Cálculo Ejemplos

$y = (\csc (x) + \cot (x)) (\csc (x) - \cot (x))$

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 $f (x) = \csc (x) + \cot (x)$ y $g (x) = \csc (x) - \cot (x)$ .

$(\csc (x) + \cot (x)) \frac{d}{d x} [\csc (x) - \cot (x)] + (\csc (x) - \cot (x)) \frac{d}{d x} [\csc (x) + \cot (x)]$

Paso 2

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

$(\csc (x) + \cot (x)) (\frac{d}{d x} [\csc (x)] + \frac{d}{d x} [- \cot (x)]) + (\csc (x) - \cot (x)) \frac{d}{d x} [\csc (x) + \cot (x)]$

Paso 3

La derivada de $\csc (x)$ con respecto a $x$ es $- \csc (x) \cot (x)$ .

$(\csc (x) + \cot (x)) (- \csc (x) \cot (x) + \frac{d}{d x} [- \cot (x)]) + (\csc (x) - \cot (x)) \frac{d}{d x} [\csc (x) + \cot (x)]$

Paso 4

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

$(\csc (x) + \cot (x)) (- \csc (x) \cot (x) - \frac{d}{d x} [\cot (x)]) + (\csc (x) - \cot (x)) \frac{d}{d x} [\csc (x) + \cot (x)]$

Paso 5

La derivada de $\cot (x)$ con respecto a $x$ es $- \csc^{2} (x)$ .

$(\csc (x) + \cot (x)) (- \csc (x) \cot (x) - - \csc^{2} (x)) + (\csc (x) - \cot (x)) \frac{d}{d x} [\csc (x) + \cot (x)]$

Paso 6

Diferencia con la regla de la suma.

Toca para ver más pasos...

Paso 6.1

Multiplica $- 1$ por $- 1$ .

$(\csc (x) + \cot (x)) (- \csc (x) \cot (x) + 1 \csc^{2} (x)) + (\csc (x) - \cot (x)) \frac{d}{d x} [\csc (x) + \cot (x)]$

Paso 6.2

Multiplica $\csc^{2} (x)$ por $1$ .

$(\csc (x) + \cot (x)) (- \csc (x) \cot (x) + \csc^{2} (x)) + (\csc (x) - \cot (x)) \frac{d}{d x} [\csc (x) + \cot (x)]$

Paso 6.3

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

$(\csc (x) + \cot (x)) (- \csc (x) \cot (x) + \csc^{2} (x)) + (\csc (x) - \cot (x)) (\frac{d}{d x} [\csc (x)] + \frac{d}{d x} [\cot (x)])$

Paso 7

La derivada de $\csc (x)$ con respecto a $x$ es $- \csc (x) \cot (x)$ .

$(\csc (x) + \cot (x)) (- \csc (x) \cot (x) + \csc^{2} (x)) + (\csc (x) - \cot (x)) (- \csc (x) \cot (x) + \frac{d}{d x} [\cot (x)])$

Paso 8

La derivada de $\cot (x)$ con respecto a $x$ es $- \csc^{2} (x)$ .

$(\csc (x) + \cot (x)) (- \csc (x) \cot (x) + \csc^{2} (x)) + (\csc (x) - \cot (x)) (- \csc (x) \cot (x) - \csc^{2} (x))$

Paso 9

Simplifica.

Toca para ver más pasos...

Paso 9.1

Reordena los términos.

$(- \csc (x) \cot (x) + \csc^{2} (x)) (\csc (x) + \cot (x)) + (- \csc (x) \cot (x) - \csc^{2} (x)) (\csc (x) - \cot (x))$

Paso 9.2

Simplifica cada término.

Toca para ver más pasos...

Paso 9.2.1

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

Toca para ver más pasos...

Paso 9.2.1.1

Aplica la propiedad distributiva.

