$$$x \left(x - 1\right)$$$의 도함수

$$$f{\left(x \right)} = x$$$와 $$$g{\left(x \right)} = x - 1$$$에 대해 곱의 미분법칙 $$$\frac{d}{dx} \left(f{\left(x \right)} g{\left(x \right)}\right) = \frac{d}{dx} \left(f{\left(x \right)}\right) g{\left(x \right)} + f{\left(x \right)} \frac{d}{dx} \left(g{\left(x \right)}\right)$$$을 적용하십시오:

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

멱법칙 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$을 $$$n = 1$$$에 대해 적용하면, 즉 $$$\frac{d}{dx} \left(x\right) = 1$$$:

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

합/차의 도함수는 도함수들의 합/차이다:

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

상수의 도함수는 $$$0$$$입니다:

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

멱법칙 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$을 $$$n = 1$$$에 대해 적용하면, 즉 $$$\frac{d}{dx} \left(x\right) = 1$$$:

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

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