Implicit Differentiation C3Ee39

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:** Given the equation $$x^2 + y^2 = 25,$$ differentiate both sides: $$\frac{d}{dx}[x^2] + \frac{d}{dx}[y^2] = \frac{d}{dx}[25].$$ 4. **Differentiate each term:** $$2x + 2y \frac{dy}{dx} = 0.$$ Here, $\frac{d}{dx}[y^2] = 2y \frac{dy}{dx}$ by the chain rule. 5. **Solve for $\frac{dy}{dx}$:** $$2y \frac{dy}{dx} = -2x$$ $$\frac{dy}{dx} = \frac{-2x}{2y}$$ 6. **Simplify the fraction:** $$\frac{dy}{dx} = \frac{\cancel{2}(-x)}{\cancel{2}y} = \frac{-x}{y}.$$ 7. **Final answer:** The derivative of $y$ with respect to $x$ is $$\boxed{\frac{dy}{dx} = -\frac{x}{y}}.$$