Chapter 3 - Derivatives - 3.8 Implicit Differentiation - 3.8 Exercises - Page 200: 32

$\frac{d^2y}{dx^2} = \frac{-2y^2-4x^2}{y^3}$

$2x^2+y^2=4$ First Derivative: $4x+2y\frac{dy}{dx}=0$ $2y\frac{dy}{dx} = -4x$ $\frac{dy}{dx} = \frac{-2x}{y}$ Second Derivative: $\frac{d^2y}{dx^2} = \frac{(-2)(y)-(-2x)(\frac{dy}{dx})}{y^2} = \frac{-2y-\frac{4x^2}{y}}{y^2} = \frac{-2y^2-4x^2}{y^3}$