Question #41175

This question tests the student's ability to handle the derivative of an implicit function where the right-hand side involves the principal argument and principal square root of a complex function.

The core trap lies in the multi-valued nature of the complex argument.

Students often incorrectly simplify $\sqrt{e^{-i 2\theta}}$ as $e^{-i\theta}$, ignoring the branch cut of the principal square root.

For $x < -1/2$, the correct principal argument of $Z(x)$ requires adding $2\pi$ to $-2\theta$, shifting the argument of the square root to $\pi - \theta$.

Failing to account for this branch selection leads to an incorrect initial value of $y(-1) = -1$ instead of $y(-1) = 2$.

Because the implicit derivative depends heavily on the coordinates of the point $(x,y)$, this algebraic sign error propagates into the final slope, leading directly to the distractor option.