Encuentra $$$\frac{d^{2}}{dx^{2}} \left(x^{3} - 3 x^{2}\right)$$$
Calculadoras relacionadas: Calculadora de diferenciación logarítmica, Calculadora de diferenciación implícita con pasos
Tu aportación
Encuentra $$$\frac{d^{2}}{dx^{2}} \left(x^{3} - 3 x^{2}\right)$$$.
Solución
Encuentra la primera derivada $$$\frac{d}{dx} \left(x^{3} - 3 x^{2}\right)$$$
La derivada de una suma/diferencia es la suma/diferencia de 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 la regla de potencia $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ con $$$n = 3$$$:
$${\color{red}\left(\frac{d}{dx} \left(x^{3}\right)\right)} - \frac{d}{dx} \left(3 x^{2}\right) = {\color{red}\left(3 x^{2}\right)} - \frac{d}{dx} \left(3 x^{2}\right)$$Aplique la regla del múltiplo constante $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ con $$$c = 3$$$ y $$$f{\left(x \right)} = x^{2}$$$:
$$3 x^{2} - {\color{red}\left(\frac{d}{dx} \left(3 x^{2}\right)\right)} = 3 x^{2} - {\color{red}\left(3 \frac{d}{dx} \left(x^{2}\right)\right)}$$Aplique la regla de potencia $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ con $$$n = 2$$$:
$$3 x^{2} - 3 {\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)} = 3 x^{2} - 3 {\color{red}\left(2 x\right)}$$Simplificar:
$$3 x^{2} - 6 x = 3 x \left(x - 2\right)$$Por lo tanto, $$$\frac{d}{dx} \left(x^{3} - 3 x^{2}\right) = 3 x \left(x - 2\right)$$$.
A continuación, $$$\frac{d^{2}}{dx^{2}} \left(x^{3} - 3 x^{2}\right) = \frac{d}{dx} \left(3 x \left(x - 2\right)\right)$$$
Aplique la regla del múltiplo constante $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ con $$$c = 3$$$ y $$$f{\left(x \right)} = x \left(x - 2\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(3 x \left(x - 2\right)\right)\right)} = {\color{red}\left(3 \frac{d}{dx} \left(x \left(x - 2\right)\right)\right)}$$Aplique 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 $$$g{\left(x \right)} = x - 2$$$:
$$3 {\color{red}\left(\frac{d}{dx} \left(x \left(x - 2\right)\right)\right)} = 3 {\color{red}\left(\frac{d}{dx} \left(x\right) \left(x - 2\right) + x \frac{d}{dx} \left(x - 2\right)\right)}$$La derivada de una suma/diferencia es la suma/diferencia de derivadas:
$$3 x {\color{red}\left(\frac{d}{dx} \left(x - 2\right)\right)} + 3 \left(x - 2\right) \frac{d}{dx} \left(x\right) = 3 x {\color{red}\left(\frac{d}{dx} \left(x\right) - \frac{d}{dx} \left(2\right)\right)} + 3 \left(x - 2\right) \frac{d}{dx} \left(x\right)$$La derivada de una constante es $$$0$$$:
$$3 x \left(- {\color{red}\left(\frac{d}{dx} \left(2\right)\right)} + \frac{d}{dx} \left(x\right)\right) + 3 \left(x - 2\right) \frac{d}{dx} \left(x\right) = 3 x \left(- {\color{red}\left(0\right)} + \frac{d}{dx} \left(x\right)\right) + 3 \left(x - 2\right) \frac{d}{dx} \left(x\right)$$Aplique la regla de potencia $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ con $$$n = 1$$$, en otras palabras, $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$3 x {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} + 3 \left(x - 2\right) {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = 3 x {\color{red}\left(1\right)} + 3 \left(x - 2\right) {\color{red}\left(1\right)}$$Por lo tanto, $$$\frac{d}{dx} \left(3 x \left(x - 2\right)\right) = 6 x - 6$$$.
Por lo tanto, $$$\frac{d^{2}}{dx^{2}} \left(x^{3} - 3 x^{2}\right) = 6 x - 6$$$.
Respuesta
$$$\frac{d^{2}}{dx^{2}} \left(x^{3} - 3 x^{2}\right) = 6 x - 6$$$A