Cálculo Ejemplos

$y = 5 (\tan (x) + \sec (x)) (\tan (x) - \sec (x))$

Paso 1

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

$5 \frac{d}{d x} [(\tan (x) + \sec (x)) (\tan (x) - \sec (x))]$

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 $f (x) = \tan (x) + \sec (x)$ y $g (x) = \tan (x) - \sec (x)$ .

$5 ((\tan (x) + \sec (x)) \frac{d}{d x} [\tan (x) - \sec (x)] + (\tan (x) - \sec (x)) \frac{d}{d x} [\tan (x) + \sec (x)])$

Paso 3

Según la regla de la suma, la derivada de $\tan (x) - \sec (x)$ con respecto a $x$ es $\frac{d}{d x} [\tan (x)] + \frac{d}{d x} [- \sec (x)]$ .

$5 ((\tan (x) + \sec (x)) (\frac{d}{d x} [\tan (x)] + \frac{d}{d x} [- \sec (x)]) + (\tan (x) - \sec (x)) \frac{d}{d x} [\tan (x) + \sec (x)])$

Paso 4

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

$5 ((\tan (x) + \sec (x)) (\sec^{2} (x) + \frac{d}{d x} [- \sec (x)]) + (\tan (x) - \sec (x)) \frac{d}{d x} [\tan (x) + \sec (x)])$

Paso 5

Como $- 1$ es constante con respecto a $x$ , la derivada de $- \sec (x)$ con respecto a $x$ es $- \frac{d}{d x} [\sec (x)]$ .

$5 ((\tan (x) + \sec (x)) (\sec^{2} (x) - \frac{d}{d x} [\sec (x)]) + (\tan (x) - \sec (x)) \frac{d}{d x} [\tan (x) + \sec (x)])$

Paso 6

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

$5 ((\tan (x) + \sec (x)) (\sec^{2} (x) - (\sec (x) \tan (x))) + (\tan (x) - \sec (x)) \frac{d}{d x} [\tan (x) + \sec (x)])$

Paso 7

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

$5 ((\tan (x) + \sec (x)) (\sec^{2} (x) - \sec (x) \tan (x)) + (\tan (x) - \sec (x)) (\frac{d}{d x} [\tan (x)] + \frac{d}{d x} [\sec (x)]))$

Paso 8

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

$5 ((\tan (x) + \sec (x)) (\sec^{2} (x) - \sec (x) \tan (x)) + (\tan (x) - \sec (x)) (\sec^{2} (x) + \frac{d}{d x} [\sec (x)]))$

Paso 9

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

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

Paso 10

Simplifica.

Toca para ver más pasos...

Paso 10.1

Aplica la propiedad distributiva.

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

Paso 10.2

Elimina los paréntesis.

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

Paso 10.3

Reordena los términos.

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

Paso 10.4

Simplifica cada término.

Toca para ver más pasos...

Paso 10.4.1

Aplica la propiedad distributiva.

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

Paso 10.4.2

Multiplica $- 1$ por $5$ .

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

Paso 10.4.3

Expande $(5 \sec^{2} (x) - 5 \sec (x) \tan (x)) (\tan (x) + \sec (x))$ con el método PEIU (primero, exterior, interior, ultimo).

Toca para ver más pasos...

Paso 10.4.3.1

Aplica la propiedad distributiva.

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

Paso 10.4.3.2

Aplica la propiedad distributiva.

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

Paso 10.4.3.3

Aplica la propiedad distributiva.

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

Paso 10.4.4

Simplifica y combina los términos similares.

Toca para ver más pasos...

Paso 10.4.4.1

Simplifica cada término.

Toca para ver más pasos...

Paso 10.4.4.1.1

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

Toca para ver más pasos...

Paso 10.4.4.1.1.1

Mueve $\sec (x)$ .

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

Paso 10.4.4.1.1.2

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

Toca para ver más pasos...

