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Segunda derivada de $$$\csc{\left(x \right)}$$$

La calculadora encontrará la segunda derivada de $$$\csc{\left(x \right)}$$$, mostrando los pasos.

Calculadoras relacionadas: Calculadora de derivadas, Calculadora de diferenciación logarítmica

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Tu entrada

Halla $$$\frac{d^{2}}{dx^{2}} \left(\csc{\left(x \right)}\right)$$$.

Solución

Calcule la primera derivada $$$\frac{d}{dx} \left(\csc{\left(x \right)}\right)$$$

La derivada de la cosecante es $$$\frac{d}{dx} \left(\csc{\left(x \right)}\right) = - \cot{\left(x \right)} \csc{\left(x \right)}$$$:

$${\color{red}\left(\frac{d}{dx} \left(\csc{\left(x \right)}\right)\right)} = {\color{red}\left(- \cot{\left(x \right)} \csc{\left(x \right)}\right)}$$

Por lo tanto, $$$\frac{d}{dx} \left(\csc{\left(x \right)}\right) = - \cot{\left(x \right)} \csc{\left(x \right)}$$$.

A continuación, $$$\frac{d^{2}}{dx^{2}} \left(\csc{\left(x \right)}\right) = \frac{d}{dx} \left(- \cot{\left(x \right)} \csc{\left(x \right)}\right)$$$

Aplica la regla del factor constante $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ con $$$c = -1$$$ y $$$f{\left(x \right)} = \cot{\left(x \right)} \csc{\left(x \right)}$$$:

$${\color{red}\left(\frac{d}{dx} \left(- \cot{\left(x \right)} \csc{\left(x \right)}\right)\right)} = {\color{red}\left(- \frac{d}{dx} \left(\cot{\left(x \right)} \csc{\left(x \right)}\right)\right)}$$

Aplica la regla del producto $$$\frac{d}{dx} \left(f{\left(x \right)} g{\left(x \right)}\right) = \frac{d}{dx} \left(f{\left(x \right)}\right) g{\left(x \right)} + f{\left(x \right)} \frac{d}{dx} \left(g{\left(x \right)}\right)$$$ con $$$f{\left(x \right)} = \cot{\left(x \right)}$$$ y $$$g{\left(x \right)} = \csc{\left(x \right)}$$$:

$$- {\color{red}\left(\frac{d}{dx} \left(\cot{\left(x \right)} \csc{\left(x \right)}\right)\right)} = - {\color{red}\left(\frac{d}{dx} \left(\cot{\left(x \right)}\right) \csc{\left(x \right)} + \cot{\left(x \right)} \frac{d}{dx} \left(\csc{\left(x \right)}\right)\right)}$$

La derivada de la cosecante es $$$\frac{d}{dx} \left(\csc{\left(x \right)}\right) = - \cot{\left(x \right)} \csc{\left(x \right)}$$$:

$$- \cot{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(\csc{\left(x \right)}\right)\right)} - \csc{\left(x \right)} \frac{d}{dx} \left(\cot{\left(x \right)}\right) = - \cot{\left(x \right)} {\color{red}\left(- \cot{\left(x \right)} \csc{\left(x \right)}\right)} - \csc{\left(x \right)} \frac{d}{dx} \left(\cot{\left(x \right)}\right)$$

La derivada de la cotangente es $$$\frac{d}{dx} \left(\cot{\left(x \right)}\right) = - \csc^{2}{\left(x \right)}$$$:

$$\cot^{2}{\left(x \right)} \csc{\left(x \right)} - \csc{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(\cot{\left(x \right)}\right)\right)} = \cot^{2}{\left(x \right)} \csc{\left(x \right)} - \csc{\left(x \right)} {\color{red}\left(- \csc^{2}{\left(x \right)}\right)}$$

Simplificar:

$$\cot^{2}{\left(x \right)} \csc{\left(x \right)} + \csc^{3}{\left(x \right)} = \left(-1 + \frac{2}{\sin^{2}{\left(x \right)}}\right) \csc{\left(x \right)}$$

Por lo tanto, $$$\frac{d}{dx} \left(- \cot{\left(x \right)} \csc{\left(x \right)}\right) = \left(-1 + \frac{2}{\sin^{2}{\left(x \right)}}\right) \csc{\left(x \right)}$$$.

Por lo tanto, $$$\frac{d^{2}}{dx^{2}} \left(\csc{\left(x \right)}\right) = \left(-1 + \frac{2}{\sin^{2}{\left(x \right)}}\right) \csc{\left(x \right)}$$$.

Respuesta

$$$\frac{d^{2}}{dx^{2}} \left(\csc{\left(x \right)}\right) = \left(-1 + \frac{2}{\sin^{2}{\left(x \right)}}\right) \csc{\left(x \right)}$$$A