  Let $\chi$ be a quadratic Dirichlet character. In some literatures, various
asymptotic formulae of $L'(1,\chi)$, under the assumption that $L(1,\chi)$
takes a small value, were derived. In this paper, we will give a new treatment
unified for the odd and even cases, not depending on Kronecker limit formula.
For imaginary quadratic fields, our result coincides with Proposition 22.10 in
[1].
