Let $M$ be a manifold and $G$ be a Lie Group acting smoothly on $M$ on the right and on the left.
Set $f:M \times G \rightarrow M$ by $f(p,g)=g^{-1}.p$.
If we denote the right action by $p.g=\mu(p,g)$, we know that $$d_{(p,g)}\mu(X,(L_g)_*(Y))=(r_g)_*(X)+\left. \frac{\mathrm{d}}{\mathrm{d}t} \right|_{t=0}((p.g).e^{tY})$$ where $r_g:M\rightarrow M$ is given by $r_g(p)=p.g$
Is there a similar expression to $d_{(p,g)} f$ ? Any help is appreciated.