What is the relation between $\frac{d^2y}{dx^2}$ and $\frac{d^2x}{dy^2}$?

Question

What is the relation between $\frac{d^2y}{dx^2}$ and $\frac{d^2x}{dy^2}$ ? For example $\frac{dy}{dx}$=$\frac{1}{\frac{dx}{dy}}$

Answer 1

you have $$\frac{dy}{dx} \cdot \frac{dx}{dy} = 1 \tag 1 $$ differentiationg $(1)$ with respect to $x,$ we get $$\begin{align} 0 &= \frac{d^2y}{dx^2} \cdot \frac{dx}{dy} + \frac{dy}{dx} \cdot \frac{d }{dx} \left(\frac{dx}{dy}\right) \\ &= \frac{d^2y}{dx^2} \cdot \frac{dx}{dy} +\frac{dy}{dx} \cdot\frac{d }{dy} \left(\frac{dx}{dy}\right) \cdot \frac{dy}{dx}\\ &=\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^3 \frac{d^2x}{dy^2}\end{align}$$