1. **Problem:** Find the derivative $y'$ of the implicit function defined by $$x^2 - xy + y^2 = 7.$$ 2. **Formula and rules:** We use implicit differentiation. Differentiate both sides with respect to $x$, remembering that $y$ is a function of $x$, so when differentiating terms with $y$, apply the chain rule: $$\frac{d}{dx}[y] = y'.$$ 3. **Differentiate each term:** - $\frac{d}{dx}[x^2] = 2x$ - $\frac{d}{dx}[-xy] = -\left(y + x y'\right)$ by product rule - $\frac{d}{dx}[y^2] = 2y y'$ by chain rule - $\frac{d}{dx}[7] = 0$ 4. **Write the differentiated equation:** $$2x - (y + x y') + 2y y' = 0$$ 5. **Group terms with $y'$ on one side:** $$2x - y - x y' + 2y y' = 0$$ $$- x y' + 2y y' = y - 2x$$ $$y'(-x + 2y) = y - 2x$$ 6. **Solve for $y'$:** $$y' = \frac{y - 2x}{-x + 2y} = \frac{y - 2x}{2y - x}$$ 7. **Final answer:** $$\boxed{y' = \frac{y - 2x}{2y - x}}$$ This matches option B.