Map between Zariski tangent spaces(?)

Well, using Hartshorne's notation, we have a map $f_x^\#:\mathcal{O}_{Y,y}\to\mathcal{O}_{X,x}$ that restricts and descends to a map $f_x^\#:{\frak{m}}_y/{\frak{m}}_y^2\to {\frak{m}}_x/{\frak{m}}_x^2$ (since $(f_x^\#)^{-1}({\frak{m}}_x)={\frak{m}}_y$). By pulling back, we get a map $$T_{X,x}=\mbox{Hom}_{k(x)}({\frak{m}}_x/{\frak{m}}_x^2,k(x))\to\mbox{Hom}_{k(y)}({\frak{m}}_y/{\frak{m}}_y^2,k(x))\simeq\mbox{Hom}_{k(y)}({\frak{m}}_y/{\frak{m}}_y^2,k(y))\otimes k(x)$$ where $k(y)$ acts on $k(x)$ by $f_x^\#$. This last $k(x)$-vector space is then $T_{Y,y}\otimes k(x)$.

Edit As Georges points out below, this last isomorphism works whenever $k(x)$ is a finite extension of $k(y)$.