Let $G$ be a Lie group, X be a tangent vector on $x \in G$ ( $X \in T \_x G$ ), and $\phi$ be an one-parameter subgroup on $G$.
I want to calculate these 2 formula; $$ \frac{d}{dt} d L_{\phi} X $$ $$ \frac{d}{dt} d R_{\phi} X $$ where $L$ and $R$ are left translation and right translation $G \rightarrow G$.
Some books describe $d L_{\phi} X$ as $\phi X$ and $d R_{\phi} X$ as $X \phi$, and calculate them; $$ \frac{d}{dt} d L_{\phi} X = \frac{d}{dt} \phi X = \frac{d \phi}{dt} X $$ $$ \frac{d}{dt} d R_{\phi} X = \frac{d}{dt} X \phi = X \frac{d \phi}{dt} $$ They look strange for me. How can those calculation be justified and how can I calculate them "formally"?