Smooth action $G\curvearrowright M$ induces smooth action $TG \curvearrowright TM$

Question

Setup: Let $G$ be a Lie group acting smoothly on a manifold $M$ by a map $$ A\colon G\times M \to M.$$

The derivative of the mutliplication $\mu\colon G\times G \to G$ defines a group structure on $TG$. Then the derivative of $A$ defines a $TG$-action on $TM$ $$ DA \colon TG \times TM \to TM. $$

Questions:

1. How would you call this construction?
2. Do you know a book that discusses properties of this?
3. Is $DA$ a proper action if $A$ is?

Thoughts:

• The group $TG$ is isomorphic to the semi-direct product of $G$ and $\mathfrak g$ wrt the Adjoint action
• (Proper action on $M$ induces proper action on $TM$?)