$- \csc (x) \cot (x) (\csc (x) + \cot (x)) + \csc^{2} (x) (\csc (x) + \cot (x)) + (- \csc (x) \cot (x) - \csc^{2} (x)) (\csc (x) - \cot (x))$

Paso 9.2.1.2

Aplica la propiedad distributiva.

$- \csc (x) \cot (x) \csc (x) - \csc (x) \cot (x) \cot (x) + \csc^{2} (x) (\csc (x) + \cot (x)) + (- \csc (x) \cot (x) - \csc^{2} (x)) (\csc (x) - \cot (x))$

Paso 9.2.1.3

Aplica la propiedad distributiva.

$- \csc (x) \cot (x) \csc (x) - \csc (x) \cot (x) \cot (x) + \csc^{2} (x) \csc (x) + \csc^{2} (x) \cot (x) + (- \csc (x) \cot (x) - \csc^{2} (x)) (\csc (x) - \cot (x))$

Paso 9.2.2

Simplifica y combina los términos similares.

Toca para ver más pasos...

Paso 9.2.2.1

Simplifica cada término.

Toca para ver más pasos...

Paso 9.2.2.1.1

Multiplica $- \csc (x) \cot (x) \csc (x)$ .

Toca para ver más pasos...

Paso 9.2.2.1.1.1

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

$- (\csc^{1} (x) \csc (x)) \cot (x) - \csc (x) \cot (x) \cot (x) + \csc^{2} (x) \csc (x) + \csc^{2} (x) \cot (x) + (- \csc (x) \cot (x) - \csc^{2} (x)) (\csc (x) - \cot (x))$

Paso 9.2.2.1.1.2

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

$- (\csc^{1} (x) \csc^{1} (x)) \cot (x) - \csc (x) \cot (x) \cot (x) + \csc^{2} (x) \csc (x) + \csc^{2} (x) \cot (x) + (- \csc (x) \cot (x) - \csc^{2} (x)) (\csc (x) - \cot (x))$

Paso 9.2.2.1.1.3

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

$- {\csc (x)}^{1 + 1} \cot (x) - \csc (x) \cot (x) \cot (x) + \csc^{2} (x) \csc (x) + \csc^{2} (x) \cot (x) + (- \csc (x) \cot (x) - \csc^{2} (x)) (\csc (x) - \cot (x))$

Paso 9.2.2.1.1.4

Suma $1$ y $1$ .

$- \csc^{2} (x) \cot (x) - \csc (x) \cot (x) \cot (x) + \csc^{2} (x) \csc (x) + \csc^{2} (x) \cot (x) + (- \csc (x) \cot (x) - \csc^{2} (x)) (\csc (x) - \cot (x))$

Paso 9.2.2.1.2

Multiplica $- \csc (x) \cot (x) \cot (x)$ .

Toca para ver más pasos...

Paso 9.2.2.1.2.1

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

$- \csc^{2} (x) \cot (x) - \csc (x) (\cot^{1} (x) \cot (x)) + \csc^{2} (x) \csc (x) + \csc^{2} (x) \cot (x) + (- \csc (x) \cot (x) - \csc^{2} (x)) (\csc (x) - \cot (x))$

Paso 9.2.2.1.2.2

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

$- \csc^{2} (x) \cot (x) - \csc (x) (\cot^{1} (x) \cot^{1} (x)) + \csc^{2} (x) \csc (x) + \csc^{2} (x) \cot (x) + (- \csc (x) \cot (x) - \csc^{2} (x)) (\csc (x) - \cot (x))$

Paso 9.2.2.1.2.3

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

$- \csc^{2} (x) \cot (x) - \csc (x) {\cot (x)}^{1 + 1} + \csc^{2} (x) \csc (x) + \csc^{2} (x) \cot (x) + (- \csc (x) \cot (x) - \csc^{2} (x)) (\csc (x) - \cot (x))$

Paso 9.2.2.1.2.4

Suma $1$ y $1$ .

