Let $M^n$ be a smooth manifold and $v : M \to TM$ a vector field. I am interested in a singularity $p$ of $v$. I know that the derivative of $v$ at $p$ maps $T_p M$ into $T_{(p,0)} (TM)$, but is the latter space isomorphic to a more treatable vector space (in terms of $M$, say)?
Moreover, let $\varphi : U \to M$ be a parametrization around $p$ and write
$v(\varphi(x)) = \sum_i a_i(x) \frac{\partial \varphi}{\partial x_i}(x)$
on $U$. How can we calculate the derivative of $v$ in terms of the functions $a_i$?