Given that for each $\mathbf{a} \in(0,1)$
$ \lim _{h \rightarrow 0^{+}} \int_h^{1-h} t^{-a}(1-t)^{a-1} d t $
exists. Let this limit be $\mathrm{g}(\mathrm{a})$. In addition, it is given that the function $\mathrm{g}(\mathrm{a})$ is differentiable on $(0,1)$.
$\lim _{h \rightarrow 0^{+}} \int_h^{1-h} t^{-a}(1-t)^{a-1} d t$

The value of $g^{\prime}\left(\frac{1}{2}\right)$ is