Encontre $$$\frac{d^{2}}{dx^{2}} \left(x^{3} - 3 x^{2}\right)$$$
Calculadoras relacionadas: Calculadora de diferenciação logarítmica, Calculadora de Diferenciação Implícita com Passos
Sua entrada
Encontre $$$\frac{d^{2}}{dx^{2}} \left(x^{3} - 3 x^{2}\right)$$$.
Solução
Encontre a primeira derivada $$$\frac{d}{dx} \left(x^{3} - 3 x^{2}\right)$$$
A derivada de uma soma/diferença é a soma/diferença das derivadas:
$${\color{red}\left(\frac{d}{dx} \left(x^{3} - 3 x^{2}\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(x^{3}\right) - \frac{d}{dx} \left(3 x^{2}\right)\right)}$$Aplique a regra de poder $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ com $$$n = 3$$$:
$${\color{red}\left(\frac{d}{dx} \left(x^{3}\right)\right)} - \frac{d}{dx} \left(3 x^{2}\right) = {\color{red}\left(3 x^{2}\right)} - \frac{d}{dx} \left(3 x^{2}\right)$$Aplique a regra múltipla constante $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ com $$$c = 3$$$ e $$$f{\left(x \right)} = x^{2}$$$:
$$3 x^{2} - {\color{red}\left(\frac{d}{dx} \left(3 x^{2}\right)\right)} = 3 x^{2} - {\color{red}\left(3 \frac{d}{dx} \left(x^{2}\right)\right)}$$Aplique a regra de poder $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ com $$$n = 2$$$:
$$3 x^{2} - 3 {\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)} = 3 x^{2} - 3 {\color{red}\left(2 x\right)}$$Simplificar:
$$3 x^{2} - 6 x = 3 x \left(x - 2\right)$$Assim, $$$\frac{d}{dx} \left(x^{3} - 3 x^{2}\right) = 3 x \left(x - 2\right)$$$.
Em seguida, $$$\frac{d^{2}}{dx^{2}} \left(x^{3} - 3 x^{2}\right) = \frac{d}{dx} \left(3 x \left(x - 2\right)\right)$$$
Aplique a regra múltipla constante $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ com $$$c = 3$$$ e $$$f{\left(x \right)} = x \left(x - 2\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(3 x \left(x - 2\right)\right)\right)} = {\color{red}\left(3 \frac{d}{dx} \left(x \left(x - 2\right)\right)\right)}$$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 $$$g{\left(x \right)} = x - 2$$$:
$$3 {\color{red}\left(\frac{d}{dx} \left(x \left(x - 2\right)\right)\right)} = 3 {\color{red}\left(\frac{d}{dx} \left(x\right) \left(x - 2\right) + x \frac{d}{dx} \left(x - 2\right)\right)}$$A derivada de uma soma/diferença é a soma/diferença das derivadas:
$$3 x {\color{red}\left(\frac{d}{dx} \left(x - 2\right)\right)} + 3 \left(x - 2\right) \frac{d}{dx} \left(x\right) = 3 x {\color{red}\left(\frac{d}{dx} \left(x\right) - \frac{d}{dx} \left(2\right)\right)} + 3 \left(x - 2\right) \frac{d}{dx} \left(x\right)$$A derivada de uma constante é $$$0$$$:
$$3 x \left(- {\color{red}\left(\frac{d}{dx} \left(2\right)\right)} + \frac{d}{dx} \left(x\right)\right) + 3 \left(x - 2\right) \frac{d}{dx} \left(x\right) = 3 x \left(- {\color{red}\left(0\right)} + \frac{d}{dx} \left(x\right)\right) + 3 \left(x - 2\right) \frac{d}{dx} \left(x\right)$$Aplique a regra de potência $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ com $$$n = 1$$$, ou seja, $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$3 x {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} + 3 \left(x - 2\right) {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = 3 x {\color{red}\left(1\right)} + 3 \left(x - 2\right) {\color{red}\left(1\right)}$$Assim, $$$\frac{d}{dx} \left(3 x \left(x - 2\right)\right) = 6 x - 6$$$.
Portanto, $$$\frac{d^{2}}{dx^{2}} \left(x^{3} - 3 x^{2}\right) = 6 x - 6$$$.
Responder
$$$\frac{d^{2}}{dx^{2}} \left(x^{3} - 3 x^{2}\right) = 6 x - 6$$$A