Kernel of a homomorphism of modules

Question

Let $R$ be a finitely generated $k$-algebra, $k$ a field. (Impose more conditions if you like.) For an element $f\in R$ consider the natural map $\Omega^1_{R/k}\rightarrow\Omega^1_{R/(f)/k}$ which takes $dr$ to $d[r]$. This is obviously surjective. Is there an explicit description of the kernel? I can see that all $di$, $i\in I=(f)$ are in the kernel.