Answer
$\dfrac{d^2y}{dx^2}=-\dfrac{4}{y^3}.$
Work Step by Step
First Derivative:
$\dfrac{d}{dx}(x^2)+\dfrac{d}{dx}(y^2)=\dfrac{d}{dx}(4)\rightarrow$
$2x+\dfrac{dy}{dx}(2y)=0\rightarrow$
$\dfrac{dy}{dx}=-\dfrac{x}{y}$
Second derivative:
$\dfrac{d}{dx}(\dfrac{dy}{dx})=\dfrac{d}{dx}(-\dfrac{x}{y})\rightarrow$
$\dfrac{d^2y}{dx^2}=-\dfrac{y-(\dfrac{dy}{dx})(x)}{y^2}=-\dfrac{y^2+x^2}{y^3}=-\dfrac{4}{y^3}.$
