Suppose $G$ is an algebraic group, and $\rho_x\colon G\to G$ is the translation map $\rho_x(g)=gx$. What is $(d\rho_x)_1\colon T_1(G)\to T_x(G)$?
If $a\in T_1(G)$, then $(d\rho_x)(a)=a\circ \rho_x^\ast$, so if $f\in k[G]$ is a function on $G$, then $$ (d\rho_x)(a)(f)=a(\rho_x^\ast(f))=a(f\circ\rho_x). $$
Is there a way to get something more computationally meaningful though? Like how the differentials of multiplication and inversion correspond to addition and multiplying by a negative sign?
I like to use dual numbers to compute with Lie algebras. If you consider an algebraic group $G$ over a commutative ring $k$ as a functor then the tangent bundle is $G(k[\epsilon]/\epsilon^2)$. The usual tangent space is the subfunctor whose elements are of the form $e + A\epsilon$ where $e\in G$ is the identity.
So the differential of multiplication can be computed via:
$(e + A\epsilon)(e + B\epsilon) = e + (A + B)\epsilon.$
Hence multiplication corresponds to addition in the tangent space.
The differential of right translate by $x$ is then
$(e + A\epsilon)\mapsto (e + A\epsilon)x = x + Ax\epsilon$
which is now in the tangent space at $x$. So the map is just $A\mapsto Ax$.