For a smooth n-manifold $M$ we have a tangent bundle $TM$, which is a vector bundle of dimension 2n. Now let $p_M: TM \rightarrow M$ be the canonical projection map $(x, v_x) \mapsto x$. I am looking at an exercise that requires me to show that the derivative of $p_M$, $D p_M$ gives a smooth map $T(T(M)) \rightarrow TM$.
My first difficulty is interpreting what is meant by derivative here? Also, could you provide me with a hint on how to go about the question itself?