Left-invariant vector field is differentiable

I am quite new to the subject, please comment if something does not make sense.

Let $$M$$ be a Lie group. $$L_x$$ be left multiplication by $$x\in M$$.
I want to show that vector space $$X:M \to TM$$ defined as $$X_x = T_1 L_x(v)$$ is differentiable. ($$v\in T_1M$$)

If I am not mistaken, I need to argue the map $$$$a \mapsto \left(a,\ T_{\varphi^{-1}(a)}\varphi (T_1L_{\varphi^{-1}(a)}(v))\right) = \left(a,\ T_1(\varphi\ \circ L_{\varphi^{-1}(a)})(v)\right)$$$$ is differentiable ($$a\in \mathbb{R}^n$$, $$\varphi$$ is a chart). I am not sure how to argue tangent map is differentiable.