Segunda derivada de $$$x \sin{\left(x \right)}$$$
Calculadoras relacionadas: Calculadora de derivados, Calculadora de diferenciación logarítmica
Tu aportación
Encuentra $$$\frac{d^{2}}{dx^{2}} \left(x \sin{\left(x \right)}\right)$$$.
Solución
Encuentra la primera derivada $$$\frac{d}{dx} \left(x \sin{\left(x \right)}\right)$$$
Aplique 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 $$$g{\left(x \right)} = \sin{\left(x \right)}$$$:
$${\color{red}\left(\frac{d}{dx} \left(x \sin{\left(x \right)}\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(x\right) \sin{\left(x \right)} + x \frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)}$$Aplique la regla de potencia $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ con $$$n = 1$$$, en otras palabras, $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$x \frac{d}{dx} \left(\sin{\left(x \right)}\right) + \sin{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = x \frac{d}{dx} \left(\sin{\left(x \right)}\right) + \sin{\left(x \right)} {\color{red}\left(1\right)}$$La derivada del seno es $$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$:
$$x {\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)} + \sin{\left(x \right)} = x {\color{red}\left(\cos{\left(x \right)}\right)} + \sin{\left(x \right)}$$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 derivadas:
$${\color{red}\left(\frac{d}{dx} \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(x \cos{\left(x \right)}\right) + \frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)}$$Aplique 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 $$$g{\left(x \right)} = \cos{\left(x \right)}$$$:
$${\color{red}\left(\frac{d}{dx} \left(x \cos{\left(x \right)}\right)\right)} + \frac{d}{dx} \left(\sin{\left(x \right)}\right) = {\color{red}\left(\frac{d}{dx} \left(x\right) \cos{\left(x \right)} + x \frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)} + \frac{d}{dx} \left(\sin{\left(x \right)}\right)$$Aplique la regla de potencia $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ con $$$n = 1$$$, en otras palabras, $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$x \frac{d}{dx} \left(\cos{\left(x \right)}\right) + \cos{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} + \frac{d}{dx} \left(\sin{\left(x \right)}\right) = x \frac{d}{dx} \left(\cos{\left(x \right)}\right) + \cos{\left(x \right)} {\color{red}\left(1\right)} + \frac{d}{dx} \left(\sin{\left(x \right)}\right)$$La derivada del seno es $$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$:
$$x \frac{d}{dx} \left(\cos{\left(x \right)}\right) + \cos{\left(x \right)} + {\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)} = x \frac{d}{dx} \left(\cos{\left(x \right)}\right) + \cos{\left(x \right)} + {\color{red}\left(\cos{\left(x \right)}\right)}$$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