Cálculo Ejemplos

$y = \frac{\sin (x) - \cos (x)}{\sin (x) + \cos (x)}$

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 $f (x) = \sin (x) - \cos (x)$ y $g (x) = \sin (x) + \cos (x)$ .

$\frac{(\sin (x) + \cos (x)) \frac{d}{d x} [\sin (x) - \cos (x)] - (\sin (x) - \cos (x)) \frac{d}{d x} [\sin (x) + \cos (x)]}{{(\sin (x) + \cos (x))}^{2}}$

Paso 2

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

$\frac{(\sin (x) + \cos (x)) (\frac{d}{d x} [\sin (x)] + \frac{d}{d x} [- \cos (x)]) - (\sin (x) - \cos (x)) \frac{d}{d x} [\sin (x) + \cos (x)]}{{(\sin (x) + \cos (x))}^{2}}$

Paso 3

La derivada de $\sin (x)$ con respecto a $x$ es $\cos (x)$ .

$\frac{(\sin (x) + \cos (x)) (\cos (x) + \frac{d}{d x} [- \cos (x)]) - (\sin (x) - \cos (x)) \frac{d}{d x} [\sin (x) + \cos (x)]}{{(\sin (x) + \cos (x))}^{2}}$

Paso 4

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

$\frac{(\sin (x) + \cos (x)) (\cos (x) - \frac{d}{d x} [\cos (x)]) - (\sin (x) - \cos (x)) \frac{d}{d x} [\sin (x) + \cos (x)]}{{(\sin (x) + \cos (x))}^{2}}$

Paso 5

La derivada de $\cos (x)$ con respecto a $x$ es $- \sin (x)$ .

$\frac{(\sin (x) + \cos (x)) (\cos (x) - - \sin (x)) - (\sin (x) - \cos (x)) \frac{d}{d x} [\sin (x) + \cos (x)]}{{(\sin (x) + \cos (x))}^{2}}$

Paso 6

Diferencia con la regla de la suma.

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Paso 6.1

Multiplica $- 1$ por $- 1$ .

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

Paso 6.2

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

$\frac{(\sin (x) + \cos (x)) (\cos (x) + \sin (x)) - (\sin (x) - \cos (x)) \frac{d}{d x} [\sin (x) + \cos (x)]}{{(\sin (x) + \cos (x))}^{2}}$

Paso 6.3

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

$\frac{(\sin (x) + \cos (x)) (\cos (x) + \sin (x)) - (\sin (x) - \cos (x)) (\frac{d}{d x} [\sin (x)] + \frac{d}{d x} [\cos (x)])}{{(\sin (x) + \cos (x))}^{2}}$

Paso 7

La derivada de $\sin (x)$ con respecto a $x$ es $\cos (x)$ .

$\frac{(\sin (x) + \cos (x)) (\cos (x) + \sin (x)) - (\sin (x) - \cos (x)) (\cos (x) + \frac{d}{d x} [\cos (x)])}{{(\sin (x) + \cos (x))}^{2}}$

Paso 8

La derivada de $\cos (x)$ con respecto a $x$ es $- \sin (x)$ .

$\frac{(\sin (x) + \cos (x)) (\cos (x) + \sin (x)) - (\sin (x) - \cos (x)) (\cos (x) - \sin (x))}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9

Simplifica.

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Paso 9.1

Aplica la propiedad distributiva.

$\frac{(\sin (x) + \cos (x)) (\cos (x) + \sin (x)) + (- \sin (x) - - \cos (x)) (\cos (x) - \sin (x))}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.2

Simplifica el numerador.

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Paso 9.2.1

Simplifica cada término.

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Paso 9.2.1.1

Expande $(\sin (x) + \cos (x)) (\cos (x) + \sin (x))$ con el método PEIU (primero, exterior, interior, ultimo).

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Paso 9.2.1.1.1

Aplica la propiedad distributiva.

