Possible solution:-
$\begin{aligned}& h(x, y)=(f(x, y), g(x, y)) \\ & J(h(x, y))=\left|\begin{array}{cc}f_{x} & f_{y} \\ g_{x} & g_{y}\end{array}\right|=\left(f_{x} g_{y}-f_{y} g_{x}\right) \end{aligned}$
$A=|J|^{-1}dudv$ $=\left(\frac{\partial u}{\partial x} \frac{\partial v}{\partial y}-\frac{\partial u}{\partial y} \frac{\partial v}{\partial x}\right)^{-1}$ $dudv$
I think the key idea is to use the Jacobian but this gives me an expression with x and y as I apply the chain rule. Is this still the right answer? [Standard notation for partial derivatives used]
Edit: Addressed in comments