Cálculo Ejemplos

$\tan (x) (4 \sin (x) + 5 \cos (x))$

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 $f (x) = \tan (x)$ y $g (x) = 4 \sin (x) + 5 \cos (x)$ .

$\tan (x) \frac{d}{d x} [4 \sin (x) + 5 \cos (x)] + (4 \sin (x) + 5 \cos (x)) \frac{d}{d x} [\tan (x)]$

Paso 2

Diferencia.

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Paso 2.1

Según la regla de la suma, la derivada de $4 \sin (x) + 5 \cos (x)$ con respecto a $x$ es $\frac{d}{d x} [4 \sin (x)] + \frac{d}{d x} [5 \cos (x)]$ .

$\tan (x) (\frac{d}{d x} [4 \sin (x)] + \frac{d}{d x} [5 \cos (x)]) + (4 \sin (x) + 5 \cos (x)) \frac{d}{d x} [\tan (x)]$

Paso 2.2

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

$\tan (x) (4 \frac{d}{d x} [\sin (x)] + \frac{d}{d x} [5 \cos (x)]) + (4 \sin (x) + 5 \cos (x)) \frac{d}{d x} [\tan (x)]$

Paso 3

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

$\tan (x) (4 \cos (x) + \frac{d}{d x} [5 \cos (x)]) + (4 \sin (x) + 5 \cos (x)) \frac{d}{d x} [\tan (x)]$

Paso 4

Como $5$ es constante con respecto a $x$ , la derivada de $5 \cos (x)$ con respecto a $x$ es $5 \frac{d}{d x} [\cos (x)]$ .

$\tan (x) (4 \cos (x) + 5 \frac{d}{d x} [\cos (x)]) + (4 \sin (x) + 5 \cos (x)) \frac{d}{d x} [\tan (x)]$

Paso 5

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

$\tan (x) (4 \cos (x) + 5 (- \sin (x))) + (4 \sin (x) + 5 \cos (x)) \frac{d}{d x} [\tan (x)]$

Paso 6

Multiplica $- 1$ por $5$ .

$\tan (x) (4 \cos (x) - 5 \sin (x)) + (4 \sin (x) + 5 \cos (x)) \frac{d}{d x} [\tan (x)]$

Paso 7

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

$\tan (x) (4 \cos (x) - 5 \sin (x)) + (4 \sin (x) + 5 \cos (x)) \sec^{2} (x)$

Paso 8

Simplifica.

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Paso 8.1

Aplica la propiedad distributiva.

$\tan (x) (4 \cos (x)) + \tan (x) (- 5 \sin (x)) + (4 \sin (x) + 5 \cos (x)) \sec^{2} (x)$

Paso 8.2

Aplica la propiedad distributiva.

$\tan (x) (4 \cos (x)) + \tan (x) (- 5 \sin (x)) + 4 \sin (x) \sec^{2} (x) + 5 \cos (x) \sec^{2} (x)$

Paso 8.3

Mueve $4$ a la izquierda de $\tan (x)$ .

$4 \tan (x) \cos (x) + \tan (x) (- 5 \sin (x)) + 4 \sin (x) \sec^{2} (x) + 5 \cos (x) \sec^{2} (x)$

Paso 8.4

Reordena los términos.

$4 \cos (x) \tan (x) - 5 \sin (x) \tan (x) + 4 \sec^{2} (x) \sin (x) + 5 \sec^{2} (x) \cos (x)$

Paso 8.5

Simplifica cada término.

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Paso 8.5.1

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

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Paso 8.5.1.1

Agrega paréntesis.

$4 (\cos (x) \tan (x)) - 5 \sin (x) \tan (x) + 4 \sec^{2} (x) \sin (x) + 5 \sec^{2} (x) \cos (x)$

Paso 8.5.1.2

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

$4 (\tan (x) \cos (x)) - 5 \sin (x) \tan (x) + 4 \sec^{2} (x) \sin (x) + 5 \sec^{2} (x) \cos (x)$

Paso 8.5.1.3

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

$4 (\frac{\sin (x)}{\cos (x)} \cos (x)) - 5 \sin (x) \tan (x) + 4 \sec^{2} (x) \sin (x) + 5 \sec^{2} (x) \cos (x)$

Paso 8.5.1.4

Cancela los factores comunes.

$4 \sin (x) - 5 \sin (x) \tan (x) + 4 \sec^{2} (x) \sin (x) + 5 \sec^{2} (x) \cos (x)$

Paso 8.5.2

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

$4 \sin (x) - 5 \sin (x) \frac{\sin (x)}{\cos (x)} + 4 \sec^{2} (x) \sin (x) + 5 \sec^{2} (x) \cos (x)$

Paso 8.5.3

Multiplica $- 5 \sin (x) \frac{\sin (x)}{\cos (x)}$ .

