I am trying to derive
$$\mathrm dx\ \mathrm dy = r\,\mathrm dr\ \mathrm d\phi.$$
I start with the following ansatz:
$$\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} r\cos\phi \\ r\sin\phi \end{pmatrix}.$$
The change $\mathrm dx$ in $x$ is equal to:
$$\mathrm dx = \frac{\partial x}{\partial r}\ \mathrm dr + \frac{\partial x}{\partial \phi}\ \mathrm d\phi$$
and similarly for $y$. This gives
$$\mathrm dx\ \mathrm dy = \left(\frac{\partial x}{\partial r} \frac{\partial y}{\partial \phi} + \frac{\partial y}{\partial r} \frac{\partial x}{\partial \phi}\right) \mathrm dr\ \mathrm d\phi = r(\cos^2\phi - \sin^2\phi)\ \mathrm dr\ \mathrm d\phi \quad\text{(ignoring terms like $(\mathrm dr)^2$ and $(\mathrm d\phi)^2$)}$$
This is evidently incorrect and differs from the Jacobian determinant (only) by the minus sign:
$$\left| \frac{\partial (x,y)}{\partial (r,\phi)} \right|\mathrm dr\ \mathrm d\phi = \left| \frac{\partial x}{\partial r} \frac{\partial y}{\partial \phi} - \frac{\partial y}{\partial r} \frac{\partial x}{\partial \phi}\right| \mathrm dr\ \mathrm d\phi.$$
Where is my ansatz incorrect? How do I derive the Jacobian determinant?