$$$a - b u$$$ 對 $$$u$$$ 的導數
您的輸入
求$$$\frac{d}{du} \left(a - b u\right)$$$。
解答
和/差的導數等於導數的和/差：
$${\color{red}\left(\frac{d}{du} \left(a - b u\right)\right)} = {\color{red}\left(\frac{da}{du} - \frac{d}{du} \left(b u\right)\right)}$$套用常數倍法則 $$$\frac{d}{du} \left(c f{\left(u \right)}\right) = c \frac{d}{du} \left(f{\left(u \right)}\right)$$$，使用 $$$c = b$$$ 與 $$$f{\left(u \right)} = u$$$：
$$- {\color{red}\left(\frac{d}{du} \left(b u\right)\right)} + \frac{da}{du} = - {\color{red}\left(b \frac{d}{du} \left(u\right)\right)} + \frac{da}{du}$$套用冪次法則 $$$\frac{d}{du} \left(u^{n}\right) = n u^{n - 1}$$$，取 $$$n = 1$$$，也就是 $$$\frac{d}{du} \left(u\right) = 1$$$：
$$- b {\color{red}\left(\frac{d}{du} \left(u\right)\right)} + \frac{da}{du} = - b {\color{red}\left(1\right)} + \frac{da}{du}$$常數的導數為$$$0$$$：
$$- b + {\color{red}\left(\frac{da}{du}\right)} = - b + {\color{red}\left(0\right)}$$因此，$$$\frac{d}{du} \left(a - b u\right) = - b$$$。
答案
$$$\frac{d}{du} \left(a - b u\right) = - b$$$A