Question Detail

To find dy/dx, we use implicit differentiation. Differentiate both sides of the equation x³ + x + y³ + 3y = 6 with respect to x:d/dx(x³) + d/dx(x) + d/dx(y³) + d/dx(3y) = d/dx(6)3x² + 1 + 3y²(dy/dx) + 3(dy/dx) = 0Factor out dy/dx:dy/dx(3y² + 3) = -(3x² + 1)Solve for dy/dx:dy/dx = -(3x² + 1) / (3y² + 3)Now, substitute the point (1, 1) into the derivative:dy/dx = -(3(1)² + 1) / (3(1)² + 3) = -(3 + 1) / (3 + 3) = -4 / 6 = -2/3.