Cálculo Ejemplos

$y = 8 x^{2} \cos (x) \cot (x)$

Paso 1

Como $8$ es constante con respecto a $x$ , la derivada de $8 x^{2} \cos (x) \cot (x)$ con respecto a $x$ es $8 \frac{d}{d x} [x^{2} \cos (x) \cot (x)]$ .

$8 \frac{d}{d x} [x^{2} \cos (x) \cot (x)]$

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

$8 (x^{2} \cos (x) \frac{d}{d x} [\cot (x)] + \cot (x) \frac{d}{d x} [x^{2} \cos (x)])$

Paso 3

La derivada de $\cot (x)$ con respecto a $x$ es $- \csc^{2} (x)$ .

$8 (x^{2} \cos (x) (- \csc^{2} (x)) + \cot (x) \frac{d}{d x} [x^{2} \cos (x)])$

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

$8 (x^{2} \cos (x) (- \csc^{2} (x)) + \cot (x) (x^{2} \frac{d}{d x} [\cos (x)] + \cos (x) \frac{d}{d x} [x^{2}]))$

Paso 5

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

$8 (x^{2} \cos (x) (- \csc^{2} (x)) + \cot (x) (x^{2} (- \sin (x)) + \cos (x) \frac{d}{d x} [x^{2}]))$

Paso 6

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

$8 (x^{2} \cos (x) (- \csc^{2} (x)) + \cot (x) (x^{2} (- \sin (x)) + \cos (x) (2 x)))$

Paso 7

Simplifica.

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Paso 7.1

Aplica la propiedad distributiva.

$8 (x^{2} \cos (x) (- \csc^{2} (x)) + \cot (x) (x^{2} (- \sin (x))) + \cot (x) (\cos (x) (2 x)))$

Paso 7.2

Aplica la propiedad distributiva.

$8 (x^{2} \cos (x) (- \csc^{2} (x))) + 8 (\cot (x) (x^{2} (- \sin (x)))) + 8 (\cot (x) (\cos (x) (2 x)))$

Paso 7.3

Combina los términos.

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Paso 7.3.1

Multiplica $- 1$ por $8$ .

$- 8 (x^{2} \cos (x) (\csc^{2} (x))) + 8 (\cot (x) (x^{2} (- \sin (x)))) + 8 (\cot (x) (\cos (x) (2 x)))$

Paso 7.3.2

Multiplica $- 1$ por $8$ .

$- 8 x^{2} \cos (x) \csc^{2} (x) - 8 (\cot (x) x^{2} (\sin (x))) + 8 (\cot (x) (\cos (x) (2 x)))$

Paso 7.3.3

Multiplica $2$ por $8$ .

$- 8 x^{2} \cos (x) \csc^{2} (x) - 8 \cot (x) x^{2} \sin (x) + 16 \cot (x) \cos (x) x$

Paso 7.4

Reordena los términos.

$- 8 x^{2} \csc^{2} (x) \cos (x) - 8 x^{2} \cot (x) \sin (x) + 16 x \cos (x) \cot (x)$

Paso 7.5

Simplifica cada término.

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Paso 7.5.1

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

$- 8 x^{2} {(\frac{1}{\sin (x)})}^{2} \cos (x) - 8 x^{2} \cot (x) \sin (x) + 16 x \cos (x) \cot (x)$

Paso 7.5.2

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

$- 8 x^{2} \frac{1^{2}}{\sin^{2} (x)} \cos (x) - 8 x^{2} \cot (x) \sin (x) + 16 x \cos (x) \cot (x)$

Paso 7.5.3

Uno elevado a cualquier potencia es uno.

$- 8 x^{2} \frac{1}{\sin^{2} (x)} \cos (x) - 8 x^{2} \cot (x) \sin (x) + 16 x \cos (x) \cot (x)$

Paso 7.5.4

Multiplica $- 8 x^{2} \frac{1}{\sin^{2} (x)}$ .

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Paso 7.5.4.1

Combina $- 8$ y $\frac{1}{\sin^{2} (x)}$ .

$x^{2} \frac{- 8}{\sin^{2} (x)} \cos (x) - 8 x^{2} \cot (x) \sin (x) + 16 x \cos (x) \cot (x)$

Paso 7.5.4.2

Combina $x^{2}$ y $\frac{- 8}{\sin^{2} (x)}$ .

