Cálculo Ejemplos

$f (x) = 4 x^{2} \sin (x) \tan (x)$

Paso 1

Como $4$ es constante con respecto a $x$ , la derivada de $4 x^{2} \sin (x) \tan (x)$ con respecto a $x$ es $4 \frac{d}{d x} [x^{2} \sin (x) \tan (x)]$ .

$4 \frac{d}{d x} [x^{2} \sin (x) \tan (x)]$

Paso 2

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

$4 (x^{2} \sin (x) \frac{d}{d x} [\tan (x)] + \tan (x) \frac{d}{d x} [x^{2} \sin (x)])$

Paso 3

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

$4 (x^{2} \sin (x) \sec^{2} (x) + \tan (x) \frac{d}{d x} [x^{2} \sin (x)])$

Paso 4

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

$4 (x^{2} \sin (x) \sec^{2} (x) + \tan (x) (x^{2} \frac{d}{d x} [\sin (x)] + \sin (x) \frac{d}{d x} [x^{2}]))$

Paso 5

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

$4 (x^{2} \sin (x) \sec^{2} (x) + \tan (x) (x^{2} \cos (x) + \sin (x) \frac{d}{d x} [x^{2}]))$

Paso 6

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

$4 (x^{2} \sin (x) \sec^{2} (x) + \tan (x) (x^{2} \cos (x) + \sin (x) (2 x)))$

Paso 7

Simplifica.

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Paso 7.1

Aplica la propiedad distributiva.

$4 (x^{2} \sin (x) \sec^{2} (x) + \tan (x) (x^{2} \cos (x)) + \tan (x) (\sin (x) (2 x)))$

Paso 7.2

Aplica la propiedad distributiva.

$4 (x^{2} \sin (x) \sec^{2} (x)) + 4 (\tan (x) (x^{2} \cos (x))) + 4 (\tan (x) (\sin (x) (2 x)))$

Paso 7.3

Multiplica $2$ por $4$ .

$4 x^{2} \sin (x) \sec^{2} (x) + 4 \tan (x) x^{2} \cos (x) + 8 \tan (x) \sin (x) x$

Paso 7.4

Reordena los términos.

$4 x^{2} \sec^{2} (x) \sin (x) + 4 x^{2} \cos (x) \tan (x) + 8 x \sin (x) \tan (x)$

Paso 7.5

Simplifica cada término.

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Paso 7.5.1

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

$4 x^{2} {(\frac{1}{\cos (x)})}^{2} \sin (x) + 4 x^{2} \cos (x) \tan (x) + 8 x \sin (x) \tan (x)$

Paso 7.5.2

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

$4 x^{2} \frac{1^{2}}{\cos^{2} (x)} \sin (x) + 4 x^{2} \cos (x) \tan (x) + 8 x \sin (x) \tan (x)$

Paso 7.5.3

Uno elevado a cualquier potencia es uno.

$4 x^{2} \frac{1}{\cos^{2} (x)} \sin (x) + 4 x^{2} \cos (x) \tan (x) + 8 x \sin (x) \tan (x)$

Paso 7.5.4

Multiplica $4 x^{2} \frac{1}{\cos^{2} (x)}$ .

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Paso 7.5.4.1

Combina $4$ y $\frac{1}{\cos^{2} (x)}$ .

$x^{2} \frac{4}{\cos^{2} (x)} \sin (x) + 4 x^{2} \cos (x) \tan (x) + 8 x \sin (x) \tan (x)$

Paso 7.5.4.2

Combina $x^{2}$ y $\frac{4}{\cos^{2} (x)}$ .

$\frac{x^{2} \cdot 4}{\cos^{2} (x)} \sin (x) + 4 x^{2} \cos (x) \tan (x) + 8 x \sin (x) \tan (x)$

Paso 7.5.5

Mueve $4$ a la izquierda de $x^{2}$ .

$\frac{4 \cdot x^{2}}{\cos^{2} (x)} \sin (x) + 4 x^{2} \cos (x) \tan (x) + 8 x \sin (x) \tan (x)$

Paso 7.5.6

Combina $\frac{4 x^{2}}{\cos^{2} (x)}$ y $\sin (x)$ .

$\frac{4 x^{2} \sin (x)}{\cos^{2} (x)} + 4 x^{2} \cos (x) \tan (x) + 8 x \sin (x) \tan (x)$

Paso 7.5.7

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

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Paso 7.5.7.1

Agrega paréntesis.

$\frac{4 x^{2} \sin (x)}{\cos^{2} (x)} + 4 x^{2} (\cos (x) \tan (x)) + 8 x \sin (x) \tan (x)$

Paso 7.5.7.2

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

$\frac{4 x^{2} \sin (x)}{\cos^{2} (x)} + 4 x^{2} (\tan (x) \cos (x)) + 8 x \sin (x) \tan (x)$

Paso 7.5.7.3

Reescribe $4 x^{2} \cos (x) \tan (x)$ en términos de senos y cosenos.

$\frac{4 x^{2} \sin (x)}{\cos^{2} (x)} + 4 x^{2} (\frac{\sin (x)}{\cos (x)} \cos (x)) + 8 x \sin (x) \tan (x)$

Paso 7.5.7.4

Cancela los factores comunes.

$\frac{4 x^{2} \sin (x)}{\cos^{2} (x)} + 4 x^{2} \sin (x) + 8 x \sin (x) \tan (x)$

Paso 7.5.8

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

$\frac{4 x^{2} \sin (x)}{\cos^{2} (x)} + 4 x^{2} \sin (x) + 8 x \sin (x) \frac{\sin (x)}{\cos (x)}$

Paso 7.5.9

Multiplica $8 x \sin (x) \frac{\sin (x)}{\cos (x)}$ .

