Suppose we have a function $u(x,y)$ which we re-define wrt another variable $\omega$, such that $u(x,y)=u(\omega(x,y))$ and there is an implicit relation between $\omega$, $x$ and $y$,
$x-p(\omega)y=0$
The double derivatives needs to be calculated. The solution I found somewhere was,
$u_{,yy}= \frac{\partial \omega}{\partial x} \frac{\partial}{\partial \omega} \left(p(\omega)^2\frac{\partial \omega}{\partial x} u'(\omega)\right)$ , where $u'(\omega)$ is differentiation wrt $\omega$
What I am getting by chain rule is,
$u_{,yy}=p(\omega) \frac{\partial \omega}{\partial x} \frac{\partial}{\partial \omega} \left(p(\omega)\frac{\partial \omega}{\partial x} u'(\omega)\right)$
Please let me know if I am missing something here. Thanks.