<s>[INST] If $\frac{dy}{dx}\frac{dx}{dy} = 1$, does $\frac{d^2 y}{dx^2} \frac{d^2 x}{dy^2} = 1$? I know $\frac{dy}{dx}\frac{dx}{dy} = 1$ because the chain rule says $1 = \frac{dy}{dy} = \frac{dy}{dx}\frac{dx}{dy}$.  But does $\frac{d^2 y}{dx^2} \frac{d^2 x}{dy^2} = 1$?  Or would that be too good to be true?
 [/INST] What is true is that (assuming always that $y$ is a one-to-one function of $x$ on some interval)
$$ \eqalign{\dfrac{d^2 x}{dy^2} &= \dfrac{d}{dy} \dfrac{dx}{dy} = \dfrac{dx}{dy} \dfrac{d}{dx} \left(\dfrac{dy}{dx}\right)^{-1}\cr
&= - \left(\dfrac{dy}{dx}\right)^{-3} \dfrac{d^2 y}{dx^2} }$$
In order to have $\dfrac{d^2 x}{dy^2} \dfrac{d^2 y}{dx^2} = 1$, you'd need
$$ \left(\dfrac {d^2 y}{dx^2}\right)^2 = - \left( \dfrac{dy}{dx}\right)^3$$
The solutions of this differential equation include 
$$ y = \dfrac{4}{a+x} + b$$
for constants $a,b$.
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