Cálculo Ejemplos

$f (x) = \frac{3 x^{2} \tan (x)}{\sec (x)}$

Paso 1

Como $3$ es constante con respecto a $x$ , la derivada de $\frac{3 x^{2} \tan (x)}{\sec (x)}$ con respecto a $x$ es $3 \frac{d}{d x} [\frac{x^{2} \tan (x)}{\sec (x)}]$ .

$3 \frac{d}{d x} [\frac{x^{2} \tan (x)}{\sec (x)}]$

Paso 2

Diferencia con la regla del cociente, que establece que $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ es $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ donde $f (x) = x^{2} \tan (x)$ y $g (x) = \sec (x)$ .

$3 \frac{\sec (x) \frac{d}{d x} [x^{2} \tan (x)] - x^{2} \tan (x) \frac{d}{d x} [\sec (x)]}{\sec^{2} (x)}$

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

$3 \frac{\sec (x) (x^{2} \frac{d}{d x} [\tan (x)] + \tan (x) \frac{d}{d x} [x^{2}]) - x^{2} \tan (x) \frac{d}{d x} [\sec (x)]}{\sec^{2} (x)}$

Paso 4

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

$3 \frac{\sec (x) (x^{2} \sec^{2} (x) + \tan (x) \frac{d}{d x} [x^{2}]) - x^{2} \tan (x) \frac{d}{d x} [\sec (x)]}{\sec^{2} (x)}$

Paso 5

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

$3 \frac{\sec (x) (x^{2} \sec^{2} (x) + \tan (x) (2 x)) - x^{2} \tan (x) \frac{d}{d x} [\sec (x)]}{\sec^{2} (x)}$

Paso 6

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

$3 \frac{\sec (x) (x^{2} \sec^{2} (x) + \tan (x) (2 x)) - x^{2} \tan (x) (\sec (x) \tan (x))}{\sec^{2} (x)}$

Paso 7

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

$3 \frac{\sec (x) (x^{2} \sec^{2} (x) + \tan (x) (2 x)) - x^{2} (\tan^{1} (x) \tan (x)) \sec (x)}{\sec^{2} (x)}$

Paso 8

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

$3 \frac{\sec (x) (x^{2} \sec^{2} (x) + \tan (x) (2 x)) - x^{2} (\tan^{1} (x) \tan^{1} (x)) \sec (x)}{\sec^{2} (x)}$

Paso 9

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

$3 \frac{\sec (x) (x^{2} \sec^{2} (x) + \tan (x) (2 x)) - x^{2} {\tan (x)}^{1 + 1} \sec (x)}{\sec^{2} (x)}$

Paso 10

Suma $1$ y $1$ .

$3 \frac{\sec (x) (x^{2} \sec^{2} (x) + \tan (x) (2 x)) - x^{2} \tan^{2} (x) \sec (x)}{\sec^{2} (x)}$

Paso 11

Factoriza $\sec (x)$ de $\sec (x) (x^{2} \sec^{2} (x) + \tan (x) (2 x)) - x^{2} \tan^{2} (x) \sec (x)$ .

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Paso 11.1

Factoriza $\sec (x)$ de $- x^{2} \tan^{2} (x) \sec (x)$ .

$3 \frac{\sec (x) (x^{2} \sec^{2} (x) + \tan (x) (2 x)) + \sec (x) (- x^{2} \tan^{2} (x))}{\sec^{2} (x)}$

Paso 11.2

Factoriza $\sec (x)$ de $\sec (x) (x^{2} \sec^{2} (x) + \tan (x) (2 x)) + \sec (x) (- x^{2} \tan^{2} (x))$ .

$3 \frac{\sec (x) (x^{2} \sec^{2} (x) + \tan (x) (2 x) - x^{2} \tan^{2} (x))}{\sec^{2} (x)}$

Paso 12

Cancela los factores comunes.

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Paso 12.1

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

$3 \frac{\sec (x) (x^{2} \sec^{2} (x) + \tan (x) (2 x) - x^{2} \tan^{2} (x))}{\sec (x) \sec (x)}$

Paso 12.2

Cancela el factor común.

$3 \frac{\sec (x) (x^{2} \sec^{2} (x) + \tan (x) (2 x) - x^{2} \tan^{2} (x))}{\sec (x) \sec (x)}$

Paso 12.3

Reescribe la expresión.

$3 \frac{x^{2} \sec^{2} (x) + \tan (x) (2 x) - x^{2} \tan^{2} (x)}{\sec (x)}$

Paso 13

Combina $3$ y $\frac{x^{2} \sec^{2} (x) + \tan (x) (2 x) - x^{2} \tan^{2} (x)}{\sec (x)}$ .

$\frac{3 (x^{2} \sec^{2} (x) + \tan (x) (2 x) - x^{2} \tan^{2} (x))}{\sec (x)}$

Paso 14

Simplifica.

