$x=r\cos \theta$ $\therefore x_{r}=\frac{\partial x}{\partial r}=\cos \theta\ and\ x_{\theta}=\dfrac{\partial x}{\partial \theta}=-r\sin \theta$ $y=r\sin \theta$ $\therefore y_{r}=\dfrac{\partial y}{\partial r}=\sin \theta\ and\ y_{\theta}=\dfrac{\partial y}{\partial \theta}=r\cos\theta$ $\therefore \dfrac{\partial (x,y)}{\partial(r,\theta)}=\begin{vmatrix} x_{r} &x_{\theta} \y_{r} &y_{\theta} \end{vmatrix}$ $=x_{r}y_{\theta}-x_{\theta}y_{r}$ $=\cos\theta\cdot r\cos\theta+r\sin \theta\cdot\sin \theta$ $=r(\cos^{2}\theta+\sin^{2}\theta)$

= r