How to prove that a given function $\varphi : \mathbb{R}^2 \to \mathbb{R}$ is weakly differentiable?

Question

For $0<r<R$, we define $\varphi : \mathbb{R}^2 \to \mathbb{R}$ by $$\varphi(w):=\left\{ \begin{array}{ll} 1, & \lvert w\rvert \leq r \\ \frac{log \lvert w\rvert - \log R}{\log r - \log R}, & r<\lvert w\rvert < R \\ 0, & \lvert w\rvert \geq R \end{array} \right. $$ Then $\varphi$ is continuous on $\mathbb{R}^2$. Since $\varphi$ is differentiable on $\mathbb{R}^2 \setminus \{ w\in \mathbb{R}^2 : \lvert w\rvert = r \text{ or } \lvert w\rvert=R\}$, an obvious choice for the weak derivatives of $\varphi$ are the the classical partial derivatives $\varphi_u, \varphi_v$, which can be continued to functions on the whole real plane by $$\varphi_u = \varphi_v = 0\text{ on } \{ w\in \mathbb{R}^2 : \lvert w\rvert = r \text{ or } \lvert w\rvert=R\}$$ I'm not sure if one can use the definition to prove that these functions are in fact the weak derivatives of $\varphi$, but I wasn't able to find a theorem which would guarantee the weak differentiability of $\varphi$.