We present a Radon-Nikodým theorem for vector measures of
bounded variation that are absolutely continuous with respect
to the Lebesgue measure on the unit interval. Traditional
Radon-Nikodým derivatives are Banach space-valued Bochner
integrable functions defined on the unit interval or some
other measure space. The derivatives we construct are
functions from $\ster[0,1]$, the nonstandard extension of the
unit interval into a nonstandard hull of the Banach space
$E$. For these generalized derivatives we have an integral
that resembles the Bochner integral. Furthermore, we can
standardize the generalized derivatives to produce the
weak*-measurable $E''$-valued derivatives that Ionescu-Tulcea,
Dinculeanu and others obtained in \cite{8} and \cite{5}.