Derivative of $$$\operatorname{atan}{\left(2 x \right)}$$$
Related calculator: Derivative Calculator
Solution
The function $$$\operatorname{atan}{\left(2 x \right)}$$$ is the composition $$$f{\left(g{\left(x \right)} \right)}$$$ of two functions $$$f{\left(u \right)} = \operatorname{atan}{\left(u \right)}$$$ and $$$g{\left(x \right)} = 2 x$$$.
Apply the chain rule $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(\operatorname{atan}{\left(2 x \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\operatorname{atan}{\left(u \right)}\right) \frac{d}{dx} \left(2 x\right)\right)}$$The derivative of the inverse tangent is $$$\frac{d}{du} \left(\operatorname{atan}{\left(u \right)}\right) = \frac{1}{u^{2} + 1}$$$:
$${\color{red}\left(\frac{d}{du} \left(\operatorname{atan}{\left(u \right)}\right)\right)} \frac{d}{dx} \left(2 x\right) = {\color{red}\left(\frac{1}{u^{2} + 1}\right)} \frac{d}{dx} \left(2 x\right)$$Return to the old variable:
$$\frac{\frac{d}{dx} \left(2 x\right)}{{\color{red}\left(u\right)}^{2} + 1} = \frac{\frac{d}{dx} \left(2 x\right)}{{\color{red}\left(2 x\right)}^{2} + 1}$$Apply the constant multiple rule $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ with $$$c = 2$$$ and $$$f{\left(x \right)} = x$$$:
$$\frac{{\color{red}\left(\frac{d}{dx} \left(2 x\right)\right)}}{4 x^{2} + 1} = \frac{{\color{red}\left(2 \frac{d}{dx} \left(x\right)\right)}}{4 x^{2} + 1}$$Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = 1$$$, in other words, $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$\frac{2 {\color{red}\left(\frac{d}{dx} \left(x\right)\right)}}{4 x^{2} + 1} = \frac{2 {\color{red}\left(1\right)}}{4 x^{2} + 1}$$Thus, $$$\frac{d}{dx} \left(\operatorname{atan}{\left(2 x \right)}\right) = \frac{2}{4 x^{2} + 1}$$$.
Answer
$$$\frac{d}{dx} \left(\operatorname{atan}{\left(2 x \right)}\right) = \frac{2}{4 x^{2} + 1}$$$A