Weak Gradient of the Unit Ball

Question

Let $\mathbf{x}\in\mathbb{R}^2,$ and $f(\mathbf{x})=1$ if $||\mathbf{x}||\leq R\in\mathbb{R}$, $f(\mathbf{x})=0$ otherwise. I want to find the weak gradient of $f$ in the dual space of compactly supported $C^{\infty}(\mathbb{R}^2)$ test functions, $\phi$. This is my attempt for the derivative with respect to $r$ by just following definitions, Fubini, and integration by parts: $$\frac{\partial}{\partial r}T_f(\phi)=-T_f\left(\frac{\partial \phi}{\partial r}\right)=-\int_{0}^{2\pi}\int_{0}^R\frac{\partial \phi}{\partial r}rdrd\theta=-\int_{0}^{2\pi}\left(R\phi(R,\theta)- \int_{0}^R\phi dr\right)d\theta$$ $$=\int_0^{2\pi}\int_{0}^R(1-r\delta(R-r))\phi drd\theta.$$ Granted that this work is correct, can I conclude that the weak derivative of $f$ with respect to $r$ is $\frac{1}{r}-\delta(R-r)$? Or do I have the wrong idea?