Derivative of $$$2 \operatorname{atan}{\left(u \right)}$$$
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Find $$$\frac{d}{du} \left(2 \operatorname{atan}{\left(u \right)}\right)$$$.
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 $$$c = 2$$$ and $$$f{\left(u \right)} = \operatorname{atan}{\left(u \right)}$$$:
$${\color{red}\left(\frac{d}{du} \left(2 \operatorname{atan}{\left(u \right)}\right)\right)} = {\color{red}\left(2 \frac{d}{du} \left(\operatorname{atan}{\left(u \right)}\right)\right)}$$The derivative of the inverse tangent is $$$\frac{d}{du} \left(\operatorname{atan}{\left(u \right)}\right) = \frac{1}{u^{2} + 1}$$$:
$$2 {\color{red}\left(\frac{d}{du} \left(\operatorname{atan}{\left(u \right)}\right)\right)} = 2 {\color{red}\left(\frac{1}{u^{2} + 1}\right)}$$Thus, $$$\frac{d}{du} \left(2 \operatorname{atan}{\left(u \right)}\right) = \frac{2}{u^{2} + 1}$$$.
Answer
$$$\frac{d}{du} \left(2 \operatorname{atan}{\left(u \right)}\right) = \frac{2}{u^{2} + 1}$$$A