Given that I have $u = x\cdot y$, and $v = \frac{y}{x}$.
How do I calculate the Jacobian determinant which should be:
$$J =\begin{vmatrix} \frac{dx}{du} & \frac{dx}{dv} \\ \frac{dy}{du} & \frac{dy}{dv} \end{vmatrix}$$
Is there any good way to do this? Or do I just have to use $x = \sqrt{\frac{u}{v}}$, and $y = \sqrt{u\cdot v}$
Just substitute and derive. Also, those are partial derivatives
$$\begin{align}\begin{vmatrix}\dfrac{\partial x}{\partial u} & \dfrac{\partial x}{\partial v} \\ \dfrac{\partial y}{\partial u} & \dfrac{\partial y}{dv} \end{vmatrix}&=\begin{vmatrix}\dfrac{\partial u}{\partial x} & \dfrac{\partial u}{\partial y} \\ \dfrac{\partial v}{\partial x} & \dfrac{\partial v}{dy} \end{vmatrix}^{-1}\\[2ex] &= \begin{vmatrix}\dfrac{\partial xy}{\partial x} & \dfrac{\partial xy}{\partial y} \\ \dfrac{\partial x^{-1}y }{\partial x} & \dfrac{\partial x^{-1}y}{\partial y} \end{vmatrix}^{-1}\\[2ex] &= \left(\phantom{\begin{vmatrix}y & x \\ -x^{-2}y & x^{-1} \end{vmatrix}}\right)^{-1}\\[2ex]&=(\phantom{2x^{-1}y})^{-1}\\[2ex]&=\phantom{\tfrac 12v^{-1}}\end{align} $$