$\frac{\mathrm{dx}}{\mathrm{dy}}=\frac{1}{\frac{\mathrm{dy}}{\mathrm{dx}}}={\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^{-1}\phantom{\rule{0ex}{0ex}}⇒\frac{\mathrm{d}}{\mathrm{dy}}\left(\frac{\mathrm{dx}}{\mathrm{dy}}\right)=\frac{\mathrm{d}}{\mathrm{dy}}{\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^{-1}=\frac{\mathrm{d}}{\mathrm{dx}}{\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^{-1}\left(\frac{\mathrm{dx}}{\mathrm{dy}}\right)\phantom{\rule{0ex}{0ex}}⇒\frac{{\mathrm{d}}^{2}\mathrm{x}}{{\mathrm{dy}}^{2}}=-\left(\frac{{\mathrm{d}}^{2}\mathrm{y}}{{\mathrm{dx}}^{2}}\right){\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^{-2}\left(\frac{\mathrm{dx}}{\mathrm{dy}}\right)=-\left(\frac{{\mathrm{d}}^{2}\mathrm{y}}{{\mathrm{dx}}^{2}}\right){\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^{-2}{\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^{-1}\phantom{\rule{0ex}{0ex}}\mathbf{⇒}\frac{{\mathbf{d}}^{\mathbf{2}}\mathbf{x}}{{\mathbf{dy}}^{\mathbf{2}}}\mathbf{=}\mathbf{-}\left(\frac{{\mathbf{d}}^{\mathbf{2}}\mathbf{y}}{{\mathbf{dx}}^{\mathbf{2}}}\right){\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^{\mathbf{-}\mathbf{3}}\phantom{\rule{0ex}{0ex}}$