Differential of a representation of a linear algebraic group

Question

I asked a question if a representation of the lie algebra of a simply connected algebraic group G induces a representation of the group itself here:

\link {Representation of the lie algebra of a simply connected algebraic group $G$ induces a representation of the group itself}

I have got an answer which was satisfactory enough for me to proceed but now I would appreciate any help to realise the following:

Further, if we take the differential of the induced representation of the group, do we get the representation $\psi$ which we started with ? If yes, how ? Please help me in realising this !