Paso 10.4.4.1.1.2.1

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

$5 \sec^{2} (x) \tan (x) + 5 (\sec^{1} (x) \sec^{2} (x)) - 5 \sec (x) \tan (x) \tan (x) - 5 \sec (x) \tan (x) \sec (x) + 5 (\sec^{2} (x) + \sec (x) \tan (x)) (\tan (x) - \sec (x))$

Paso 10.4.4.1.1.2.2

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

$5 \sec^{2} (x) \tan (x) + 5 {\sec (x)}^{1 + 2} - 5 \sec (x) \tan (x) \tan (x) - 5 \sec (x) \tan (x) \sec (x) + 5 (\sec^{2} (x) + \sec (x) \tan (x)) (\tan (x) - \sec (x))$

Paso 10.4.4.1.1.3

Suma $1$ y $2$ .

$5 \sec^{2} (x) \tan (x) + 5 \sec^{3} (x) - 5 \sec (x) \tan (x) \tan (x) - 5 \sec (x) \tan (x) \sec (x) + 5 (\sec^{2} (x) + \sec (x) \tan (x)) (\tan (x) - \sec (x))$

Paso 10.4.4.1.2

Multiplica $- 5 \sec (x) \tan (x) \tan (x)$ .

Toca para ver más pasos...

Paso 10.4.4.1.2.1

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

$5 \sec^{2} (x) \tan (x) + 5 \sec^{3} (x) - 5 \sec (x) (\tan^{1} (x) \tan (x)) - 5 \sec (x) \tan (x) \sec (x) + 5 (\sec^{2} (x) + \sec (x) \tan (x)) (\tan (x) - \sec (x))$

Paso 10.4.4.1.2.2

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

$5 \sec^{2} (x) \tan (x) + 5 \sec^{3} (x) - 5 \sec (x) (\tan^{1} (x) \tan^{1} (x)) - 5 \sec (x) \tan (x) \sec (x) + 5 (\sec^{2} (x) + \sec (x) \tan (x)) (\tan (x) - \sec (x))$

Paso 10.4.4.1.2.3

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

$5 \sec^{2} (x) \tan (x) + 5 \sec^{3} (x) - 5 \sec (x) {\tan (x)}^{1 + 1} - 5 \sec (x) \tan (x) \sec (x) + 5 (\sec^{2} (x) + \sec (x) \tan (x)) (\tan (x) - \sec (x))$

Paso 10.4.4.1.2.4

Suma $1$ y $1$ .

$5 \sec^{2} (x) \tan (x) + 5 \sec^{3} (x) - 5 \sec (x) \tan^{2} (x) - 5 \sec (x) \tan (x) \sec (x) + 5 (\sec^{2} (x) + \sec (x) \tan (x)) (\tan (x) - \sec (x))$

Paso 10.4.4.1.3

Multiplica $- 5 \sec (x) \tan (x) \sec (x)$ .

Toca para ver más pasos...

Paso 10.4.4.1.3.1

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

$5 \sec^{2} (x) \tan (x) + 5 \sec^{3} (x) - 5 \sec (x) \tan^{2} (x) - 5 (\sec^{1} (x) \sec (x)) \tan (x) + 5 (\sec^{2} (x) + \sec (x) \tan (x)) (\tan (x) - \sec (x))$

Paso 10.4.4.1.3.2

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

$5 \sec^{2} (x) \tan (x) + 5 \sec^{3} (x) - 5 \sec (x) \tan^{2} (x) - 5 (\sec^{1} (x) \sec^{1} (x)) \tan (x) + 5 (\sec^{2} (x) + \sec (x) \tan (x)) (\tan (x) - \sec (x))$

Paso 10.4.4.1.3.3

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

$5 \sec^{2} (x) \tan (x) + 5 \sec^{3} (x) - 5 \sec (x) \tan^{2} (x) - 5 {\sec (x)}^{1 + 1} \tan (x) + 5 (\sec^{2} (x) + \sec (x) \tan (x)) (\tan (x) - \sec (x))$

Paso 10.4.4.1.3.4

Suma $1$ y $1$ .

$5 \sec^{2} (x) \tan (x) + 5 \sec^{3} (x) - 5 \sec (x) \tan^{2} (x) - 5 \sec^{2} (x) \tan (x) + 5 (\sec^{2} (x) + \sec (x) \tan (x)) (\tan (x) - \sec (x))$

Paso 10.4.4.2

Resta $5 \sec^{2} (x) \tan (x)$ de $5 \sec^{2} (x) \tan (x)$ .

