Let $M$ be a smooth manifold, let $p\in M$ and $v\in T_pM\ v \neq 0$ show there exists a smooth chart $(U, \varphi=(x_1,\dots,x_n))$ such that: $p \in U$ and $$ \frac{\partial}{\partial x_1}\bigg\rvert_p = v $$
Even more this can be generalized to given any linearly independet set $\{v_1,...,v_k\}$ in $T_pM$ there exists a smooth chart $(U, \varphi=(x_1,\dots,x_n))$ such that: $p \in U$ and $$ \frac{\partial}{\partial x_1}\bigg\rvert_p = v_1 ,\ \frac{\partial}{\partial x_2}\bigg\rvert_p = v_2\ , ...,\ \frac{\partial}{\partial x_k}\bigg\rvert_p = v_k $$
any help in solving this problem is appreciated.