Answer is: option3
\( \frac{6 - \pi}{9} \)Solution:
You use implicit differentiation to obtain:
\[ \pi \cos(\pi x) - 9\pi y' \sin(\pi y) = 2xy + x^2 y' \] \[ \pi \cos(\pi x) - 2xy = y'(9\pi \sin(\pi y) - x^2) \] \[ y' = \frac{\pi \cos(\pi x) - 2xy}{9\pi \sin(\pi y) - x^2} \]
When you evaluate this last expression at \( (3, -1) \) you obtain:
\[ \frac{6 - \pi}{9} Ans \]