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Paso 14.1

Aplica la propiedad distributiva.

$\frac{3 (x^{2} \sec^{2} (x)) + 3 (\tan (x) (2 x)) + 3 (- x^{2} \tan^{2} (x))}{\sec (x)}$

Paso 14.2

Simplifica el numerador.

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Paso 14.2.1

Factoriza $3 x$ de $3 (x^{2} \sec^{2} (x)) + 3 (\tan (x) (2 x)) + 3 (- x^{2} \tan^{2} (x))$ .

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Paso 14.2.1.1

Factoriza $3 x$ de $3 (x^{2} \sec^{2} (x))$ .

$\frac{3 x (x \sec^{2} (x)) + 3 (\tan (x) (2 x)) + 3 (- x^{2} \tan^{2} (x))}{\sec (x)}$

Paso 14.2.1.2

Factoriza $3 x$ de $3 (\tan (x) (2 x))$ .

$\frac{3 x (x \sec^{2} (x)) + 3 x (\tan (x) \cdot (2)) + 3 (- x^{2} \tan^{2} (x))}{\sec (x)}$

Paso 14.2.1.3

Factoriza $3 x$ de $3 (- x^{2} \tan^{2} (x))$ .

$\frac{3 x (x \sec^{2} (x)) + 3 x (\tan (x) \cdot (2)) + 3 x (- x \tan^{2} (x))}{\sec (x)}$

Paso 14.2.1.4

Factoriza $3 x$ de $3 x (x \sec^{2} (x)) + 3 x (\tan (x) \cdot (2))$ .

$\frac{3 x (x \sec^{2} (x) + \tan (x) \cdot (2)) + 3 x (- x \tan^{2} (x))}{\sec (x)}$

Paso 14.2.1.5

Factoriza $3 x$ de $3 x (x \sec^{2} (x) + \tan (x) \cdot (2)) + 3 x (- x \tan^{2} (x))$ .

$\frac{3 x (x \sec^{2} (x) + \tan (x) \cdot (2) - x \tan^{2} (x))}{\sec (x)}$

Paso 14.2.2

Mueve $- x \tan^{2} (x)$ .

$\frac{3 x (x \sec^{2} (x) - x \tan^{2} (x) + \tan (x) \cdot (2))}{\sec (x)}$

Paso 14.2.3

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

$\frac{3 x (x (\sec^{2} (x)) - x \tan^{2} (x) + \tan (x) \cdot (2))}{\sec (x)}$

Paso 14.2.4

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

$\frac{3 x (x (\sec^{2} (x)) + x (- 1 \tan^{2} (x)) + \tan (x) \cdot (2))}{\sec (x)}$

Paso 14.2.5

Factoriza $x$ de $x (\sec^{2} (x)) + x (- 1 \tan^{2} (x))$ .

$\frac{3 x (x (\sec^{2} (x) - 1 \tan^{2} (x)) + \tan (x) \cdot (2))}{\sec (x)}$

Paso 14.2.6

Aplica la identidad pitagórica.

$\frac{3 x (x \cdot 1 + \tan (x) \cdot (2))}{\sec (x)}$

Paso 14.2.7

Simplifica cada término.

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Paso 14.2.7.1

Multiplica $x$ por $1$ .

$\frac{3 x (x + \tan (x) \cdot (2))}{\sec (x)}$

Paso 14.2.7.2

Mueve $2$ a la izquierda de $\tan (x)$ .

$\frac{3 x (x + 2 \tan (x))}{\sec (x)}$

Paso 14.2.8

Aplica la propiedad distributiva.

$\frac{3 x \cdot x + 3 x (2 \tan (x))}{\sec (x)}$

Paso 14.2.9

Multiplica $x$ por $x$ sumando los exponentes.

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Paso 14.2.9.1

Mueve $x$ .

$\frac{3 (x \cdot x) + 3 x (2 \tan (x))}{\sec (x)}$

Paso 14.2.9.2

Multiplica $x$ por $x$ .

$\frac{3 x^{2} + 3 x (2 \tan (x))}{\sec (x)}$

Paso 14.2.10

Multiplica $2$ por $3$ .

$\frac{3 x^{2} + 6 x \tan (x)}{\sec (x)}$

Paso 14.3

Factoriza $3 x$ de $3 x^{2} + 6 x \tan (x)$ .

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Paso 14.3.1

Factoriza $3 x$ de $3 x^{2}$ .

$\frac{3 x (x) + 6 x \tan (x)}{\sec (x)}$

Paso 14.3.2

Factoriza $3 x$ de $6 x \tan (x)$ .

$\frac{3 x (x) + 3 x (2 \tan (x))}{\sec (x)}$

Paso 14.3.3

Factoriza $3 x$ de $3 x (x) + 3 x (2 \tan (x))$ .

$\frac{3 x (x + 2 \tan (x))}{\sec (x)}$