$\frac{x^{2} \cdot - 8}{\sin^{2} (x)} \cos (x) - 8 x^{2} \cot (x) \sin (x) + 16 x \cos (x) \cot (x)$

Paso 7.5.5

Mueve $- 8$ a la izquierda de $x^{2}$ .

$\frac{- 8 \cdot x^{2}}{\sin^{2} (x)} \cos (x) - 8 x^{2} \cot (x) \sin (x) + 16 x \cos (x) \cot (x)$

Paso 7.5.6

Mueve el negativo al frente de la fracción.

$- \frac{8 \cdot x^{2}}{\sin^{2} (x)} \cos (x) - 8 x^{2} \cot (x) \sin (x) + 16 x \cos (x) \cot (x)$

Paso 7.5.7

Combina $\cos (x)$ y $\frac{8 x^{2}}{\sin^{2} (x)}$ .

$- \frac{\cos (x) (8 x^{2})}{\sin^{2} (x)} - 8 x^{2} \cot (x) \sin (x) + 16 x \cos (x) \cot (x)$

Paso 7.5.8

Mueve $8$ a la izquierda de $\cos (x)$ .

$- \frac{8 \cdot \cos (x) x^{2}}{\sin^{2} (x)} - 8 x^{2} \cot (x) \sin (x) + 16 x \cos (x) \cot (x)$

Paso 7.5.9

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

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Paso 7.5.9.1

Agrega paréntesis.

$- \frac{8 \cos (x) x^{2}}{\sin^{2} (x)} - 8 x^{2} (\cot (x) \sin (x)) + 16 x \cos (x) \cot (x)$

Paso 7.5.9.2

Reescribe $- 8 x^{2} \cot (x) \sin (x)$ en términos de senos y cosenos.

$- \frac{8 \cos (x) x^{2}}{\sin^{2} (x)} - 8 x^{2} (\frac{\cos (x)}{\sin (x)} \sin (x)) + 16 x \cos (x) \cot (x)$

Paso 7.5.9.3

Cancela los factores comunes.

$- \frac{8 \cos (x) x^{2}}{\sin^{2} (x)} - 8 x^{2} \cos (x) + 16 x \cos (x) \cot (x)$

Paso 7.5.10

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

$- \frac{8 \cos (x) x^{2}}{\sin^{2} (x)} - 8 x^{2} \cos (x) + 16 x \cos (x) \frac{\cos (x)}{\sin (x)}$

Paso 7.5.11

Multiplica $16 x \cos (x) \frac{\cos (x)}{\sin (x)}$ .

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Paso 7.5.11.1

Combina $16$ y $\frac{\cos (x)}{\sin (x)}$ .

$- \frac{8 \cos (x) x^{2}}{\sin^{2} (x)} - 8 x^{2} \cos (x) + x \cos (x) \frac{16 \cos (x)}{\sin (x)}$

Paso 7.5.11.2

Combina $\frac{16 \cos (x)}{\sin (x)}$ y $x$ .

$- \frac{8 \cos (x) x^{2}}{\sin^{2} (x)} - 8 x^{2} \cos (x) + \frac{16 \cos (x) x}{\sin (x)} \cos (x)$

Paso 7.5.11.3

Combina $\frac{16 \cos (x) x}{\sin (x)}$ y $\cos (x)$ .

$- \frac{8 \cos (x) x^{2}}{\sin^{2} (x)} - 8 x^{2} \cos (x) + \frac{16 \cos (x) x \cos (x)}{\sin (x)}$

Paso 7.5.11.4

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

$- \frac{8 \cos (x) x^{2}}{\sin^{2} (x)} - 8 x^{2} \cos (x) + \frac{16 x (\cos^{1} (x) \cos (x))}{\sin (x)}$

Paso 7.5.11.5

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

$- \frac{8 \cos (x) x^{2}}{\sin^{2} (x)} - 8 x^{2} \cos (x) + \frac{16 x (\cos^{1} (x) \cos^{1} (x))}{\sin (x)}$

Paso 7.5.11.6

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

$- \frac{8 \cos (x) x^{2}}{\sin^{2} (x)} - 8 x^{2} \cos (x) + \frac{16 x {\cos (x)}^{1 + 1}}{\sin (x)}$

Paso 7.5.11.7

Suma $1$ y $1$ .

