$\frac{d^2 x}{d y^2}$ equals

$-\left(\frac{d^2 y}{d x^2}\right)\left(\frac{d y}{d x}\right)^{-3}$

We have,

$\frac{d x}{d y}=\left(\frac{d y}{d x}\right)^{-1}$

$\Rightarrow \frac{d}{d y}\left(\frac{d x}{d y}\right)=\frac{d}{d y}\left\{\left(\frac{d y}{d x}\right)^{-1}\right\}$

$\Rightarrow \frac{d^2 x}{d y^2}=\frac{d}{d x}\left\{\left(\frac{d y}{d x}\right)^{-1}\right\} \frac{d x}{d y}$

$\Rightarrow \frac{d^2 x}{d y^2}=-\left(\frac{d y}{d x}\right)^{-2} \frac{d}{d x}\left(\frac{d y}{d x}\right) . \frac{d x}{d y}$

$\Rightarrow \frac{d^2 x}{d y^2}=-\left(\frac{d y}{d x}\right)^{-3}\left(\frac{d^2 y}{d x^2}\right)$