1. **State the problem:** We need to find the derivative $\frac{dy}{dx}$ given the implicit equation $$xy = y^2 + 1.$$ 2. **Use implicit differentiation:** Differentiate both sides with respect to $x$. Remember, $y$ is a function of $x$, so use the product rule on $xy$ and chain rule on $y^2$. $$\frac{d}{dx}(xy) = \frac{d}{dx}(y^2 + 1)$$ $$x \frac{dy}{dx} + y = 2y \frac{dy}{dx} + 0$$ 3. **Rearrange terms to isolate $\frac{dy}{dx}$:** $$x \frac{dy}{dx} - 2y \frac{dy}{dx} = -y$$ $$\left(x - 2y\right) \frac{dy}{dx} = -y$$ 4. **Solve for $\frac{dy}{dx}$:** $$\frac{dy}{dx} = \frac{-y}{x - 2y}$$ 5. **Interpretation:** The derivative $\frac{dy}{dx}$ is the slope of the curve defined implicitly by $xy = y^2 + 1$. The correct answer matches option (b).