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Paso 7.5.9.1

Combina $8$ y $\frac{\sin (x)}{\cos (x)}$ .

$\frac{4 x^{2} \sin (x)}{\cos^{2} (x)} + 4 x^{2} \sin (x) + x \sin (x) \frac{8 \sin (x)}{\cos (x)}$

Paso 7.5.9.2

Combina $\frac{8 \sin (x)}{\cos (x)}$ y $x$ .

$\frac{4 x^{2} \sin (x)}{\cos^{2} (x)} + 4 x^{2} \sin (x) + \frac{8 \sin (x) x}{\cos (x)} \sin (x)$

Paso 7.5.9.3

Combina $\frac{8 \sin (x) x}{\cos (x)}$ y $\sin (x)$ .

$\frac{4 x^{2} \sin (x)}{\cos^{2} (x)} + 4 x^{2} \sin (x) + \frac{8 \sin (x) x \sin (x)}{\cos (x)}$

Paso 7.5.9.4

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

$\frac{4 x^{2} \sin (x)}{\cos^{2} (x)} + 4 x^{2} \sin (x) + \frac{8 x (\sin^{1} (x) \sin (x))}{\cos (x)}$

Paso 7.5.9.5

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

$\frac{4 x^{2} \sin (x)}{\cos^{2} (x)} + 4 x^{2} \sin (x) + \frac{8 x (\sin^{1} (x) \sin^{1} (x))}{\cos (x)}$

Paso 7.5.9.6

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

$\frac{4 x^{2} \sin (x)}{\cos^{2} (x)} + 4 x^{2} \sin (x) + \frac{8 x {\sin (x)}^{1 + 1}}{\cos (x)}$

Paso 7.5.9.7

Suma $1$ y $1$ .

$\frac{4 x^{2} \sin (x)}{\cos^{2} (x)} + 4 x^{2} \sin (x) + \frac{8 x \sin^{2} (x)}{\cos (x)}$

Paso 7.6

Simplifica cada término.

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Paso 7.6.1

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

$\frac{4 x^{2} \sin (x)}{\cos (x) \cos (x)} + 4 x^{2} \sin (x) + \frac{8 x \sin^{2} (x)}{\cos (x)}$

Paso 7.6.2

Separa las fracciones.

$\frac{4 x^{2}}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)} + 4 x^{2} \sin (x) + \frac{8 x \sin^{2} (x)}{\cos (x)}$

Paso 7.6.3

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

$\frac{4 x^{2}}{\cos (x)} \tan (x) + 4 x^{2} \sin (x) + \frac{8 x \sin^{2} (x)}{\cos (x)}$

Paso 7.6.4

Combina $\frac{4 x^{2}}{\cos (x)}$ y $\tan (x)$ .

$\frac{4 x^{2} \tan (x)}{\cos (x)} + 4 x^{2} \sin (x) + \frac{8 x \sin^{2} (x)}{\cos (x)}$

Paso 7.6.5

Separa las fracciones.

$\frac{4 x^{2}}{1} \cdot \frac{\tan (x)}{\cos (x)} + 4 x^{2} \sin (x) + \frac{8 x \sin^{2} (x)}{\cos (x)}$

Paso 7.6.6

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

$\frac{4 x^{2}}{1} \cdot \frac{\frac{\sin (x)}{\cos (x)}}{\cos (x)} + 4 x^{2} \sin (x) + \frac{8 x \sin^{2} (x)}{\cos (x)}$

Paso 7.6.7

Reescribe $\frac{\frac{\sin (x)}{\cos (x)}}{\cos (x)}$ como un producto.

$\frac{4 x^{2}}{1} (\frac{\sin (x)}{\cos (x)} \cdot \frac{1}{\cos (x)}) + 4 x^{2} \sin (x) + \frac{8 x \sin^{2} (x)}{\cos (x)}$

Paso 7.6.8

Simplifica.

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Paso 7.6.8.1

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

$\frac{4 x^{2}}{1} (\tan (x) \frac{1}{\cos (x)}) + 4 x^{2} \sin (x) + \frac{8 x \sin^{2} (x)}{\cos (x)}$

Paso 7.6.8.2

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

$\frac{4 x^{2}}{1} (\tan (x) \sec (x)) + 4 x^{2} \sin (x) + \frac{8 x \sin^{2} (x)}{\cos (x)}$

Paso 7.6.9

Divide $4 x^{2}$ por $1$ .

$4 x^{2} (\tan (x) \sec (x)) + 4 x^{2} \sin (x) + \frac{8 x \sin^{2} (x)}{\cos (x)}$

Paso 7.6.10

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

$4 x^{2} (\tan (x) \sec (x)) + 4 x^{2} \sin (x) + \frac{8 x (\sin (x) \sin (x))}{\cos (x)}$

Paso 7.6.11

Separa las fracciones.

$4 x^{2} (\tan (x) \sec (x)) + 4 x^{2} \sin (x) + \frac{8 x (\sin (x))}{1} \cdot \frac{\sin (x)}{\cos (x)}$

Paso 7.6.12

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

$4 x^{2} (\tan (x) \sec (x)) + 4 x^{2} \sin (x) + \frac{8 x (\sin (x))}{1} \tan (x)$

Paso 7.6.13

Divide $8 x (\sin (x))$ por $1$ .

$4 x^{2} (\tan (x) \sec (x)) + 4 x^{2} \sin (x) + 8 x (\sin (x)) \tan (x)$

Paso 7.7

Reordena los factores en $4 x^{2} (\tan (x) \sec (x)) + 4 x^{2} \sin (x) + 8 x (\sin (x)) \tan (x)$ .

$4 x^{2} \tan (x) \sec (x) + 4 x^{2} \sin (x) + 8 x (\sin (x)) \tan (x)$