Zeros of differentials along one-fibered ideals

Let $\left( {R,m} \right)$ be a complete local domain containing the rationals. If $I \subseteq R$ is a one-fibered ideal then there is a constant $l$, depending only on $R$ and $I$, such that if $f \in m$ and $f \notin {I^n}$, then there exists a derivation $d$ such that $d\left( f \right) \notin {I^{n + l}}$.