Question

If $y=f\left(x\right)$ is a differentiable function, then show that $\frac{d^{2}x}{d{y}^2}=-{\left(\frac{dy}{dx}\right)}^{-3}\frac{d^{2}y}{d{x}^2}$

Answer 1

$\frac{dx}{dy}=\frac{1}{(\frac{dy}{dx})}$

$\frac{d^{2}x}{d{y}^2}=\frac{d(\frac{dx}{dy})}{dy}=\frac{\frac{d(\frac{dx}{dy})}{dx}}{(\frac{dy}{dx})}=\frac{\frac{d}{dx}(\frac{dy}{dx}{)}^{-1}}{\frac{dy}{dx}}=(\frac{dy}{dx}{)}^{-3}\frac{d^{2}y}{d{x}^2}$