$$$x \left(2 - 3 x\right)$$$の導関数
入力内容
$$$\frac{d}{dx} \left(x \left(2 - 3 x\right)\right)$$$ を求めよ。
解答
積の微分法 $$$\frac{d}{dx} \left(f{\left(x \right)} g{\left(x \right)}\right) = \frac{d}{dx} \left(f{\left(x \right)}\right) g{\left(x \right)} + f{\left(x \right)} \frac{d}{dx} \left(g{\left(x \right)}\right)$$$ を $$$f{\left(x \right)} = x$$$ と $$$g{\left(x \right)} = 2 - 3 x$$$ に適用する:
$${\color{red}\left(\frac{d}{dx} \left(x \left(2 - 3 x\right)\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(x\right) \left(2 - 3 x\right) + x \frac{d}{dx} \left(2 - 3 x\right)\right)}$$和/差の導関数は、導関数の和/差である:
$$x {\color{red}\left(\frac{d}{dx} \left(2 - 3 x\right)\right)} + \left(2 - 3 x\right) \frac{d}{dx} \left(x\right) = x {\color{red}\left(\frac{d}{dx} \left(2\right) - \frac{d}{dx} \left(3 x\right)\right)} + \left(2 - 3 x\right) \frac{d}{dx} \left(x\right)$$$$$n = 1$$$ を用いて冪法則 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ を適用すると、すなわち $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$x \left(\frac{d}{dx} \left(2\right) - \frac{d}{dx} \left(3 x\right)\right) + \left(2 - 3 x\right) {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = x \left(\frac{d}{dx} \left(2\right) - \frac{d}{dx} \left(3 x\right)\right) + \left(2 - 3 x\right) {\color{red}\left(1\right)}$$定数の導数は$$$0$$$です:
$$x \left({\color{red}\left(\frac{d}{dx} \left(2\right)\right)} - \frac{d}{dx} \left(3 x\right)\right) - 3 x + 2 = x \left({\color{red}\left(0\right)} - \frac{d}{dx} \left(3 x\right)\right) - 3 x + 2$$定数倍の法則 $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ を $$$c = 3$$$ と $$$f{\left(x \right)} = x$$$ に対して適用します:
$$- x {\color{red}\left(\frac{d}{dx} \left(3 x\right)\right)} - 3 x + 2 = - x {\color{red}\left(3 \frac{d}{dx} \left(x\right)\right)} - 3 x + 2$$$$$n = 1$$$ を用いて冪法則 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ を適用すると、すなわち $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$- 3 x {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} - 3 x + 2 = - 3 x {\color{red}\left(1\right)} - 3 x + 2$$したがって、$$$\frac{d}{dx} \left(x \left(2 - 3 x\right)\right) = 2 - 6 x$$$。
解答
$$$\frac{d}{dx} \left(x \left(2 - 3 x\right)\right) = 2 - 6 x$$$A