$$$x \left(x - 1\right)$$$ 的導數

將乘積法則 $$$\frac{d}{dx} \left(f{\left(x \right)} g{\left(x \right)}\right) = \frac{d}{dx} \left(f{\left(x \right)}\right) g{\left(x \right)} + f{\left(x \right)} \frac{d}{dx} \left(g{\left(x \right)}\right)$$$ 應用於 $$$f{\left(x \right)} = x$$$ 和 $$$g{\left(x \right)} = x - 1$$$：

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

套用冪次法則 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$，取 $$$n = 1$$$，也就是 $$$\frac{d}{dx} \left(x\right) = 1$$$：

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

和/差的導數等於導數的和/差：

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

常數的導數為$$$0$$$：

$$x \left(- {\color{red}\left(\frac{d}{dx} \left(1\right)\right)} + \frac{d}{dx} \left(x\right)\right) + x - 1 = x \left(- {\color{red}\left(0\right)} + \frac{d}{dx} \left(x\right)\right) + x - 1$$

套用冪次法則 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$，取 $$$n = 1$$$，也就是 $$$\frac{d}{dx} \left(x\right) = 1$$$：

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

因此，$$$\frac{d}{dx} \left(x \left(x - 1\right)\right) = 2 x - 1$$$。