1. **Problem:** Find $\frac{dy}{dx}$ by implicit differentiation for the equation $$x^2 + y^2 = 100$$. 2. **Formula and rules:** When differentiating implicitly, treat $y$ as a function of $x$, so use the chain rule: $$\frac{d}{dx}[y^2] = 2y \frac{dy}{dx}$$. 3. **Differentiate both sides:** $$\frac{d}{dx}[x^2] + \frac{d}{dx}[y^2] = \frac{d}{dx}[100]$$ $$2x + 2y \frac{dy}{dx} = 0$$ 4. **Solve for $\frac{dy}{dx}$:** $$2y \frac{dy}{dx} = -2x$$ $$\frac{dy}{dx} = \frac{-2x}{2y} = \frac{-x}{y}$$ 5. **Explanation:** We differentiated $x^2$ normally, and for $y^2$ we used the chain rule because $y$ depends on $x$. Then we isolated $\frac{dy}{dx}$ to find the slope of the curve at any point. **Final answer:** $$\frac{dy}{dx} = \frac{-x}{y}$$