The derivative of \(\arctan{f{\left ( x \right )}}\) is
\(d \frac{\arctan{f{\left ( x \right )}}}{dx} = \frac{f{'} \left ( x \right )}{1 + \left ( f{\left ( x \right )} \right )^{2}}\)
where \(f{\left ( x \right )} = x^{2}\) hence
\(d \frac{\arctan{x}^{2}}{dx} = \frac{2 x}{1 + \left ( x^{4} \right )}\)