Answer is: option1

Solution:

Given equation: \[ 3xy + x^2 - 2y^2 = 2 \] Differentiating both sides implicitly: \[ 3x \frac{dy}{dx} + 3y + 2x - 4y \frac{dy}{dx} = 0 \] \[ 3x \frac{dy}{dx} - 4y \frac{dy}{dx} = -3y - 2x \] \[ (3x - 4y) \frac{dy}{dx} = -(3y + 2x) \] \[ \frac{dy}{dx} = \frac{-(3y + 2x)}{3x - 4y} \] Substituting \( x = 1, y = 1 \): \[ \frac{dy}{dx} = \frac{-(3(1) + 2(1))}{3(1) - 4(1)} \] \[ = \frac{-(3 + 2)}{3 - 4} \] \[ = \frac{-5}{-1} = 5 \] Final Answer: \[ \boxed{5} \]