That's not an equation- you are just saying that those two operators are the same. But I don't understand what it is that you want to take the second derivative of. The second derivative of the left side would be $\displaystyle \frac{\partial^4}{\partial y^4}$ and the second derivative of the right $\displaystyle \frac{\partial^3}{\partial y^3}\left(\xi_y\frac{\partial}{\partial \xi}+ \eta_y\frac{\partial}{\partial \eta}\right)$
but I doubt that is what you are asking.
just use the product rule for derivatives
$\displaystyle \frac{\partial}{\partial y}\left(\xi_y \frac{\partial}{\partial \xi}+\eta_y \frac{\partial}{\partial \eta}\right)=\frac{\partial \xi_y}{\partial y}\frac{\partial}{\partial \xi}+\xi_y \frac{\partial}{\partial y \partial \xi}+\frac{\partial \eta_y}{\partial y}\frac{\partial}{\partial \eta}+\eta_y \frac{\partial }{\partial y \partial \eta}$
I hope you understand that these are operators.