Derivative of $$$a - b u$$$ with respect to $$$u$$$
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Your Input
Find $$$\frac{d}{du} \left(a - b u\right)$$$.
Solution
The derivative of a sum/difference is the sum/difference of derivatives:
$${\color{red}\left(\frac{d}{du} \left(a - b u\right)\right)} = {\color{red}\left(\frac{da}{du} - \frac{d}{du} \left(b u\right)\right)}$$Apply the constant multiple rule $$$\frac{d}{du} \left(c f{\left(u \right)}\right) = c \frac{d}{du} \left(f{\left(u \right)}\right)$$$ with $$$c = b$$$ and $$$f{\left(u \right)} = u$$$:
$$- {\color{red}\left(\frac{d}{du} \left(b u\right)\right)} + \frac{da}{du} = - {\color{red}\left(b \frac{d}{du} \left(u\right)\right)} + \frac{da}{du}$$Apply the power rule $$$\frac{d}{du} \left(u^{n}\right) = n u^{n - 1}$$$ with $$$n = 1$$$, in other words, $$$\frac{d}{du} \left(u\right) = 1$$$:
$$- b {\color{red}\left(\frac{d}{du} \left(u\right)\right)} + \frac{da}{du} = - b {\color{red}\left(1\right)} + \frac{da}{du}$$The derivative of a constant is $$$0$$$:
$$- b + {\color{red}\left(\frac{da}{du}\right)} = - b + {\color{red}\left(0\right)}$$Thus, $$$\frac{d}{du} \left(a - b u\right) = - b$$$.
Answer
$$$\frac{d}{du} \left(a - b u\right) = - b$$$A