$y=\sin^{-1}\left(\frac{2x}{1+x^2}\right)$

This problem is really about recognizing a hidden trigonometric identity. The expression $\frac{2x}{1+x^2}$ looks complicated, but it's actually the sine double-angle formula in disguise—if you let $x = \tan(\theta)$, the numerator and denominator match $\sin(2\theta)$ and $\sec^2(\theta)$ patterns. Once you make that substitution and simplify, the inverse sine collapses into something much cleaner, and you can differentiate or analyze it far more easily.

This problem is really about recognizing a hidden trigonometric identity. The expression $\frac{2x}{1+x^2}$ looks complicated, but it's actually the sine double-angle formula in disguise—if you let $x = \tan(\theta)$, the numerator and denominator match $\sin(2\theta)$ and $\sec^2(\theta)$ patterns. Once you make that substitution and simplify, the inverse sine collapses into something much cleaner, and you can differentiate or analyze it far more easily.