Question #40398

This question tests the 'Quadrant Adjustment Logic' within the differentiation of implicit inverse trigonometric functions.

Students often blindly apply the chain rule or assume $\sin^{-1}(\sin \theta) = \theta$ and $\cos^{-1}(\cos \theta) = \theta$, which leads to an incorrect derivative of $3(1) - 2(1) = 1$.

The trap lies in the hidden domain constraints: the compound angles $x+\alpha$ and $x+\beta$ at $x = 3\pi/4$ fall into the second/third quadrants where the principal branches shift.

Specifically, $x+\alpha \in (\pi/2, 3\pi/2)$ forces $\sin^{-1}(\sin \theta) = \pi - \theta$, and $x+\beta \in (\pi, 2\pi)$ forces $\cos^{-1}(\cos \theta) = 2\pi - \theta$.

Recognizing these boundary crossings is essential to extract the correct negative slopes before differentiation.