1. **Problem:** Find $\frac{dy}{dx}$ using implicit differentiation for the equation $x^2 + y^2 = 25$. 2. **Formula:** Use the rule $\frac{d}{dx}[y^n] = n y^{n-1} \frac{dy}{dx}$ when differentiating terms with $y$ implicitly. 3. **Step 1:** Differentiate both sides: $\frac{d}{dx}[x^2] + \frac{d}{dx}[y^2] = \frac{d}{dx}[25]$. 4. **Step 2:** Compute derivatives: $2x + 2y \frac{dy}{dx} = 0$. 5. **Step 3:** Solve for $\frac{dy}{dx}$: $2y \frac{dy}{dx} = -2x$ so $\frac{dy}{dx} = -\frac{x}{y}$. 6. **Explanation:** We treat $y$ as a function of $x$, so when differentiating $y^2$, we apply the chain rule. 7. **Final answer:** $$\frac{dy}{dx} = -\frac{x}{y}$$