Weak Derivative of $x \mapsto \ln(|x|)$ doesn't exists in $(-1,1)$ but in $B_1(0)\in \mathbb{R}^2$

Question

How can I show that $f: (-1,1) \to \mathbb{R}, \ x \mapsto \ln(|x|)$ has no weak derivative but in $B_1(0)\in \mathbb{R}^2$ it has? I know that every classical solution is also a weak solution in this case.

Any help is greatly appreciated.