= Solution
Differentiate $u_y=y^{-1/2}(U_\xi-U_\eta)$, retaining the <derivative> of the prefactor. The <product rule> and <multivariable chain rule> give
$$
u_{yy}=-\frac1{2y^{3/2}}(U_\xi-U_\eta)
+\frac1y(U_{\xi\xi}-2U_{\xi\eta}+U_{\eta\eta}).
$$
Using $y=(\xi-\eta)^2/16$, this becomes
$$
\boxed{u_{yy}=\frac{16}{(\xi-\eta)^2}(U_{\xi\xi}-2U_{\xi\eta}+U_{\eta\eta})
-\frac{32}{(\xi-\eta)^3}(U_\xi-U_\eta)}.
$$
The last term is essential because the coordinate change is nonlinear.
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