Derivative of $$$6 u + v$$$ with respect to $$$u$$$
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Your Input
Find $$$\frac{d}{du} \left(6 u + v\right)$$$.
Solution
The derivative of a sum/difference is the sum/difference of derivatives:
$${\color{red}\left(\frac{d}{du} \left(6 u + v\right)\right)} = {\color{red}\left(\frac{d}{du} \left(6 u\right) + \frac{dv}{du}\right)}$$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 $$$c = 6$$$ and $$$f{\left(u \right)} = u$$$:
$${\color{red}\left(\frac{d}{du} \left(6 u\right)\right)} + \frac{dv}{du} = {\color{red}\left(6 \frac{d}{du} \left(u\right)\right)} + \frac{dv}{du}$$Apply the power rule $$$\frac{d}{du} \left(u^{n}\right) = n u^{n - 1}$$$ with $$$n = 1$$$, in other words, $$$\frac{d}{du} \left(u\right) = 1$$$:
$$6 {\color{red}\left(\frac{d}{du} \left(u\right)\right)} + \frac{dv}{du} = 6 {\color{red}\left(1\right)} + \frac{dv}{du}$$The derivative of a constant is $$$0$$$:
$${\color{red}\left(\frac{dv}{du}\right)} + 6 = {\color{red}\left(0\right)} + 6$$Thus, $$$\frac{d}{du} \left(6 u + v\right) = 6$$$.
Answer
$$$\frac{d}{du} \left(6 u + v\right) = 6$$$A