Official solution
\[\frac{dx}{dy}=\left(\frac{dy}{dx}\right)^{-1}\Rightarrow\frac{d^2x}{dy^2}=\frac{d}{dx}\left[\left(\frac{dy}{dx}\right)^{-1}\right]\cdot\frac{dx}{dy}=-\left(\frac{dy}{dx}\right)^{-2}\frac{d^2y}{dx^2}\cdot\left(\frac{dy}{dx}\right)^{-1}=-\left(\frac{d^2y}{dx^2}\right)\left(\frac{dy}{dx}\right)^{-3}\]