$\frac{\sin (x) (\cos (x) + \sin (x)) + \cos (x) (\cos (x) + \sin (x)) + (- \sin (x) - - \cos (x)) (\cos (x) - \sin (x))}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.2.1.1.2

Aplica la propiedad distributiva.

$\frac{\sin (x) \cos (x) + \sin (x) \sin (x) + \cos (x) (\cos (x) + \sin (x)) + (- \sin (x) - - \cos (x)) (\cos (x) - \sin (x))}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.2.1.1.3

Aplica la propiedad distributiva.

$\frac{\sin (x) \cos (x) + \sin (x) \sin (x) + \cos (x) \cos (x) + \cos (x) \sin (x) + (- \sin (x) - - \cos (x)) (\cos (x) - \sin (x))}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.2.1.2

Simplifica y combina los términos similares.

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Paso 9.2.1.2.1

Simplifica cada término.

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Paso 9.2.1.2.1.1

Multiplica $\sin (x) \sin (x)$ .

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Paso 9.2.1.2.1.1.1

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

$\frac{\sin (x) \cos (x) + \sin^{1} (x) \sin (x) + \cos (x) \cos (x) + \cos (x) \sin (x) + (- \sin (x) - - \cos (x)) (\cos (x) - \sin (x))}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.2.1.2.1.1.2

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

$\frac{\sin (x) \cos (x) + \sin^{1} (x) \sin^{1} (x) + \cos (x) \cos (x) + \cos (x) \sin (x) + (- \sin (x) - - \cos (x)) (\cos (x) - \sin (x))}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.2.1.2.1.1.3

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

$\frac{\sin (x) \cos (x) + {\sin (x)}^{1 + 1} + \cos (x) \cos (x) + \cos (x) \sin (x) + (- \sin (x) - - \cos (x)) (\cos (x) - \sin (x))}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.2.1.2.1.1.4

Suma $1$ y $1$ .

$\frac{\sin (x) \cos (x) + \sin^{2} (x) + \cos (x) \cos (x) + \cos (x) \sin (x) + (- \sin (x) - - \cos (x)) (\cos (x) - \sin (x))}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.2.1.2.1.2

Multiplica $\cos (x) \cos (x)$ .

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Paso 9.2.1.2.1.2.1

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

$\frac{\sin (x) \cos (x) + \sin^{2} (x) + \cos^{1} (x) \cos (x) + \cos (x) \sin (x) + (- \sin (x) - - \cos (x)) (\cos (x) - \sin (x))}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.2.1.2.1.2.2

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

$\frac{\sin (x) \cos (x) + \sin^{2} (x) + \cos^{1} (x) \cos^{1} (x) + \cos (x) \sin (x) + (- \sin (x) - - \cos (x)) (\cos (x) - \sin (x))}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.2.1.2.1.2.3

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

$\frac{\sin (x) \cos (x) + \sin^{2} (x) + {\cos (x)}^{1 + 1} + \cos (x) \sin (x) + (- \sin (x) - - \cos (x)) (\cos (x) - \sin (x))}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.2.1.2.1.2.4

Suma $1$ y $1$ .

$\frac{\sin (x) \cos (x) + \sin^{2} (x) + \cos^{2} (x) + \cos (x) \sin (x) + (- \sin (x) - - \cos (x)) (\cos (x) - \sin (x))}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.2.1.2.2

Reordena los factores de $\sin (x) \cos (x)$ .

$\frac{\cos (x) \sin (x) + \sin^{2} (x) + \cos^{2} (x) + \cos (x) \sin (x) + (- \sin (x) - - \cos (x)) (\cos (x) - \sin (x))}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.2.1.2.3

Suma $\cos (x) \sin (x)$ y $\cos (x) \sin (x)$ .

$\frac{2 \cos (x) \sin (x) + \sin^{2} (x) + \cos^{2} (x) + (- \sin (x) - - \cos (x)) (\cos (x) - \sin (x))}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.2.1.3

Aplica la identidad pitagórica.

