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