$- \csc^{2} (x) \cot (x) - \csc (x) \cot^{2} (x) + \csc^{2} (x) \csc (x) + \csc^{2} (x) \cot (x) + (- \csc (x) \cot (x) - \csc^{2} (x)) (\csc (x) - \cot (x))$

Paso 9.2.2.1.3

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

Toca para ver más pasos...

Paso 9.2.2.1.3.1

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

Toca para ver más pasos...

Paso 9.2.2.1.3.1.1

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

$- \csc^{2} (x) \cot (x) - \csc (x) \cot^{2} (x) + \csc^{2} (x) \csc^{1} (x) + \csc^{2} (x) \cot (x) + (- \csc (x) \cot (x) - \csc^{2} (x)) (\csc (x) - \cot (x))$

Paso 9.2.2.1.3.1.2

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

$- \csc^{2} (x) \cot (x) - \csc (x) \cot^{2} (x) + {\csc (x)}^{2 + 1} + \csc^{2} (x) \cot (x) + (- \csc (x) \cot (x) - \csc^{2} (x)) (\csc (x) - \cot (x))$

Paso 9.2.2.1.3.2

Suma $2$ y $1$ .

$- \csc^{2} (x) \cot (x) - \csc (x) \cot^{2} (x) + \csc^{3} (x) + \csc^{2} (x) \cot (x) + (- \csc (x) \cot (x) - \csc^{2} (x)) (\csc (x) - \cot (x))$

Paso 9.2.2.2

Suma $- \csc^{2} (x) \cot (x)$ y $\csc^{2} (x) \cot (x)$ .

$0 - \csc (x) \cot^{2} (x) + \csc^{3} (x) + (- \csc (x) \cot (x) - \csc^{2} (x)) (\csc (x) - \cot (x))$

Paso 9.2.2.3

Resta $\csc (x) \cot^{2} (x)$ de $0$ .

$- \csc (x) \cot^{2} (x) + \csc^{3} (x) + (- \csc (x) \cot (x) - \csc^{2} (x)) (\csc (x) - \cot (x))$

Paso 9.2.3

Factoriza $\csc (x)$ de $- \csc (x) \cot^{2} (x) + \csc^{3} (x)$ .

Toca para ver más pasos...

Paso 9.2.3.1

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

$\csc (x) (- \cot^{2} (x)) + \csc^{3} (x) + (- \csc (x) \cot (x) - \csc^{2} (x)) (\csc (x) - \cot (x))$

Paso 9.2.3.2

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

$\csc (x) (- \cot^{2} (x)) + \csc (x) \csc^{2} (x) + (- \csc (x) \cot (x) - \csc^{2} (x)) (\csc (x) - \cot (x))$

Paso 9.2.3.3

Factoriza $\csc (x)$ de $\csc (x) (- \cot^{2} (x)) + \csc (x) \csc^{2} (x)$ .

$\csc (x) (- \cot^{2} (x) + \csc^{2} (x)) + (- \csc (x) \cot (x) - \csc^{2} (x)) (\csc (x) - \cot (x))$

Paso 9.2.4

Reordena $- \cot^{2} (x)$ y $\csc^{2} (x)$ .

$\csc (x) (\csc^{2} (x) - \cot^{2} (x)) + (- \csc (x) \cot (x) - \csc^{2} (x)) (\csc (x) - \cot (x))$

Paso 9.2.5

Aplica la identidad pitagórica.

$\csc (x) \cdot 1 + (- \csc (x) \cot (x) - \csc^{2} (x)) (\csc (x) - \cot (x))$

Paso 9.2.6

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

$\csc (x) + (- \csc (x) \cot (x) - \csc^{2} (x)) (\csc (x) - \cot (x))$

Paso 9.2.7

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

Toca para ver más pasos...

Paso 9.2.7.1

Aplica la propiedad distributiva.

