If y = sin^ - 1 ( 2x 1 + x^ 2 ), then (dy)/(dx) is

\dfrac{dy}{dx} = \dfrac{1}{\sqrt{1 - \dfrac{4{x}^{2}}{{\left(1 + {x}^{2}\right)}^{2}}}}\dfrac{d}{dx}\left(\dfrac{2x}{1 + {x}^{2}}\right)$<br />$⇒ \dfrac{dy}{dx} = \dfrac{1 + {x}^{2}}{\sqrt{{\left(1 - {x}^{2}\right)}^{2}}}\dfrac{2\left(1 + {x}^{2}\right) - 4{x}^{2}}{{\left(1 + {x}^{2}\right)}^{2}}$<br />$⇒ \dfrac{dy}{dx} = 2\dfrac{\left(1 + {x}^{2}\right)}{\left|1 - {x}^{2}\right|}\dfrac{1 - {x}^{2}}{{\left(1 + {x}^{2}\right)}^{2}}$<br />$⇒ \dfrac{dy}{dx} = 2\left(\dfrac{1 - {x}^{2}}{\left|1 - {x}^{2}\right|}\right)\left(\dfrac{1}{1 + {x}^{2}}\right)$<br />$⇒ \dfrac{dy}{dx} = \left\{$\begin{cases}\dfrac{- 2}{1 + {x}^{2}}, & if\left|x\right| > 1 \\ \dfrac{2}{1 + {x}^{2}}, & if\left|x\right| < 1\end{cases}$\right)