  The main goal of this paper is to characterize the module of K\"ahler
differentials for an extension of valuation rings. More precisely, we consider
a simple algebraic valued field extension $(L/K,v)$ and the corresponding
valuation rings $\VR_L$ and $\VR_K$. In the case when $e(L/K,v)=1$ we present a
characterization for $\Omega_{\VR_L/\VR_K}$ in terms of a given sequence of key
polynomials for the extension. Moreover, we use our main result to present a
characterization for when $\Omega_{\VR_L/\VR_K}=\{0\}$.
