Writing the Different in terms of the trace

This question is based on Chapter 6 of Field Arithmetic by Fried and Jarden.

Let $R$ be a Dedekind domain with a quotient field $K$. Let $L/K$ be a Galois extension and let $S$ be the integral closure of $R$ in $L$. For a given $z\in S$ we have that $f$ is its irreducible polynomial in $K$.

They then define the different of $S$ over $R$ as the gcd of all the principal ideals $f'(z)S$, with $z\in S$ and then give the following formula

$(\mathrm{Diff}(S/R)^{-1}=\{x\in L|\mathrm{trace}_{L/K}(xS)\subseteq R\}$.

I am not sure how this follows from the way it has been defined. Any hints would be appreciated.