Cálculo Ejemplos

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

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

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

Paso 2

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

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

Paso 3

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

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

Paso 4

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

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

Paso 5

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

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

Paso 6

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

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

Paso 7

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

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

Paso 8

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

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

Paso 9

Simplifica.

Toca para ver más pasos...

Paso 9.1

Simplifica cada término.

Toca para ver más pasos...

Paso 9.1.1

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

Toca para ver más pasos...

Paso 9.1.1.1

Aplica la propiedad distributiva.

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

Paso 9.1.1.2

Aplica la propiedad distributiva.

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

Paso 9.1.1.3

Aplica la propiedad distributiva.

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

Paso 9.1.2

Simplifica y combina los términos similares.

Toca para ver más pasos...

Paso 9.1.2.1

Simplifica cada término.

Toca para ver más pasos...

Paso 9.1.2.1.1

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

Toca para ver más pasos...

Paso 9.1.2.1.1.1

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

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

Paso 9.1.2.1.1.2

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

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

Paso 9.1.2.1.1.3

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

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

Paso 9.1.2.1.1.4

Suma $1$ y $1$ .

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

Paso 9.1.2.1.2

Reescribe con la propiedad conmutativa de la multiplicación.

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

Paso 9.1.2.1.3

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

Toca para ver más pasos...

Paso 9.1.2.1.3.1

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

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

Paso 9.1.2.1.3.2

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

Toca para ver más pasos...

Paso 9.1.2.1.3.2.1

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

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

Paso 9.1.2.1.3.2.2

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

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

Paso 9.1.2.1.3.3

Suma $2$ y $1$ .

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

Paso 9.1.2.1.4

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

Toca para ver más pasos...

Paso 9.1.2.1.4.1

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

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

Paso 9.1.2.1.4.2

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

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

Paso 9.1.2.1.4.3

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

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

Paso 9.1.2.1.4.4

Suma $1$ y $1$ .

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

Paso 9.1.2.1.5

Reescribe con la propiedad conmutativa de la multiplicación.

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

Paso 9.1.2.2

Reordena los factores de $- \tan (x) \sec^{2} (x)$ .

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

Paso 9.1.2.3

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

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

Paso 9.1.2.4

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

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

Paso 9.1.3

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

Toca para ver más pasos...

Paso 9.1.3.1

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

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

Paso 9.1.3.2

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

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

Paso 9.1.3.3

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

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

Paso 9.1.4

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

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

Paso 9.1.5

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

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

Paso 9.1.6

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

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

Paso 9.1.7

Aplica la identidad pitagórica.

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

Paso 9.1.8

Multiplica $- 1$ por $1$ .

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

Paso 9.1.9

Mueve $- 1$ a la izquierda de $\sec (x)$ .

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

Paso 9.1.10

Reescribe $- 1 \sec (x)$ como $- \sec (x)$ .

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

Paso 9.1.11

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

Toca para ver más pasos...

Paso 9.1.11.1

Aplica la propiedad distributiva.

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

Paso 9.1.11.2

Aplica la propiedad distributiva.

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

Paso 9.1.11.3

Aplica la propiedad distributiva.

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

Paso 9.1.12

Simplifica y combina los términos similares.

Toca para ver más pasos...

Paso 9.1.12.1

Simplifica cada término.

Toca para ver más pasos...

Paso 9.1.12.1.1

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

Toca para ver más pasos...

Paso 9.1.12.1.1.1

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

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

Paso 9.1.12.1.1.2

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

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

Paso 9.1.12.1.1.3

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

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

Paso 9.1.12.1.1.4

Suma $1$ y $1$ .

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

Paso 9.1.12.1.2

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

Toca para ver más pasos...

Paso 9.1.12.1.2.1

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

Toca para ver más pasos...

Paso 9.1.12.1.2.1.1

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

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

Paso 9.1.12.1.2.1.2

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

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

Paso 9.1.12.1.2.2

Suma $1$ y $2$ .

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

Paso 9.1.12.1.3

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

Toca para ver más pasos...

Paso 9.1.12.1.3.1

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

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

Paso 9.1.12.1.3.2

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

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

Paso 9.1.12.1.3.3

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

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

Paso 9.1.12.1.3.4

Suma $1$ y $1$ .

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

Paso 9.1.12.2

Reordena los factores de $- \tan (x) \sec^{2} (x)$ .

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

Paso 9.1.12.3

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

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

Paso 9.1.12.4

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

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

Paso 9.1.13

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

Toca para ver más pasos...

Paso 9.1.13.1

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

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

Paso 9.1.13.2

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

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

Paso 9.1.13.3

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

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

Paso 9.1.14

Aplica la identidad pitagórica.

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

Paso 9.1.15

Multiplica $\sec (x)$ por $1$ .

$- \sec (x) + \sec (x)$

Paso 9.2

Suma $- \sec (x)$ y $\sec (x)$ .

$0$