1. We are given the implicit equation $$3y^2 + x^2 - xy = 1$$ and asked to find $$\frac{dy}{dx}$$. 2. To find $$\frac{dy}{dx}$$ for implicit functions, we use implicit differentiation: differentiate both sides with respect to $$x$$, treating $$y$$ as a function of $$x$$ (so $$\frac{dy}{dx}$$ appears when differentiating terms with $$y$$). 3. Differentiate each term: - $$\frac{d}{dx}(3y^2) = 3 \cdot 2y \cdot \frac{dy}{dx} = 6y \frac{dy}{dx}$$ (chain rule) - $$\frac{d}{dx}(x^2) = 2x$$ - $$\frac{d}{dx}(-xy) = -\left(y + x \frac{dy}{dx}\right)$$ (product rule) - $$\frac{d}{dx}(1) = 0$$ 4. Substitute these into the differentiated equation: $$6y \frac{dy}{dx} + 2x - \left(y + x \frac{dy}{dx}\right) = 0$$ 5. Distribute the minus sign: $$6y \frac{dy}{dx} + 2x - y - x \frac{dy}{dx} = 0$$ 6. Group terms with $$\frac{dy}{dx}$$ on one side and others on the opposite side: $$6y \frac{dy}{dx} - x \frac{dy}{dx} = y - 2x$$ 7. Factor out $$\frac{dy}{dx}$$: $$\frac{dy}{dx} (6y - x) = y - 2x$$ 8. Solve for $$\frac{dy}{dx}$$: $$\frac{dy}{dx} = \frac{y - 2x}{6y - x}$$ 9. This matches answer choice A. Final answer: $$\boxed{\frac{dy}{dx} = \frac{y - 2x}{6y - x}}$$