I am trying to express the second derivatives of a function $U(x,y)$ in terms of a change of variable $\alpha (x,y)$ and $\beta(x,y)$ using the chain rule and that: $$U_x=\frac{\partial\alpha}{\partial x}U_{ \alpha}+\frac{\partial\beta}{\partial x}U_{ \beta} \quad \quad U_y=\frac{\partial\alpha}{\partial y}U_{ \alpha}+\frac{\partial\beta}{\partial y}U_{ \beta}, $$ Notice that: $$U_{xx}=\frac{\partial^2\alpha}{\partial x^2}U_{ \alpha}+\frac{\partial U_\alpha}{\partial x} \frac{\partial \alpha}{\partial x}+ \frac{\partial^2\beta}{\partial x^2}U_{ \beta}+\frac{\partial U_\beta}{\partial x} \frac{\partial \beta}{\partial x}, $$ and $$ \frac{\partial U_\alpha}{\partial x}=\frac{\partial}{\partial x}\left( \frac{\partial U}{\partial x}\frac{\partial x}{\partial \alpha} \right)+\frac{\partial}{\partial x}\left( \frac{\partial U}{\partial y}\frac{\partial y}{\partial \alpha} \right)$$
And there I am not sure how to continue, please some help.
Remember, you're trying to get $\partial U_\alpha /\partial x$ in terms of $\alpha$- and $\beta$-derivatives. What you're looking for is $$ \frac{\partial U_\alpha}{\partial x} = \frac{\partial \alpha}{\partial x}U_{\alpha\alpha} + \frac{\partial \beta}{\partial x} U_{\alpha\beta} $$ so that $$U_{xx}=\frac{\partial^2\alpha}{\partial x^2}U_{ \alpha}+\left(\frac{\partial \alpha}{\partial x}U_{\alpha\alpha} + \frac{\partial \beta}{\partial x} U_{\alpha\beta}\right) \frac{\partial \alpha}{\partial x}+ \frac{\partial^2\beta}{\partial x^2}U_{ \beta}+\left( \frac{\partial \alpha}{\partial x}U_{\alpha\beta} + \frac{\partial \beta}{\partial x} U_{\beta\beta}\right)\frac{\partial \beta}{\partial x} \\ = \frac{\partial^2\alpha}{\partial x^2}U_{ \alpha}+ \frac{\partial^2\beta}{\partial x^2}U_{ \beta} +\left(\frac{\partial \alpha}{\partial x}\right)^2U_{\alpha\alpha} + 2\frac{\partial \alpha}{\partial x}\frac{\partial \beta}{\partial x} U_{\alpha\beta}+ \left(\frac{\partial \beta}{\partial x}\right)^2 U_{\beta\beta}$$