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

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

Calculadoras relacionadas: Calculadora de derivadas, Calculadora de diferenciación logarítmica

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Tu entrada

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

Solución

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

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 $$$f{\left(x \right)} = x$$$ y $$$g{\left(x \right)} = \sin{\left(x \right)}$$$:

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

La derivada del seno es $$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$:

$$x {\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)} + \sin{\left(x \right)} \frac{d}{dx} \left(x\right) = x {\color{red}\left(\cos{\left(x \right)}\right)} + \sin{\left(x \right)} \frac{d}{dx} \left(x\right)$$

Aplica la regla de la potencia $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ con $$$n = 1$$$, en otras palabras, $$$\frac{d}{dx} \left(x\right) = 1$$$:

$$x \cos{\left(x \right)} + \sin{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = x \cos{\left(x \right)} + \sin{\left(x \right)} {\color{red}\left(1\right)}$$

Por lo tanto, $$$\frac{d}{dx} \left(x \sin{\left(x \right)}\right) = x \cos{\left(x \right)} + \sin{\left(x \right)}$$$.

A continuación, $$$\frac{d^{2}}{dx^{2}} \left(x \sin{\left(x \right)}\right) = \frac{d}{dx} \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)$$$

La derivada de una suma/diferencia es la suma/diferencia de las derivadas:

$${\color{red}\left(\frac{d}{dx} \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(x \cos{\left(x \right)}\right) + \frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)}$$

La derivada del seno es $$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$:

$${\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)} + \frac{d}{dx} \left(x \cos{\left(x \right)}\right) = {\color{red}\left(\cos{\left(x \right)}\right)} + \frac{d}{dx} \left(x \cos{\left(x \right)}\right)$$

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 $$$f{\left(x \right)} = x$$$ y $$$g{\left(x \right)} = \cos{\left(x \right)}$$$:

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

Aplica la regla de la potencia $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ con $$$n = 1$$$, en otras palabras, $$$\frac{d}{dx} \left(x\right) = 1$$$:

$$x \frac{d}{dx} \left(\cos{\left(x \right)}\right) + \cos{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} + \cos{\left(x \right)} = x \frac{d}{dx} \left(\cos{\left(x \right)}\right) + \cos{\left(x \right)} {\color{red}\left(1\right)} + \cos{\left(x \right)}$$

La derivada del coseno es $$$\frac{d}{dx} \left(\cos{\left(x \right)}\right) = - \sin{\left(x \right)}$$$:

$$x {\color{red}\left(\frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)} + 2 \cos{\left(x \right)} = x {\color{red}\left(- \sin{\left(x \right)}\right)} + 2 \cos{\left(x \right)}$$

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

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

Respuesta

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


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