Define a function $f(x)$ on $\mathbb{R}$ as:
$f(x) =\begin{cases}0&\text{for }x \le 0\\\\\frac{1}{\sqrt{x}}&\text{for }x > 0\;.\end{cases}$
Then, how can I calculate the weak derivative(of course, distribution sense) of it? I roughly guess in $x>0$ region it becomes $\dfrac{-1}{2(\sqrt{x})^3}$; but I don't know how to deal with the neighborhood of $x=0$.