Let $V$ be a smooth scheme over a field $k$, and let $\{I_n, n\geq 0\}$ be a
filtration of sheaves of ideals in $\mathcal{O}_V$, such that
$I_0=\mathcal{O}_V$, and $I_s\cdot I_t\subset I_{s+t}$. In such case $\bigoplus
I_n$ is called a Rees algebra. A Rees algebra is said to be a differential
algebra if, for any two integers $N > n$ and any differential operator
$D$ of order $n$, $D(I_N)\subset I_{N-n}$. Any Rees algebra extends to a
smallest differential algebra. There are two extensions of Rees algebras of
interest in singularity theory: one defined by taking integral closures, and
another by extending the algebra to a differential algebra. We study here some
relations between these two extensions, with particular emphasis on the behavior
of higher order differentials over arbitrary fields.