Cálculo Ejemplos

$y = (\sin (x) + \cos (x)) \sec (x)$

Paso 1

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 $f (x) = \sin (x) + \cos (x)$ y $g (x) = \sec (x)$ .

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

Paso 2

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

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

Paso 3

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

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

Paso 4

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

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

Paso 5

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

$(\sin (x) + \cos (x)) \sec (x) \tan (x) + \sec (x) (\cos (x) - \sin (x))$

Paso 6

Simplifica.

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Paso 6.1

Aplica la propiedad distributiva.

$(\sin (x) \sec (x) + \cos (x) \sec (x)) \tan (x) + \sec (x) (\cos (x) - \sin (x))$

Paso 6.2

Aplica la propiedad distributiva.

$\sin (x) \sec (x) \tan (x) + \cos (x) \sec (x) \tan (x) + \sec (x) (\cos (x) - \sin (x))$

Paso 6.3

Aplica la propiedad distributiva.

$\sin (x) \sec (x) \tan (x) + \cos (x) \sec (x) \tan (x) + \sec (x) \cos (x) + \sec (x) (- \sin (x))$

Paso 6.4

Reordena los términos.

$\sec (x) \sin (x) \tan (x) + \cos (x) \sec (x) \tan (x) + \cos (x) \sec (x) - \sec (x) \sin (x)$

Paso 6.5

Simplifica cada término.

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Paso 6.5.1

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

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

Paso 6.5.2

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

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

Paso 6.5.3

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

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

Paso 6.5.4

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

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Paso 6.5.4.1

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

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

Paso 6.5.4.2

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

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

Paso 6.5.4.3

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

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

Paso 6.5.4.4

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

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

Paso 6.5.4.5

Suma $1$ y $1$ .

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

Paso 6.5.4.6

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

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

Paso 6.5.4.7

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

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

Paso 6.5.4.8

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

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

Paso 6.5.4.9

Suma $1$ y $1$ .

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

Paso 6.5.5

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

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Paso 6.5.5.1

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

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

Paso 6.5.5.2

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

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

Paso 6.5.5.3

Cancela los factores comunes.

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

Paso 6.5.6

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

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

Paso 6.5.7

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

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

Paso 6.5.8

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

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Paso 6.5.8.1

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

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

Paso 6.5.8.2

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

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

Paso 6.5.8.3

Cancela los factores comunes.

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

Paso 6.5.9

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

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

Paso 6.5.10

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

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

Paso 6.6

Combina los términos opuestos en $\frac{\sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin (x)}{\cos (x)} + 1 - \frac{\sin (x)}{\cos (x)}$ .

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Paso 6.6.1

Resta $\frac{\sin (x)}{\cos (x)}$ de $\frac{\sin (x)}{\cos (x)}$ .

$\frac{\sin^{2} (x)}{\cos^{2} (x)} + 0 + 1$

Paso 6.6.2

Suma $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ y $0$ .

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

Paso 6.7

Convierte de $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ a $\tan^{2} (x)$ .

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

Paso 6.8

Aplica la identidad pitagórica.

$\sec^{2} (x)$