Answer is: option4
\( -\frac{y}{x} \)Solution:
Given equation: \[ xy + \tan(xy) = \pi \] Differentiating both sides implicitly: \[ x \frac{dy}{dx} + y + \sec^2(xy) \cdot (x \frac{dy}{dx} + y) = 0 \] \[ (x + x \sec^2(xy)) \frac{dy}{dx} = - (y + y \sec^2(xy)) \] \[ \frac{dy}{dx} = \frac{-y(1 + \sec^2(xy))}{x(1 + \sec^2(xy))} \] \[ \frac{dy}{dx} = -\frac{y}{x} \] Final Answer: \[ \boxed{-\frac{y}{x}} \]