\cos (x) \cos (x)} - \frac{2 \sin (x)}{\cos (x)}}{{(2 - \cos (x))}^{2}}$

Paso 5.3.5.2

Separa las fracciones.

$\frac{\frac{2}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)} - \frac{2 \sin (x)}{\cos (x)}}{{(2 - \cos (x))}^{2}}$

Paso 5.3.5.3

Convierte de $\frac{\sin (x)}{\cos (x)}$ a $\tan (x)$ .

$\frac{\frac{2}{\cos (x)} \tan (x) - \frac{2 \sin (x)}{\cos (x)}}{{(2 - \cos (x))}^{2}}$

Paso 5.3.5.4

Separa las fracciones.

$\frac{\frac{2}{1} \cdot \frac{1}{\cos (x)} \tan (x) - \frac{2 \sin (x)}{\cos (x)}}{{(2 - \cos (x))}^{2}}$

Paso 5.3.5.5

Convierte de $\frac{1}{\cos (x)}$ a $\sec (x)$ .

$\frac{\frac{2}{1} \sec (x) \tan (x) - \frac{2 \sin (x)}{\cos (x)}}{{(2 - \cos (x))}^{2}}$

Paso 5.3.5.6

Divide $2$ por $1$ .

$\frac{2 \sec (x) \tan (x) - \frac{2 \sin (x)}{\cos (x)}}{{(2 - \cos (x))}^{2}}$

Paso 5.3.5.7

Separa las fracciones.

$\frac{2 \sec (x) \tan (x) - (\frac{2}{1} \cdot \frac{\sin (x)}{\cos (x)})}{{(2 - \cos (x))}^{2}}$

Paso 5.3.5.8

Convierte de $\frac{\sin (x)}{\cos (x)}$ a $\tan (x)$ .

$\frac{2 \sec (x) \tan (x) - (\frac{2}{1} \tan (x))}{{(2 - \cos (x))}^{2}}$

Paso 5.3.5.9

Divide $2$ por $1$ .

$\frac{2 \sec (x) \tan (x) - (2 \tan (x))}{{(2 - \cos (x))}^{2}}$

Paso 5.3.5.10

Multiplica $2$ por $- 1$ .

$\frac{2 \sec (x) \tan (x) - 2 \tan (x)}{{(2 - \cos (x))}^{2}}$

Paso 5.4

Simplifica el numerador.

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Paso 5.4.1

Factoriza $2 \tan (x)$ de $2 \sec (x) \tan (x) - 2 \tan (x)$ .

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Paso 5.4.1.1

Factoriza $2 \tan (x)$ de $2 \sec (x) \tan (x)$ .

$\frac{2 \tan (x) (\sec (x)) - 2 \tan (x)}{{(2 - \cos (x))}^{2}}$

Paso 5.4.1.2

Factoriza $2 \tan (x)$ de $- 2 \tan (x)$ .

$\frac{2 \tan (x) (\sec (x)) + 2 \tan (x) (- 1)}{{(2 - \cos (x))}^{2}}$

Paso 5.4.1.3

Factoriza $2 \tan (x)$ de $2 \tan (x) (\sec (x)) + 2 \tan (x) (- 1)$ .

$\frac{2 \tan (x) (\sec (x) - 1)}{{(2 - \cos (x))}^{2}}$

Paso 5.4.2

Reescribe $\sec (x)$ en términos de senos y cosenos.

$\frac{2 \tan (x) (\frac{1}{\cos (x)} - 1)}{{(2 - \cos (x))}^{2}}$

Paso 5.4.3

Reescribe $\tan (x)$ en términos de senos y cosenos.

$\frac{2 \frac{\sin (x)}{\cos (x)} (\frac{1}{\cos (x)} - 1)}{{(2 - \cos (x))}^{2}}$

Paso 5.4.4

Combina $2$ y $\frac{\sin (x)}{\cos (x)}$ .

$\frac{\frac{2 \sin (x)}{\cos (x)} (\frac{1}{\cos (x)} - 1)}{{(2 - \cos (x))}^{2}}$

Paso 5.5

Factoriza $\frac{2 \sin (x)}{\cos (x)}$ de $\frac{\frac{2 \sin (x)}{\cos (x)} (\frac{1}{\cos (x)} - 1)}{{(2 - \cos (x))}^{2}}$ .

$\frac{2 \sin (x)}{\cos (x)} \cdot \frac{\frac{1}{\cos (x)} - 1}{{(2 - \cos (x))}^{2}}$

Paso 5.6

Multiplica $\frac{2 \sin (x)}{\cos (x)}$ por $\frac{\frac{1}{\cos (x)} - 1}{{(2 - \cos (x))}^{2}}$ .

$\frac{2 \sin (x) (\frac{1}{\cos (x)} - 1)}{\cos (x) {(2 - \cos (x))}^{2}}$

Paso 5.7

Separa las fracciones.

$\frac{2 (\frac{1}{\cos (x)} - 1)}{{(2 - \cos (x))}^{2}} \cdot \frac{\sin (x)}{\cos (x)}$

Paso 5.8

Convierte de $\frac{\sin (x)}{\cos (x)}$ a $\tan (x)$ .

$\frac{2 (\frac{1}{\cos (x)} - 1)}{{(2 - \cos (x))}^{2}} \tan (x)$

Paso 5.9

Convierte de $\frac{1}{\cos (x)}$ a $\sec (x)$ .

$\frac{2 (\sec (x) - 1)}{{(2 - \cos (x))}^{2}} \tan (x)$

Paso 5.10

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

$\frac{2 (\sec (x) - 1) \tan (x)}{{(2 - \cos (x))}^{2}}$

Paso 5.11

Reordena los factores en $\frac{2 (\sec (x) - 1) \tan (x)}{{(2 - \cos (x))}^{2}}$ .

$\frac{2 \tan (x) (\sec (x) - 1)}{{(2 - \cos (x))}^{2}}$