$0 + 5 \sec^{3} (x) - 5 \sec (x) \tan^{2} (x) + 5 (\sec^{2} (x) + \sec (x) \tan (x)) (\tan (x) - \sec (x))$

Paso 10.4.4.3

Suma $0$ y $5 \sec^{3} (x)$ .

$5 \sec^{3} (x) - 5 \sec (x) \tan^{2} (x) + 5 (\sec^{2} (x) + \sec (x) \tan (x)) (\tan (x) - \sec (x))$

Paso 10.4.5

Factoriza $5 \sec (x)$ de $5 \sec^{3} (x) - 5 \sec (x) \tan^{2} (x)$ .

Toca para ver más pasos...

Paso 10.4.5.1

Factoriza $5 \sec (x)$ de $5 \sec^{3} (x)$ .

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

Paso 10.4.5.2

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

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

Paso 10.4.5.3

Factoriza $5 \sec (x)$ de $5 \sec (x) (\sec^{2} (x)) + 5 \sec (x) (- \tan^{2} (x))$ .

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

Paso 10.4.6

Aplica la identidad pitagórica.

$5 \sec (x) \cdot 1 + 5 (\sec^{2} (x) + \sec (x) \tan (x)) (\tan (x) - \sec (x))$

Paso 10.4.7

Multiplica $5$ por $1$ .

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

Paso 10.4.8

Aplica la propiedad distributiva.

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

Paso 10.4.9

Expande $(5 \sec^{2} (x) + 5 \sec (x) \tan (x)) (\tan (x) - \sec (x))$ con el método PEIU (primero, exterior, interior, ultimo).

Toca para ver más pasos...

Paso 10.4.9.1

Aplica la propiedad distributiva.

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

Paso 10.4.9.2

Aplica la propiedad distributiva.

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

Paso 10.4.9.3

Aplica la propiedad distributiva.

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

Paso 10.4.10

Simplifica y combina los términos similares.

Toca para ver más pasos...

Paso 10.4.10.1

Simplifica cada término.

Toca para ver más pasos...

Paso 10.4.10.1.1

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

Toca para ver más pasos...

Paso 10.4.10.1.1.1

Mueve $\sec (x)$ .

$5 \sec (x) + 5 \sec^{2} (x) \tan (x) + 5 (\sec (x) \sec^{2} (x)) \cdot - 1 + 5 \sec (x) \tan (x) \tan (x) + 5 \sec (x) \tan (x) (- \sec (x))$

Paso 10.4.10.1.1.2

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

Toca para ver más pasos...

Paso 10.4.10.1.1.2.1

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

$5 \sec (x) + 5 \sec^{2} (x) \tan (x) + 5 (\sec^{1} (x) \sec^{2} (x)) \cdot - 1 + 5 \sec (x) \tan (x) \tan (x) + 5 \sec (x) \tan (x) (- \sec (x))$

Paso 10.4.10.1.1.2.2

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

$5 \sec (x) + 5 \sec^{2} (x) \tan (x) + 5 {\sec (x)}^{1 + 2} \cdot - 1 + 5 \sec (x) \tan (x) \tan (x) + 5 \sec (x) \tan (x) (- \sec (x))$

Paso 10.4.10.1.1.3

Suma $1$ y $2$ .

$5 \sec (x) + 5 \sec^{2} (x) \tan (x) + 5 \sec^{3} (x) \cdot - 1 + 5 \sec (x) \tan (x) \tan (x) + 5 \sec (x) \tan (x) (- \sec (x))$

Paso 10.4.10.1.2

Multiplica $- 1$ por $5$ .

$5 \sec (x) + 5 \sec^{2} (x) \tan (x) - 5 \sec^{3} (x) + 5 \sec (x) \tan (x) \tan (x) + 5 \sec (x) \tan (x) (- \sec (x))$

Paso 10.4.10.1.3

Multiplica $5 \sec (x) \tan (x) \tan (x)$ .

Toca para ver más pasos...

