Cálculo Ejemplos

$\frac{\sec (x)}{1 + \tan (x)}$

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 $g (x) = 1 + \tan (x)$ .

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

Paso 2

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

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

Paso 3

Diferencia.

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Paso 3.1

Según la regla de la suma, la derivada de $1 + \tan (x)$ con respecto a $x$ es $\frac{d}{d x} [1] + \frac{d}{d x} [\tan (x)]$ .

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

Paso 3.2

Como $1$ es constante con respecto a $x$ , la derivada de $1$ con respecto a $x$ es $0$ .

$\frac{(1 + \tan (x)) (\sec (x) \tan (x)) - \sec (x) (0 + \frac{d}{d x} [\tan (x)])}{{(1 + \tan (x))}^{2}}$

Paso 4

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

$\frac{(1 + \tan (x)) (\sec (x) \tan (x)) - \sec (x) (0 + \sec^{2} (x))}{{(1 + \tan (x))}^{2}}$

Paso 5

Suma $0$ y $\sec^{2} (x)$ .

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

Paso 6

Simplifica.

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Paso 6.1

Aplica la propiedad distributiva.

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

Paso 6.2

Simplifica el numerador.

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Paso 6.2.1

Simplifica cada término.

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Paso 6.2.1.1

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

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

Paso 6.2.1.2

Multiplica $\tan (x) (\sec (x) \tan (x))$ .

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Paso 6.2.1.2.1

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

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

Paso 6.2.1.2.2

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

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

Paso 6.2.1.2.3

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

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

Paso 6.2.1.2.4

Suma $1$ y $1$ .

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

Paso 6.2.1.3

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

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Paso 6.2.1.3.1

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

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

Paso 6.2.1.3.2

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

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Paso 6.2.1.3.2.1

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

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

Paso 6.2.1.3.2.2

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

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

Paso 6.2.1.3.3

Suma $2$ y $1$ .

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

Paso 6.2.2

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

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Paso 6.2.2.1

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

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

Paso 6.2.2.2

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

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

Paso 6.2.2.3

Factoriza $\sec (x)$ de $- \sec^{3} (x)$ .

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

Paso 6.2.2.4

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

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

Paso 6.2.2.5

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

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

Paso 6.2.3

Reordena $\tan^{2} (x)$ y $- \sec^{2} (x)$ .

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

Paso 6.2.4

Factoriza $- 1$ de $- \sec^{2} (x)$ .

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

Paso 6.2.5

Factoriza $- 1$ de $\tan^{2} (x)$ .

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

Paso 6.2.6

Factoriza $- 1$ de $- (\sec^{2} (x)) - 1 (- \tan^{2} (x))$ .

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

Paso 6.2.7

Aplica la identidad pitagórica.

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

Paso 6.2.8

Multiplica $- 1$ por $1$ .

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

Paso 6.2.9

Aplica la propiedad distributiva.

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

Paso 6.2.10

Mueve $- 1$ a la izquierda de $\sec (x)$ .

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

Paso 6.2.11

Reescribe $- 1 \sec (x)$ como $- \sec (x)$ .

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

Paso 6.3

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

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Paso 6.3.1

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

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

Paso 6.3.2

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

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

Paso 6.3.3

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

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