Cálculo Ejemplos

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

Paso 1

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 $f (x) = 3 x^{2} \tan (x)$ y $g (x) = \sec (x)$ .

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

Paso 2

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

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

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

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

Paso 4

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

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

Paso 5

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

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

Paso 6

Simplifica.

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Paso 6.1

Aplica la propiedad distributiva.

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

Paso 6.2

Multiplica $2$ por $3$ .

$\frac{\sec (x) (3 x^{2} \sec^{2} (x) + 6 \tan (x) x) - 1 \cdot 3 x^{2} \tan (x) \frac{d}{d x} [\sec (x)]}{\sec^{2} (x)}$

Paso 6.3

Reordena los términos.

$\frac{\sec (x) (3 x^{2} \sec^{2} (x) + 6 x \tan (x)) - 1 \cdot 3 x^{2} \tan (x) \frac{d}{d x} [\sec (x)]}{\sec^{2} (x)}$

Paso 7

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

$\frac{\sec (x) (3 x^{2} \sec^{2} (x) + 6 x \tan (x)) - 1 \cdot 3 x^{2} \tan (x) (\sec (x) \tan (x))}{\sec^{2} (x)}$

Paso 8

Simplifica.

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Paso 8.1

Aplica la propiedad distributiva.

$\frac{\sec (x) (3 x^{2} \sec^{2} (x)) + \sec (x) (6 x \tan (x)) - 1 \cdot 3 x^{2} \tan (x) (\sec (x) \tan (x))}{\sec^{2} (x)}$

Paso 8.2

Simplifica el numerador.

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Paso 8.2.1

Simplifica cada término.

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Paso 8.2.1.1

Reescribe con la propiedad conmutativa de la multiplicación.

$\frac{3 \sec (x) x^{2} \sec^{2} (x) + \sec (x) (6 x \tan (x)) - 1 \cdot 3 x^{2} \tan (x) (\sec (x) \tan (x))}{\sec^{2} (x)}$

Paso 8.2.1.2

Multiplica $\sec (x)$ por $\sec^{2} (x)$ sumando los exponentes.

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Paso 8.2.1.2.1

Mueve $\sec^{2} (x)$ .

$\frac{3 (\sec^{2} (x) \sec (x)) x^{2} + \sec (x) (6 x \tan (x)) - 1 \cdot 3 x^{2} \tan (x) (\sec (x) \tan (x))}{\sec^{2} (x)}$

Paso 8.2.1.2.2

Multiplica $\sec^{2} (x)$ por $\sec (x)$ .

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Paso 8.2.1.2.2.1

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

$\frac{3 (\sec^{2} (x) \sec^{1} (x)) x^{2} + \sec (x) (6 x \tan (x)) - 1 \cdot 3 x^{2} \tan (x) (\sec (x) \tan (x))}{\sec^{2} (x)}$

Paso 8.2.1.2.2.2

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

$\frac{3 {\sec (x)}^{2 + 1} x^{2} + \sec (x) (6 x \tan (x)) - 1 \cdot 3 x^{2} \tan (x) (\sec (x) \tan (x))}{\sec^{2} (x)}$

Paso 8.2.1.2.3

Suma $2$ y $1$ .

$\frac{3 \sec^{3} (x) x^{2} + \sec (x) (6 x \tan (x)) - 1 \cdot 3 x^{2} \tan (x) (\sec (x) \tan (x))}{\sec^{2} (x)}$

Paso 8.2.1.3

Reescribe con la propiedad conmutativa de la multiplicación.

$\frac{3 \sec^{3} (x) x^{2} + 6 \sec (x) x \tan (x) - 1 \cdot 3 x^{2} \tan (x) (\sec (x) \tan (x))}{\sec^{2} (x)}$

Paso 8.2.1.4

Multiplica $- 1$ por $3$ .

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

Paso 8.2.1.5

Multiplica $- 3 x^{2} \tan (x) (\sec (x) \tan (x))$ .

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Paso 8.2.1.5.1

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

$\frac{3 \sec^{3} (x) x^{2} + 6 \sec (x) x \tan (x) - 3 x^{2} (\tan^{1} (x) \tan (x)) \sec (x)}{\sec^{2} (x)}$

Paso 8.2.1.5.2

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

$\frac{3 \sec^{3} (x) x^{2} + 6 \sec (x) x \tan (x) - 3 x^{2} (\tan^{1} (x) \tan^{1} (x)) \sec (x)}{\sec^{2} (x)}$

Paso 8.2.1.5.3

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

$\frac{3 \sec^{3} (x) x^{2} + 6 \sec (x) x \tan (x) - 3 x^{2} {\tan (x)}^{1 + 1} \sec (x)}{\sec^{2} (x)}$

Paso 8.2.1.5.4

Suma $1$ y $1$ .

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

Paso 8.2.2

Reordena los factores en $3 \sec^{3} (x) x^{2} + 6 \sec (x) x \tan (x) - 3 x^{2} \tan^{2} (x) \sec (x)$ .

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

Paso 8.3

Simplifica el numerador.

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Paso 8.3.1

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

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Paso 8.3.1.1

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

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

Paso 8.3.1.2

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

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

Paso 8.3.1.3

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

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

Paso 8.3.1.4

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

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

Paso 8.3.1.5

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

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

Paso 8.3.2

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

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

Paso 8.3.3

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

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

Paso 8.3.4

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

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

Paso 8.3.5

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

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

Paso 8.3.6

Aplica la identidad pitagórica.

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

Paso 8.3.7

Multiplica $x$ por $1$ .

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

Paso 8.4

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

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Paso 8.4.1

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

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

Paso 8.4.2

Cancela los factores comunes.

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Paso 8.4.2.1

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

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

Paso 8.4.2.2

Cancela el factor común.

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

Paso 8.4.2.3

Reescribe la expresión.

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