Let $G$ be a (finite-dimensional) Lie group with Lie algebra $\mathfrak g$. Then for $f\in C^\infty(G)$, $X,Y\in\mathfrak g$ and $g\in G$ one can define $f(ge^{tX})\in C^\infty(\mathbb R)$ which satisfies
$$
\frac{d}{dt}f(ge^{tX}h)=(D(ge^{tX}))(t)(f\circ R_{h^{-1}})=X_{ge^{tX}}(f\circ R_{h^{-1}})=(X(f\circ R_{h^{-1}}))(ge^{tX})\tag{1}
$$
for all $t\in\mathbb R$ simply due to properties of the exponential map (where $R_h(\cdot)=(\cdot)h^{-1}$ is the right multiplication). However, when trying to compute
$$
\frac{d}{dt}f(ge^{tX}e^{tY})=\textbf{?}
$$
(unless of course $[X,Y]=0$) I don't see how to proceed. This at least roughly looks like a hybrid-chain-product rule version but I don't see the formalism necessary to evaluate this properly.
I just know that evaluating this expression at $t=0$ should (allegedly) give $X_g(f)+Y_g(f)=(X+Y)(f)(g)$ so I'd appreciate and be thankful for any answer or comment!
Best Answer
Thanks to the helpful comments of Ted and user10354138, I was able to puzzle this together now:
First of all, for a smooth map $\phi:\mathbb R^2\to\mathbb R$, $(r,s)\mapsto\phi(r,s)$ one may define the path $\gamma:\mathbb R\to\mathbb R^2$, $t\mapsto(t,t)$ to get the derivative in question via the chain rule: $$ \frac{d}{dt}\phi(t,t)=\frac{d}{dt}(\phi\circ\gamma)(t)=\langle \operatorname{grad}\phi(\gamma(t)),\gamma'(t)\rangle=\Big(\frac{\partial\phi}{\partial r}\Big)(t,t)+\Big(\frac{\partial\phi}{\partial s}\Big)(t,t). $$ This one may concretely apply to the smooth map $\phi(r,s)=f(ge^{rX}e^{sY})$ to obtain $$ \frac{d}{dt}f(ge^{tX}e^{tY}) =\Big(\frac{d}{dr}f(ge^{rX}e^{sY})\Big)(t,t)+\Big(\frac{d}{ds}f(ge^{rX}e^{sY})\Big)(t,t)\\ \overset{\text{Eq. (1)}}= (X(f\circ R_{e^{-tY}}))(ge^{tX})+(Yf)(ge^{tX}e^{tY}) $$ Evaluating this at $t=0$ (which is what the motivation behind my question was originally) yields $$ \frac{d}{dt}\Big|_{t=0}f(ge^{tX}e^{tY})=(Xf)(g)+(Yf)(g)=((X+Y)(f))(g) $$ using the basic fact $e^{tX}|_{t=0}=e$ (the latter being the identity element in $G$). This of course works the same if $\phi$ has $n\in\mathbb N$ input variables.
If there are any mistakes in my considerations feel free to let me know--and again thank you for the comments as those were of great use!