<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] If you want an example where the derivatives don't vanish, consider $y=x^3$ (so $x=y^{1/3}$).  Then:
$$
\frac{dy}{dx}=3x^2
$$ 
and
$$
\frac{dx}{dy}=\frac{1}{3}y^{-2/3}=\frac{1}{3}x^{-2}
$$
and
$$
\frac{dy}{dx}\frac{dx}{dy}=3x^2\cdot\frac{1}{3x^2}=1.
$$
On the other hand,
$$
\frac{d^2y}{dx^2}=6x
$$
and
$$
\frac{d^2x}{dy^2}=-\frac{2}{9}y^{-5/3}=-\frac{2}{9x^5}.
$$
However, 
$$
\frac{d^2y}{dx^2}\frac{d^2x}{dy^2}=6x\cdot\left(-\frac{2}{9x^5}\right)=-\frac{4}{3x^4}\not=1.
$$
</s>