$- \frac{8 \cos (x) x^{2}}{\sin^{2} (x)} - 8 x^{2} \cos (x) + \frac{16 x \cos^{2} (x)}{\sin (x)}$

Paso 7.6

Simplifica cada término.

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Paso 7.6.1

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

$- \frac{8 \cos (x) x^{2}}{\sin (x) \sin (x)} - 8 x^{2} \cos (x) + \frac{16 x \cos^{2} (x)}{\sin (x)}$

Paso 7.6.2

Separa las fracciones.

$- (\frac{8 x^{2}}{\sin (x)} \cdot \frac{\cos (x)}{\sin (x)}) - 8 x^{2} \cos (x) + \frac{16 x \cos^{2} (x)}{\sin (x)}$

Paso 7.6.3

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

$- (\frac{8 x^{2}}{\sin (x)} \cot (x)) - 8 x^{2} \cos (x) + \frac{16 x \cos^{2} (x)}{\sin (x)}$

Paso 7.6.4

Combina $\frac{8 x^{2}}{\sin (x)}$ y $\cot (x)$ .

$- \frac{8 x^{2} \cot (x)}{\sin (x)} - 8 x^{2} \cos (x) + \frac{16 x \cos^{2} (x)}{\sin (x)}$

Paso 7.6.5

Separa las fracciones.

$- (\frac{8 x^{2}}{1} \cdot \frac{\cot (x)}{\sin (x)}) - 8 x^{2} \cos (x) + \frac{16 x \cos^{2} (x)}{\sin (x)}$

Paso 7.6.6

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

$- (\frac{8 x^{2}}{1} \cdot \frac{\frac{\cos (x)}{\sin (x)}}{\sin (x)}) - 8 x^{2} \cos (x) + \frac{16 x \cos^{2} (x)}{\sin (x)}$

Paso 7.6.7

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

$- (\frac{8 x^{2}}{1} (\frac{\cos (x)}{\sin (x)} \cdot \frac{1}{\sin (x)})) - 8 x^{2} \cos (x) + \frac{16 x \cos^{2} (x)}{\sin (x)}$

Paso 7.6.8

Simplifica.

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Paso 7.6.8.1

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

$- (\frac{8 x^{2}}{1} (\cot (x) \frac{1}{\sin (x)})) - 8 x^{2} \cos (x) + \frac{16 x \cos^{2} (x)}{\sin (x)}$

Paso 7.6.8.2

Convierte de $\frac{1}{\sin (x)}$ a $\csc (x)$ .

$- (\frac{8 x^{2}}{1} (\cot (x) \csc (x))) - 8 x^{2} \cos (x) + \frac{16 x \cos^{2} (x)}{\sin (x)}$

Paso 7.6.9

Divide $8 x^{2}$ por $1$ .

$- (8 x^{2} (\cot (x) \csc (x))) - 8 x^{2} \cos (x) + \frac{16 x \cos^{2} (x)}{\sin (x)}$

Paso 7.6.10

Multiplica $8$ por $- 1$ .

$- 8 (x^{2} (\cot (x) \csc (x))) - 8 x^{2} \cos (x) + \frac{16 x \cos^{2} (x)}{\sin (x)}$

Paso 7.6.11

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

$- 8 x^{2} (\cot (x) \csc (x)) - 8 x^{2} \cos (x) + \frac{16 x (\cos (x) \cos (x))}{\sin (x)}$

Paso 7.6.12

Separa las fracciones.

$- 8 x^{2} (\cot (x) \csc (x)) - 8 x^{2} \cos (x) + \frac{16 x (\cos (x))}{1} \cdot \frac{\cos (x)}{\sin (x)}$

Paso 7.6.13

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

$- 8 x^{2} (\cot (x) \csc (x)) - 8 x^{2} \cos (x) + \frac{16 x (\cos (x))}{1} \cot (x)$

Paso 7.6.14

Divide $16 x (\cos (x))$ por $1$ .

$- 8 x^{2} (\cot (x) \csc (x)) - 8 x^{2} \cos (x) + 16 x (\cos (x)) \cot (x)$

$- 8 x^{2} \cot (x) \csc (x) - 8 x^{2} \cos (x) + 16 x (\cos (x)) \cot (x)$