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Paso 8.5.3.1

Combina $\frac{\sin (x)}{\cos (x)}$ y $- 5$ .

$4 \sin (x) + \frac{\sin (x) \cdot - 5}{\cos (x)} \sin (x) + 4 \sec^{2} (x) \sin (x) + 5 \sec^{2} (x) \cos (x)$

Paso 8.5.3.2

Combina $\frac{\sin (x) \cdot - 5}{\cos (x)}$ y $\sin (x)$ .

$4 \sin (x) + \frac{\sin (x) \cdot - 5 \sin (x)}{\cos (x)} + 4 \sec^{2} (x) \sin (x) + 5 \sec^{2} (x) \cos (x)$

Paso 8.5.3.3

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

$4 \sin (x) + \frac{- 5 (\sin^{1} (x) \sin (x))}{\cos (x)} + 4 \sec^{2} (x) \sin (x) + 5 \sec^{2} (x) \cos (x)$

Paso 8.5.3.4

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

$4 \sin (x) + \frac{- 5 (\sin^{1} (x) \sin^{1} (x))}{\cos (x)} + 4 \sec^{2} (x) \sin (x) + 5 \sec^{2} (x) \cos (x)$

Paso 8.5.3.5

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

$4 \sin (x) + \frac{- 5 {\sin (x)}^{1 + 1}}{\cos (x)} + 4 \sec^{2} (x) \sin (x) + 5 \sec^{2} (x) \cos (x)$

Paso 8.5.3.6

Suma $1$ y $1$ .

$4 \sin (x) + \frac{- 5 \sin^{2} (x)}{\cos (x)} + 4 \sec^{2} (x) \sin (x) + 5 \sec^{2} (x) \cos (x)$

Paso 8.5.4

Mueve el negativo al frente de la fracción.

$4 \sin (x) - \frac{5 \sin^{2} (x)}{\cos (x)} + 4 \sec^{2} (x) \sin (x) + 5 \sec^{2} (x) \cos (x)$

Paso 8.5.5

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

$4 \sin (x) - \frac{5 \sin^{2} (x)}{\cos (x)} + 4 {(\frac{1}{\cos (x)})}^{2} \sin (x) + 5 \sec^{2} (x) \cos (x)$

Paso 8.5.6

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

$4 \sin (x) - \frac{5 \sin^{2} (x)}{\cos (x)} + 4 \frac{1^{2}}{\cos^{2} (x)} \sin (x) + 5 \sec^{2} (x) \cos (x)$

Paso 8.5.7

Uno elevado a cualquier potencia es uno.

$4 \sin (x) - \frac{5 \sin^{2} (x)}{\cos (x)} + 4 \frac{1}{\cos^{2} (x)} \sin (x) + 5 \sec^{2} (x) \cos (x)$

Paso 8.5.8

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

$4 \sin (x) - \frac{5 \sin^{2} (x)}{\cos (x)} + \frac{4}{\cos^{2} (x)} \sin (x) + 5 \sec^{2} (x) \cos (x)$

Paso 8.5.9

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

$4 \sin (x) - \frac{5 \sin^{2} (x)}{\cos (x)} + \frac{4 \sin (x)}{\cos^{2} (x)} + 5 \sec^{2} (x) \cos (x)$

Paso 8.5.10

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

$4 \sin (x) - \frac{5 \sin^{2} (x)}{\cos (x)} + \frac{4 \sin (x)}{\cos^{2} (x)} + 5 {(\frac{1}{\cos (x)})}^{2} \cos (x)$

Paso 8.5.11

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

$4 \sin (x) - \frac{5 \sin^{2} (x)}{\cos (x)} + \frac{4 \sin (x)}{\cos^{2} (x)} + 5 \frac{1^{2}}{\cos^{2} (x)} \cos (x)$

Paso 8.5.12

Uno elevado a cualquier potencia es uno.

$4 \sin (x) - \frac{5 \sin^{2} (x)}{\cos (x)} + \frac{4 \sin (x)}{\cos^{2} (x)} + 5 \frac{1}{\cos^{2} (x)} \cos (x)$

Paso 8.5.13

Combina $5$ y $\frac{1}{\cos^{2} (x)}$ .

$4 \sin (x) - \frac{5 \sin^{2} (x)}{\cos (x)} + \frac{4 \sin (x)}{\cos^{2} (x)} + \frac{5}{\cos^{2} (x)} \cos (x)$

Paso 8.5.14

Cancela el factor común de $\cos (x)$ .

