Cálculo Ejemplos

$y = (x^{2} + \sin (x)) \sec (x)$

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

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

Paso 2

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

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

Paso 3

Diferencia.

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Paso 3.1

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

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

Paso 3.2

Diferencia con la regla de la potencia, que establece que $\frac{d}{d x} [x^{n}]$ es $n x^{n - 1}$ donde $n = 2$ .

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

Paso 4

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

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

Paso 5

Simplifica.

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Paso 5.1

Aplica la propiedad distributiva.

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

Paso 5.2

Aplica la propiedad distributiva.

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

Paso 5.3

Aplica la propiedad distributiva.

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

Paso 5.4

Reordena los términos.

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

Paso 5.5

Simplifica cada término.

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Paso 5.5.1

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

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

Paso 5.5.2

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

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

Paso 5.5.3

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

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

Paso 5.5.4

Combinar.

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

Paso 5.5.5

Simplifica el denominador.

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Paso 5.5.5.1

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

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

Paso 5.5.5.2

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

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

Paso 5.5.5.3

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

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

Paso 5.5.5.4

Suma $1$ y $1$ .

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

Paso 5.5.6

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

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

Paso 5.5.7

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

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

Paso 5.5.8

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

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

Paso 5.5.9

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

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Paso 5.5.9.1

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

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

Paso 5.5.9.2

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

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

Paso 5.5.9.3

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

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

Paso 5.5.9.4

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

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

Paso 5.5.9.5

Suma $1$ y $1$ .

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

Paso 5.5.9.6

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

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

Paso 5.5.9.7

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

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

Paso 5.5.9.8

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

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

Paso 5.5.9.9

Suma $1$ y $1$ .

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

Paso 5.5.10

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

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

Paso 5.5.11

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

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Paso 5.5.11.1

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

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

Paso 5.5.11.2

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

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

Paso 5.5.12

Mueve $2$ a la izquierda de $x$ .

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

Paso 5.5.13

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

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Paso 5.5.13.1

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

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

Paso 5.5.13.2

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

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

Paso 5.5.13.3

Cancela los factores comunes.

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

Paso 5.6

Simplifica cada término.

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Paso 5.6.1

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

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

Paso 5.6.2

Separa las fracciones.

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

Paso 5.6.3

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

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

Paso 5.6.4

Separa las fracciones.

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

Paso 5.6.5

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

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

Paso 5.6.6

Divide $x^{2}$ por $1$ .

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

Paso 5.6.7

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

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

Paso 5.7

Mueve $1$ .

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

Paso 5.8

Aplica la identidad pitagórica.

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