  Let $z_{1},\ldots,z_{K}$ be distinct grid points. If $f_{k,0}$ is the
prescribed value of a function at the grid point $z_{k}$, and $f_{k,r}$ the
prescribed value of the $r$\foreignlanguage{american}{-th} derivative, for
$1\leq r\leq n_{k}-1$, the Hermite interpolant is the unique polynomial of
degree $N-1$ ($N=n_{1}+\cdots+n_{K}$) which interpolates the prescribed
function values and function derivatives. We obtain another derivation of a
method for Hermite interpolation recently proposed by Butcher et al.
{[}\emph{Numerical Algorithms, vol. 56 (2011), p. 319-347}{]}. One advantage of
our derivation is that it leads to an efficient method for updating the
barycentric weights. If an additional derivative is prescribed at one of the
interpolation points, we show how to update the barycentric coefficients using
only $\mathcal{O}\left(N\right)$ operations. Even in the context of confluent
Newton series, a comparably efficient and general method to update the
coefficients appears not to be known. If the method is properly implemented, it
computes the barycentric weights with fewer operations than other methods and
has very good numerical stability even when derivatives of high order are
involved. We give a partial explanation of its numerical stability.