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Paso 8.5.14.1

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

$4 \sin (x) - \frac{5 \sin^{2} (x)}{\cos (x)} + \frac{4 \sin (x)}{\cos^{2} (x)} + \frac{5}{\cos (x) \cos (x)} \cos (x)$

Paso 8.5.14.2

Cancela el factor común.

$4 \sin (x) - \frac{5 \sin^{2} (x)}{\cos (x)} + \frac{4 \sin (x)}{\cos^{2} (x)} + \frac{5}{\cos (x) \cos (x)} \cos (x)$

Paso 8.5.14.3

Reescribe la expresión.

$4 \sin (x) - \frac{5 \sin^{2} (x)}{\cos (x)} + \frac{4 \sin (x)}{\cos^{2} (x)} + \frac{5}{\cos (x)}$

Paso 8.6

Combina los numeradores sobre el denominador común.

$4 \sin (x) + \frac{- 5 \sin^{2} (x) + 5}{\cos (x)} + \frac{4 \sin (x)}{\cos^{2} (x)}$

Paso 8.7

Reordena $- 5 \sin^{2} (x)$ y $5$ .

$4 \sin (x) + \frac{5 - 5 \sin^{2} (x)}{\cos (x)} + \frac{4 \sin (x)}{\cos^{2} (x)}$

Paso 8.8

Factoriza $5$ de $5$ .

$4 \sin (x) + \frac{5 (1) - 5 \sin^{2} (x)}{\cos (x)} + \frac{4 \sin (x)}{\cos^{2} (x)}$

Paso 8.9

Factoriza $5$ de $- 5 \sin^{2} (x)$ .

$4 \sin (x) + \frac{5 (1) + 5 (- \sin^{2} (x))}{\cos (x)} + \frac{4 \sin (x)}{\cos^{2} (x)}$

Paso 8.10

Factoriza $5$ de $5 (1) + 5 (- \sin^{2} (x))$ .

$4 \sin (x) + \frac{5 (1 - \sin^{2} (x))}{\cos (x)} + \frac{4 \sin (x)}{\cos^{2} (x)}$

Paso 8.11

Aplica la identidad pitagórica.

$4 \sin (x) + \frac{5 \cos^{2} (x)}{\cos (x)} + \frac{4 \sin (x)}{\cos^{2} (x)}$

Paso 8.12

Cancela el factor común de $\cos^{2} (x)$ y $\cos (x)$ .

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Paso 8.12.1

Factoriza $\cos (x)$ de $5 \cos^{2} (x)$ .

$4 \sin (x) + \frac{\cos (x) (5 \cos (x))}{\cos (x)} + \frac{4 \sin (x)}{\cos^{2} (x)}$

Paso 8.12.2

Cancela los factores comunes.

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Paso 8.12.2.1

Multiplica por $1$ .

$4 \sin (x) + \frac{\cos (x) (5 \cos (x))}{\cos (x) \cdot 1} + \frac{4 \sin (x)}{\cos^{2} (x)}$

Paso 8.12.2.2

Cancela el factor común.

$4 \sin (x) + \frac{\cos (x) (5 \cos (x))}{\cos (x) \cdot 1} + \frac{4 \sin (x)}{\cos^{2} (x)}$

Paso 8.12.2.3

Reescribe la expresión.

$4 \sin (x) + \frac{5 \cos (x)}{1} + \frac{4 \sin (x)}{\cos^{2} (x)}$

Paso 8.12.2.4

Divide $5 \cos (x)$ por $1$ .

$4 \sin (x) + 5 \cos (x) + \frac{4 \sin (x)}{\cos^{2} (x)}$

Paso 8.13

Simplifica cada término.

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Paso 8.13.1

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

$4 \sin (x) + 5 \cos (x) + \frac{4 \sin (x)}{\cos (x) \cos (x)}$

Paso 8.13.2

Separa las fracciones.

$4 \sin (x) + 5 \cos (x) + \frac{4}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$

Paso 8.13.3

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

$4 \sin (x) + 5 \cos (x) + \frac{4}{\cos (x)} \tan (x)$

Paso 8.13.4

Separa las fracciones.

$4 \sin (x) + 5 \cos (x) + \frac{4}{1} \cdot \frac{1}{\cos (x)} \tan (x)$

Paso 8.13.5

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

$4 \sin (x) + 5 \cos (x) + \frac{4}{1} \sec (x) \tan (x)$

Paso 8.13.6

Divide $4$ por $1$ .

$4 \sin (x) + 5 \cos (x) + 4 \sec (x) \tan (x)$