$\frac{2 \cos (x) \sin (x) + 1 + (- \sin (x) - - \cos (x)) (\cos (x) - \sin (x))}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.2.1.4

Simplifica cada término.

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Paso 9.2.1.4.1

Reordena $2 \cos (x)$ y $\sin (x)$ .

$\frac{\sin (x) (2 \cos (x)) + 1 + (- \sin (x) - - \cos (x)) (\cos (x) - \sin (x))}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.2.1.4.2

Reordena $\sin (x)$ y $2$ .

$\frac{2 \cdot \sin (x) \cos (x) + 1 + (- \sin (x) - - \cos (x)) (\cos (x) - \sin (x))}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.2.1.4.3

Aplica la razón del ángulo doble sinusoidal.

$\frac{\sin (2 x) + 1 + (- \sin (x) - - \cos (x)) (\cos (x) - \sin (x))}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.2.1.5

Multiplica $- - \cos (x)$ .

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Paso 9.2.1.5.1

Multiplica $- 1$ por $- 1$ .

$\frac{\sin (2 x) + 1 + (- \sin (x) + 1 \cos (x)) (\cos (x) - \sin (x))}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.2.1.5.2

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

$\frac{\sin (2 x) + 1 + (- \sin (x) + \cos (x)) (\cos (x) - \sin (x))}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.2.1.6

Expande $(- \sin (x) + \cos (x)) (\cos (x) - \sin (x))$ con el método PEIU (primero, exterior, interior, ultimo).

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Paso 9.2.1.6.1

Aplica la propiedad distributiva.

$\frac{\sin (2 x) + 1 - \sin (x) (\cos (x) - \sin (x)) + \cos (x) (\cos (x) - \sin (x))}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.2.1.6.2

Aplica la propiedad distributiva.

$\frac{\sin (2 x) + 1 - \sin (x) \cos (x) - \sin (x) (- \sin (x)) + \cos (x) (\cos (x) - \sin (x))}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.2.1.6.3

Aplica la propiedad distributiva.

$\frac{\sin (2 x) + 1 - \sin (x) \cos (x) - \sin (x) (- \sin (x)) + \cos (x) \cos (x) + \cos (x) (- \sin (x))}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.2.1.7

Simplifica y combina los términos similares.

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Paso 9.2.1.7.1

Simplifica cada término.

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Paso 9.2.1.7.1.1

Multiplica $- \sin (x) (- \sin (x))$ .

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Paso 9.2.1.7.1.1.1

Multiplica $- 1$ por $- 1$ .

$\frac{\sin (2 x) + 1 - \sin (x) \cos (x) + 1 \sin (x) \sin (x) + \cos (x) \cos (x) + \cos (x) (- \sin (x))}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.2.1.7.1.1.2

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

$\frac{\sin (2 x) + 1 - \sin (x) \cos (x) + \sin (x) \sin (x) + \cos (x) \cos (x) + \cos (x) (- \sin (x))}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.2.1.7.1.1.3

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

$\frac{\sin (2 x) + 1 - \sin (x) \cos (x) + \sin^{1} (x) \sin (x) + \cos (x) \cos (x) + \cos (x) (- \sin (x))}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.2.1.7.1.1.4

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

$\frac{\sin (2 x) + 1 - \sin (x) \cos (x) + \sin^{1} (x) \sin^{1} (x) + \cos (x) \cos (x) + \cos (x) (- \sin (x))}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.2.1.7.1.1.5

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

$\frac{\sin (2 x) + 1 - \sin (x) \cos (x) + {\sin (x)}^{1 + 1} + \cos (x) \cos (x) + \cos (x) (- \sin (x))}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.2.1.7.1.1.6

Suma $1$ y $1$ .

