1. **Problem Statement:** Find the derivative $\frac{dy}{dx}$ using implicit differentiation for an equation involving both $x$ and $y$. 2. **Formula and Rules:** When differentiating implicitly, treat $y$ as a function of $x$, so use the chain rule: $\frac{d}{dx}[y] = \frac{dy}{dx}$. Differentiate both sides of the equation with respect to $x$. 3. **Example:** Suppose the equation is $x^2 + y^2 = 25$. 4. Differentiate both sides: $$\frac{d}{dx}[x^2] + \frac{d}{dx}[y^2] = \frac{d}{dx}[25]$$ 5. Compute derivatives: $$2x + 2y \frac{dy}{dx} = 0$$ 6. Solve for $\frac{dy}{dx}$: $$2y \frac{dy}{dx} = -2x$$ $$\frac{dy}{dx} = -\frac{x}{y}$$ 7. **Explanation:** We differentiated $y^2$ as $2y \frac{dy}{dx}$ because $y$ depends on $x$. Then we isolated $\frac{dy}{dx}$ to find the slope of the curve implicitly defined by the equation. **Final answer:** $$\frac{dy}{dx} = -\frac{x}{y}$$