Cálculo Ejemplos

$x^{2} \sin (x) \tan (x)$

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

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

Paso 2

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

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

Paso 3

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

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

Paso 4

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

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

Paso 5

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

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

Paso 6

Simplifica.

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Paso 6.1

Aplica la propiedad distributiva.

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

Paso 6.2

Elimina los paréntesis innecesarios.

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

Paso 6.3

Reordena los términos.

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

Paso 6.4

Simplifica cada término.

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Paso 6.4.1

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

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

Paso 6.4.2

Aplica la regla del producto a $\frac{1}{\cos (x)}$ .

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

Paso 6.4.3

Uno elevado a cualquier potencia es uno.

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

Paso 6.4.4

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

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

Paso 6.4.5

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

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

Paso 6.4.6

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

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Paso 6.4.6.1

Agrega paréntesis.

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

Paso 6.4.6.2

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

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

Paso 6.4.6.3

Reescribe $x^{2} \cos (x) \tan (x)$ en términos de senos y cosenos.

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

Paso 6.4.6.4

Cancela los factores comunes.

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

Paso 6.4.7

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

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

Paso 6.4.8

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

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Paso 6.4.8.1

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

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

Paso 6.4.8.2

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

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

Paso 6.4.8.3

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

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

Paso 6.4.8.4

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

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

Paso 6.4.8.5

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

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

Paso 6.4.8.6

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

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

Paso 6.4.8.7

Suma $1$ y $1$ .

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

Paso 6.5

Simplifica cada término.

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Paso 6.5.1

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

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

Paso 6.5.2

Separa las fracciones.

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

Paso 6.5.3

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

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

Paso 6.5.4

Separa las fracciones.

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

Paso 6.5.5

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

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

Paso 6.5.6

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

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

Paso 6.5.7

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

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

Paso 6.5.8

Separa las fracciones.

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

Paso 6.5.9

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

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

Paso 6.5.10

Divide $2 x (\sin (x))$ por $1$ .

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