Derivative of $$$\frac{u}{v}$$$ with respect to $$$u$$$
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Find $$$\frac{d}{du} \left(\frac{u}{v}\right)$$$.
Solution
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 $$$c = \frac{1}{v}$$$ and $$$f{\left(u \right)} = u$$$:
$${\color{red}\left(\frac{d}{du} \left(\frac{u}{v}\right)\right)} = {\color{red}\left(\frac{\frac{d}{du} \left(u\right)}{v}\right)}$$Apply the power rule $$$\frac{d}{du} \left(u^{n}\right) = n u^{n - 1}$$$ with $$$n = 1$$$, in other words, $$$\frac{d}{du} \left(u\right) = 1$$$:
$$\frac{{\color{red}\left(\frac{d}{du} \left(u\right)\right)}}{v} = \frac{{\color{red}\left(1\right)}}{v}$$Thus, $$$\frac{d}{du} \left(\frac{u}{v}\right) = \frac{1}{v}$$$.
Answer
$$$\frac{d}{du} \left(\frac{u}{v}\right) = \frac{1}{v}$$$A