I was looking for a method to compute the explicit differential of a regular map between algebraic groups. More precisely if $X$ is a sub-variety in an algebraic group $G$ (say over a finite field $k$), I'd like to compute the differential of the map \begin{align} \phi:X\times X&\longrightarrow G\\ (x_1,x_2)&\longmapsto (x_1*x_2), \end{align} where * is the group low.
I found this related question, "What is the differential of a translation map on an algebraic group?" but I'm a bit confused about the re-interpretation of the tangent bundle and the dual numbers. Can someone explain me a bit more what is going on?
Thank you very much!