Implicit Differentiation 187A06

1. **Problem Statement:** Differentiate implicit functions, where the function is given in an implicit form rather than explicitly as $y=f(x)$. 2. **Formula and Rules:** For an implicit function involving $x$ and $y$, differentiate both sides of the equation with respect to $x$, treating $y$ as a function of $x$ (i.e., $y=y(x)$). Use the chain rule: $$\frac{d}{dx}[y] = \frac{dy}{dx}$$ and for terms involving $y$, multiply by $\frac{dy}{dx}$. 3. **Example:** Suppose the implicit equation is $$x^2 + y^2 = 25$$. Differentiate both sides with respect to $x$: $$\frac{d}{dx}[x^2] + \frac{d}{dx}[y^2] = \frac{d}{dx}[25]$$ 4. Compute derivatives: $$2x + 2y \frac{dy}{dx} = 0$$ 5. Solve for $\frac{dy}{dx}$: $$2y \frac{dy}{dx} = -2x$$ $$\frac{dy}{dx} = \frac{-2x}{2y}$$ 6. Simplify: $$\frac{dy}{dx} = \frac{-x}{y}$$ **Final answer:** The derivative of $y$ with respect to $x$ for the implicit function $x^2 + y^2 = 25$ is $$\frac{dy}{dx} = \frac{-x}{y}$$.