Segunda derivada de $$$x \sin{\left(x \right)}$$$
Calculadoras relacionadas: Calculadora de Derivativos, Calculadora de diferenciação logarítmica
Sua entrada
Encontre $$$\frac{d^{2}}{dx^{2}} \left(x \sin{\left(x \right)}\right)$$$.
Solução
Encontre a primeira derivada $$$\frac{d}{dx} \left(x \sin{\left(x \right)}\right)$$$
Aplique a regra do produto $$$\frac{d}{dx} \left(f{\left(x \right)} g{\left(x \right)}\right) = \frac{d}{dx} \left(f{\left(x \right)}\right) g{\left(x \right)} + f{\left(x \right)} \frac{d}{dx} \left(g{\left(x \right)}\right)$$$ com $$$f{\left(x \right)} = x$$$ e $$$g{\left(x \right)} = \sin{\left(x \right)}$$$:
$${\color{red}\left(\frac{d}{dx} \left(x \sin{\left(x \right)}\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(x\right) \sin{\left(x \right)} + x \frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)}$$A derivada do seno é $$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$:
$$x {\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)} + \sin{\left(x \right)} \frac{d}{dx} \left(x\right) = x {\color{red}\left(\cos{\left(x \right)}\right)} + \sin{\left(x \right)} \frac{d}{dx} \left(x\right)$$Aplique a regra de potência $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ com $$$n = 1$$$, ou seja, $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$x \cos{\left(x \right)} + \sin{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = x \cos{\left(x \right)} + \sin{\left(x \right)} {\color{red}\left(1\right)}$$Assim, $$$\frac{d}{dx} \left(x \sin{\left(x \right)}\right) = x \cos{\left(x \right)} + \sin{\left(x \right)}$$$.
Em seguida, $$$\frac{d^{2}}{dx^{2}} \left(x \sin{\left(x \right)}\right) = \frac{d}{dx} \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)$$$
A derivada de uma soma/diferença é a soma/diferença das derivadas:
$${\color{red}\left(\frac{d}{dx} \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(x \cos{\left(x \right)}\right) + \frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)}$$A derivada do seno é $$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$:
$${\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)} + \frac{d}{dx} \left(x \cos{\left(x \right)}\right) = {\color{red}\left(\cos{\left(x \right)}\right)} + \frac{d}{dx} \left(x \cos{\left(x \right)}\right)$$Aplique a regra do produto $$$\frac{d}{dx} \left(f{\left(x \right)} g{\left(x \right)}\right) = \frac{d}{dx} \left(f{\left(x \right)}\right) g{\left(x \right)} + f{\left(x \right)} \frac{d}{dx} \left(g{\left(x \right)}\right)$$$ com $$$f{\left(x \right)} = x$$$ e $$$g{\left(x \right)} = \cos{\left(x \right)}$$$:
$$\cos{\left(x \right)} + {\color{red}\left(\frac{d}{dx} \left(x \cos{\left(x \right)}\right)\right)} = \cos{\left(x \right)} + {\color{red}\left(\frac{d}{dx} \left(x\right) \cos{\left(x \right)} + x \frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)}$$A derivada do cosseno é $$$\frac{d}{dx} \left(\cos{\left(x \right)}\right) = - \sin{\left(x \right)}$$$:
$$x {\color{red}\left(\frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)} + \cos{\left(x \right)} \frac{d}{dx} \left(x\right) + \cos{\left(x \right)} = x {\color{red}\left(- \sin{\left(x \right)}\right)} + \cos{\left(x \right)} \frac{d}{dx} \left(x\right) + \cos{\left(x \right)}$$Aplique a regra de potência $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ com $$$n = 1$$$, ou seja, $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$- x \sin{\left(x \right)} + \cos{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} + \cos{\left(x \right)} = - x \sin{\left(x \right)} + \cos{\left(x \right)} {\color{red}\left(1\right)} + \cos{\left(x \right)}$$Assim, $$$\frac{d}{dx} \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) = - x \sin{\left(x \right)} + 2 \cos{\left(x \right)}$$$.
Portanto, $$$\frac{d^{2}}{dx^{2}} \left(x \sin{\left(x \right)}\right) = - x \sin{\left(x \right)} + 2 \cos{\left(x \right)}$$$.
Responder
$$$\frac{d^{2}}{dx^{2}} \left(x \sin{\left(x \right)}\right) = - x \sin{\left(x \right)} + 2 \cos{\left(x \right)}$$$A