$\csc (x) - \csc (x) \cot (x) (\csc (x) - \cot (x)) - \csc^{2} (x) (\csc (x) - \cot (x))$

Paso 9.2.7.2

Aplica la propiedad distributiva.

$\csc (x) - \csc (x) \cot (x) \csc (x) - \csc (x) \cot (x) (- \cot (x)) - \csc^{2} (x) (\csc (x) - \cot (x))$

Paso 9.2.7.3

Aplica la propiedad distributiva.

$\csc (x) - \csc (x) \cot (x) \csc (x) - \csc (x) \cot (x) (- \cot (x)) - \csc^{2} (x) \csc (x) - \csc^{2} (x) (- \cot (x))$

Paso 9.2.8

Simplifica y combina los términos similares.

Toca para ver más pasos...

Paso 9.2.8.1

Simplifica cada término.

Toca para ver más pasos...

Paso 9.2.8.1.1

Multiplica $- \csc (x) \cot (x) \csc (x)$ .

Toca para ver más pasos...

Paso 9.2.8.1.1.1

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

$\csc (x) - (\csc^{1} (x) \csc (x)) \cot (x) - \csc (x) \cot (x) (- \cot (x)) - \csc^{2} (x) \csc (x) - \csc^{2} (x) (- \cot (x))$

Paso 9.2.8.1.1.2

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

$\csc (x) - (\csc^{1} (x) \csc^{1} (x)) \cot (x) - \csc (x) \cot (x) (- \cot (x)) - \csc^{2} (x) \csc (x) - \csc^{2} (x) (- \cot (x))$

Paso 9.2.8.1.1.3

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

$\csc (x) - {\csc (x)}^{1 + 1} \cot (x) - \csc (x) \cot (x) (- \cot (x)) - \csc^{2} (x) \csc (x) - \csc^{2} (x) (- \cot (x))$

Paso 9.2.8.1.1.4

Suma $1$ y $1$ .

$\csc (x) - \csc^{2} (x) \cot (x) - \csc (x) \cot (x) (- \cot (x)) - \csc^{2} (x) \csc (x) - \csc^{2} (x) (- \cot (x))$

Paso 9.2.8.1.2

Multiplica $- \csc (x) \cot (x) (- \cot (x))$ .

Toca para ver más pasos...

Paso 9.2.8.1.2.1

Multiplica $- 1$ por $- 1$ .

$\csc (x) - \csc^{2} (x) \cot (x) + 1 \csc (x) \cot (x) \cot (x) - \csc^{2} (x) \csc (x) - \csc^{2} (x) (- \cot (x))$

Paso 9.2.8.1.2.2

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

$\csc (x) - \csc^{2} (x) \cot (x) + \csc (x) \cot (x) \cot (x) - \csc^{2} (x) \csc (x) - \csc^{2} (x) (- \cot (x))$

Paso 9.2.8.1.2.3

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

$\csc (x) - \csc^{2} (x) \cot (x) + \csc (x) (\cot^{1} (x) \cot (x)) - \csc^{2} (x) \csc (x) - \csc^{2} (x) (- \cot (x))$

Paso 9.2.8.1.2.4

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

$\csc (x) - \csc^{2} (x) \cot (x) + \csc (x) (\cot^{1} (x) \cot^{1} (x)) - \csc^{2} (x) \csc (x) - \csc^{2} (x) (- \cot (x))$

Paso 9.2.8.1.2.5

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

$\csc (x) - \csc^{2} (x) \cot (x) + \csc (x) {\cot (x)}^{1 + 1} - \csc^{2} (x) \csc (x) - \csc^{2} (x) (- \cot (x))$

Paso 9.2.8.1.2.6

Suma $1$ y $1$ .

$\csc (x) - \csc^{2} (x) \cot (x) + \csc (x) \cot^{2} (x) - \csc^{2} (x) \csc (x) - \csc^{2} (x) (- \cot (x))$

Paso 9.2.8.1.3

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

Toca para ver más pasos...

