Let $p:X\to S$ be an $S$-scheme and let $f:X\to \mathbb A$ be a regular function on $X$. Then $f\in \mathcal O(X)$, and we may apply the universal derivation $d_{X/S}:\mathcal O_X\to \Omega_{X/S}$ to $f$. Is it true that $d(f)=0$ if and only if $f:X\to \mathbb A$ is constant on the fibers of $X\to S$ in the sense that $f$ factors as $X\to S\to \mathbb A$ for some regular function $S\to \mathbb A$ on the scheme $S$? One direction follows because $d_{X/S}$ is the universal $p^{-1}\mathcal O_S$-derivation, but what about the other?
In other words: How does the kernel of $d_{X/S}$ look like?