Paso 10.4.10.1.3.1

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

$5 \sec (x) + 5 \sec^{2} (x) \tan (x) - 5 \sec^{3} (x) + 5 \sec (x) (\tan^{1} (x) \tan (x)) + 5 \sec (x) \tan (x) (- \sec (x))$

Paso 10.4.10.1.3.2

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

$5 \sec (x) + 5 \sec^{2} (x) \tan (x) - 5 \sec^{3} (x) + 5 \sec (x) (\tan^{1} (x) \tan^{1} (x)) + 5 \sec (x) \tan (x) (- \sec (x))$

Paso 10.4.10.1.3.3

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

$5 \sec (x) + 5 \sec^{2} (x) \tan (x) - 5 \sec^{3} (x) + 5 \sec (x) {\tan (x)}^{1 + 1} + 5 \sec (x) \tan (x) (- \sec (x))$

Paso 10.4.10.1.3.4

Suma $1$ y $1$ .

$5 \sec (x) + 5 \sec^{2} (x) \tan (x) - 5 \sec^{3} (x) + 5 \sec (x) \tan^{2} (x) + 5 \sec (x) \tan (x) (- \sec (x))$

Paso 10.4.10.1.4

Multiplica $5 \sec (x) \tan (x) (- \sec (x))$ .

Toca para ver más pasos...

Paso 10.4.10.1.4.1

Multiplica $- 1$ por $5$ .

$5 \sec (x) + 5 \sec^{2} (x) \tan (x) - 5 \sec^{3} (x) + 5 \sec (x) \tan^{2} (x) - 5 \sec (x) \tan (x) \sec (x)$

Paso 10.4.10.1.4.2

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

$5 \sec (x) + 5 \sec^{2} (x) \tan (x) - 5 \sec^{3} (x) + 5 \sec (x) \tan^{2} (x) - 5 (\sec^{1} (x) \sec (x)) \tan (x)$

Paso 10.4.10.1.4.3

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

$5 \sec (x) + 5 \sec^{2} (x) \tan (x) - 5 \sec^{3} (x) + 5 \sec (x) \tan^{2} (x) - 5 (\sec^{1} (x) \sec^{1} (x)) \tan (x)$

Paso 10.4.10.1.4.4

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

$5 \sec (x) + 5 \sec^{2} (x) \tan (x) - 5 \sec^{3} (x) + 5 \sec (x) \tan^{2} (x) - 5 {\sec (x)}^{1 + 1} \tan (x)$

Paso 10.4.10.1.4.5

Suma $1$ y $1$ .

$5 \sec (x) + 5 \sec^{2} (x) \tan (x) - 5 \sec^{3} (x) + 5 \sec (x) \tan^{2} (x) - 5 \sec^{2} (x) \tan (x)$

Paso 10.4.10.2

Resta $5 \sec^{2} (x) \tan (x)$ de $5 \sec^{2} (x) \tan (x)$ .

$5 \sec (x) + 0 - 5 \sec^{3} (x) + 5 \sec (x) \tan^{2} (x)$

Paso 10.4.10.3

Resta $5 \sec^{3} (x)$ de $0$ .

$5 \sec (x) - 5 \sec^{3} (x) + 5 \sec (x) \tan^{2} (x)$

Paso 10.4.11

Factoriza $5 \sec (x)$ de $- 5 \sec^{3} (x) + 5 \sec (x) \tan^{2} (x)$ .

Toca para ver más pasos...

Paso 10.4.11.1

Factoriza $5 \sec (x)$ de $- 5 \sec^{3} (x)$ .

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

Paso 10.4.11.2

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

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

Paso 10.4.11.3

Factoriza $5 \sec (x)$ de $5 \sec (x) (- \sec^{2} (x)) + 5 \sec (x) (\tan^{2} (x))$ .

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

Paso 10.4.12

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

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

Paso 10.4.13

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

$5 \sec (x) + 5 \sec (x) (- (\sec^{2} (x)) - 1 (- \tan^{2} (x)))$

Paso 10.4.14

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

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

Paso 10.4.15

Aplica la identidad pitagórica.

$5 \sec (x) + 5 \sec (x) (- 1 \cdot 1)$

Paso 10.4.16

Multiplica $- 1$ por $1$ .

$5 \sec (x) + 5 \sec (x) \cdot - 1$

Paso 10.4.17

Multiplica $- 1$ por $5$ .

$5 \sec (x) - 5 \sec (x)$

Paso 10.5

Resta $5 \sec (x)$ de $5 \sec (x)$ .

$0$