If $f(x) = \sin^{-1}\left(\frac{2x}{1 + x^2}\right)$, then $f'\left(\frac{1}{2}\right) =$
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Expressions like $\frac{2x}{1+x^2}$, $\frac{1-x^2}{1+x^2}$, and $\frac{2x}{1-x^2}$ are classic signatures for the substitution $x = \tan\theta$, transforming them into $\sin 2\theta$, $\cos 2\theta$, and $\tan 2\theta$ respectively.