1. **State the problem:** We are given the equation of a circle: $$x^2 + y^2 = 16$$ and need to find the derivative $$\frac{dy}{dx}$$. 2. **Formula and rules:** Since $$y$$ is implicitly defined as a function of $$x$$, we use implicit differentiation. Differentiate both sides with respect to $$x$$. 3. **Differentiate each term:** - $$\frac{d}{dx}(x^2) = 2x$$ - $$\frac{d}{dx}(y^2) = 2y \frac{dy}{dx}$$ (by chain rule) - $$\frac{d}{dx}(16) = 0$$ 4. **Write the differentiated equation:** $$2x + 2y \frac{dy}{dx} = 0$$ 5. **Solve for $$\frac{dy}{dx}$$:** $$2y \frac{dy}{dx} = -2x$$ 6. **Cancel common factor 2:** $$\cancel{2}y \frac{dy}{dx} = -\cancel{2}x$$ 7. **Isolate $$\frac{dy}{dx}$$:** $$\frac{dy}{dx} = -\frac{x}{y}$$ **Final answer:** $$\boxed{\frac{dy}{dx} = -\frac{x}{y}}$$