Let $G, H$ be algebraic groups and $K$ an algebraic closed field. Suppose that we have a morhpism $f:G \to H$ and the corresponding morphism of functions $f^*:K[H] \to K[G]$ given by $\varphi \mapsto \varphi \circ f$. Here $K[G]$ is the group algebra of $G$ over $K$.
A tangent vector at the identity is a differential $\delta: K[G]\to K_e$, where $K_e$ is the field $K$ and $K[G]$ acts via counit homomorphism. Then the tangent map $\psi: T_eG \to T_eH$ is given by $\delta \mapsto \delta \circ f$. How to verify that in coordinates $\psi$ is given by the Jacobian matrix? Thank you very much.