Consider the Dirac delta distribution on $\mathbb{R}^2$.
For an exercise I need to calculate the partial derivative $\frac{\partial}{\partial x}\delta(x,y) $ and $\frac{\partial}{\partial y}\delta(x,y)$
Let $\phi \in \mathcal{D}(\mathbb{R}^2)$
$\int_{\mathbb{R}^2}(\frac{\partial}{\partial x} \delta(x,y)) \phi(x,y)\, dx\,dy$
Using polar coordinates I get
$\int_{\mathbb{R}^2} ( \frac{\partial}{\partial r} \delta(r) ) \phi(r,\alpha)\, dr\,d\alpha=\delta(0)\phi(0,0)-\int_{\mathbb{R}^2} \delta(r) \partial_r \phi(r,\alpha)\,dr\,d\alpha$
I am not really sure on how to Continue