Paso 9.2.8.1.3.1

Mueve $\csc (x)$ .

$\csc (x) - \csc^{2} (x) \cot (x) + \csc (x) \cot^{2} (x) - (\csc (x) \csc^{2} (x)) - \csc^{2} (x) (- \cot (x))$

Paso 9.2.8.1.3.2

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

Toca para ver más pasos...

Paso 9.2.8.1.3.2.1

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

$\csc (x) - \csc^{2} (x) \cot (x) + \csc (x) \cot^{2} (x) - (\csc^{1} (x) \csc^{2} (x)) - \csc^{2} (x) (- \cot (x))$

Paso 9.2.8.1.3.2.2

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

$\csc (x) - \csc^{2} (x) \cot (x) + \csc (x) \cot^{2} (x) - {\csc (x)}^{1 + 2} - \csc^{2} (x) (- \cot (x))$

Paso 9.2.8.1.3.3

Suma $1$ y $2$ .

$\csc (x) - \csc^{2} (x) \cot (x) + \csc (x) \cot^{2} (x) - \csc^{3} (x) - \csc^{2} (x) (- \cot (x))$

Paso 9.2.8.1.4

Multiplica $- \csc^{2} (x) (- \cot (x))$ .

Toca para ver más pasos...

Paso 9.2.8.1.4.1

Multiplica $- 1$ por $- 1$ .

$\csc (x) - \csc^{2} (x) \cot (x) + \csc (x) \cot^{2} (x) - \csc^{3} (x) + 1 \csc^{2} (x) \cot (x)$

Paso 9.2.8.1.4.2

Multiplica $\csc^{2} (x)$ por $1$ .

$\csc (x) - \csc^{2} (x) \cot (x) + \csc (x) \cot^{2} (x) - \csc^{3} (x) + \csc^{2} (x) \cot (x)$

Paso 9.2.8.2

Suma $- \csc^{2} (x) \cot (x)$ y $\csc^{2} (x) \cot (x)$ .

$\csc (x) + 0 + \csc (x) \cot^{2} (x) - \csc^{3} (x)$

Paso 9.2.8.3

Suma $0$ y $\csc (x) \cot^{2} (x)$ .

$\csc (x) + \csc (x) \cot^{2} (x) - \csc^{3} (x)$

Paso 9.2.9

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

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Paso 9.2.9.1

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

$\csc (x) + \csc (x) (\cot^{2} (x)) - \csc^{3} (x)$

Paso 9.2.9.2

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

$\csc (x) + \csc (x) (\cot^{2} (x)) + \csc (x) (- \csc^{2} (x))$

Paso 9.2.9.3

Factoriza $\csc (x)$ de $\csc (x) (\cot^{2} (x)) + \csc (x) (- \csc^{2} (x))$ .

$\csc (x) + \csc (x) (\cot^{2} (x) - \csc^{2} (x))$

Paso 9.2.10

Reordena $\cot^{2} (x)$ y $- \csc^{2} (x)$ .

$\csc (x) + \csc (x) (- \csc^{2} (x) + \cot^{2} (x))$

Paso 9.2.11

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

$\csc (x) + \csc (x) (- (\csc^{2} (x)) + \cot^{2} (x))$

Paso 9.2.12

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

$\csc (x) + \csc (x) (- (\csc^{2} (x)) - 1 (- \cot^{2} (x)))$

Paso 9.2.13

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

$\csc (x) + \csc (x) (- (\csc^{2} (x) - \cot^{2} (x)))$

Paso 9.2.14

Aplica la identidad pitagórica.

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

Paso 9.2.15

Multiplica $- 1$ por $1$ .

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

Paso 9.2.16

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

$\csc (x) - 1 \cdot \csc (x)$

Paso 9.2.17

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

$\csc (x) - \csc (x)$

Paso 9.3

Resta $\csc (x)$ de $\csc (x)$ .

$0$