$\frac{\sin (2 x) + 1 - \sin (x) \cos (x) + \sin^{2} (x) + \cos (x) \cos (x) + \cos (x) (- \sin (x))}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.2.1.7.1.2

Multiplica $\cos (x) \cos (x)$ .

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Paso 9.2.1.7.1.2.1

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

$\frac{\sin (2 x) + 1 - \sin (x) \cos (x) + \sin^{2} (x) + \cos^{1} (x) \cos (x) + \cos (x) (- \sin (x))}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.2.1.7.1.2.2

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

$\frac{\sin (2 x) + 1 - \sin (x) \cos (x) + \sin^{2} (x) + \cos^{1} (x) \cos^{1} (x) + \cos (x) (- \sin (x))}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.2.1.7.1.2.3

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

$\frac{\sin (2 x) + 1 - \sin (x) \cos (x) + \sin^{2} (x) + {\cos (x)}^{1 + 1} + \cos (x) (- \sin (x))}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.2.1.7.1.2.4

Suma $1$ y $1$ .

$\frac{\sin (2 x) + 1 - \sin (x) \cos (x) + \sin^{2} (x) + \cos^{2} (x) + \cos (x) (- \sin (x))}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.2.1.7.1.3

Reescribe con la propiedad conmutativa de la multiplicación.

$\frac{\sin (2 x) + 1 - \sin (x) \cos (x) + \sin^{2} (x) + \cos^{2} (x) - \cos (x) \sin (x)}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.2.1.7.2

Reordena los factores de $- \sin (x) \cos (x)$ .

$\frac{\sin (2 x) + 1 - \cos (x) \sin (x) + \sin^{2} (x) + \cos^{2} (x) - \cos (x) \sin (x)}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.2.1.7.3

Resta $\cos (x) \sin (x)$ de $- \cos (x) \sin (x)$ .

$\frac{\sin (2 x) + 1 - 2 \cos (x) \sin (x) + \sin^{2} (x) + \cos^{2} (x)}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.2.1.8

Aplica la identidad pitagórica.

$\frac{\sin (2 x) + 1 - 2 \cos (x) \sin (x) + 1}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.2.2

Suma $1$ y $1$ .

$\frac{\sin (2 x) - 2 \cos (x) \sin (x) + 2}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.3

Reordena los términos.

$\frac{- 2 \cos (x) \sin (x) + 2 + \sin (2 x)}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.4

Factoriza $- 1$ de $- 2 \cos (x) \sin (x)$ .

$\frac{- (2 \cos (x) \sin (x)) + 2 + \sin (2 x)}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.5

Reescribe $2$ como $- 1 (- 2)$ .

$\frac{- (2 \cos (x) \sin (x)) - 1 (- 2) + \sin (2 x)}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.6

Factoriza $- 1$ de $- (2 \cos (x) \sin (x)) - 1 (- 2)$ .

$\frac{- (2 \cos (x) \sin (x) - 2) + \sin (2 x)}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.7

Factoriza $- 1$ de $\sin (2 x)$ .

$\frac{- (2 \cos (x) \sin (x) - 2) - 1 (- \sin (2 x))}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.8

Factoriza $- 1$ de $- (2 \cos (x) \sin (x) - 2) - 1 (- \sin (2 x))$ .

$\frac{- (2 \cos (x) \sin (x) - 2 - \sin (2 x))}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.9

Reescribe $- (2 \cos (x) \sin (x) - 2 - \sin (2 x))$ como $- 1 (2 \cos (x) \sin (x) - 2 - \sin (2 x))$ .

$\frac{- 1 (2 \cos (x) \sin (x) - 2 - \sin (2 x))}{{(\sin (x) + \cos (x))}^{2}}$

Paso 9.10

Mueve el negativo al frente de la fracción.

$- \frac{2 \cos (x) \sin (x) - 2 - \sin (2 x)}{{(\sin (x) + \cos (x))}^{2}}$