  In the paper, by establishing a new and explicit formula for computing the
$n$-th derivative of the reciprocal of the logarithmic function, the author
presents new and explicit formulas for calculating Bernoulli numbers of the
second kind and Stirling numbers of the first kind. As consequences of these
formulas, a recursion for Stirling numbers of the first kind and a new
representation of the reciprocal of the factorial $n!$ are derived. Finally,
the author finds several identities and integral representations relating to
Stirling numbers of the first kind.
