Question #41221

This question tests the student's ability to correctly apply piecewise definitions of inverse trigonometric functions before implicit differentiation.

The 'Lethal Collision' occurs because students often blindly differentiate using the standard chain rule (forgetting that $\sqrt{(1-y^2)^2} = |1-y^2|$) or by incorrectly assuming both inverse terms equal $2\tan^{-1}y$ for all $y > 0$.

This ignores the strictness of the interval; for $y > 1$, the derivative of the sine inverse term flips sign, causing the inverse trigonometric terms to completely cancel out.

Failing to recognize this piecewise strictness leads to an incorrect algebraic derivative expression $\frac{dx}{dy} = \frac{4}{1+y^2} + \frac{1}{y}$, resulting in the trap answer $\frac{3-\sqrt{3}}{2}$.