If $\tau$ is a hereditary torsion theory on ${\mathrm{𝐌𝐨𝐝}}_R$ and $Q_{\tau }:{\mathrm{𝐌𝐨𝐝}}_R\to {\mathrm{𝐌𝐨𝐝}}_R$ is the localization functor, then we show that every $f$-derivation $d:M\to N$ has a unique extension to an $f_{\tau }$-derivation $d_{\tau }:Q_{\tau }\left(M\right)\to Q_{\tau }\left(N\right)$ when $\tau$ is a differential torsion theory on ${\mathrm{𝐌𝐨𝐝}}_R$. Dually, it is shown that if $\tau$ is cohereditary and $C_{\tau }:{\mathrm{𝐌𝐨𝐝}}_R\to {\mathrm{𝐌𝐨𝐝}}_R$ is the colocalization functor, then every $f$-derivation $d:M\to N$ can be lifted uniquely to an $f_{\tau }$-derivation $d_{\tau }:C_{\tau }\left(M\right)\to C_{\tau }\left(N\right)$.