 Solution

For a local parametrization $\varphi :U\subset {\mathbb{R}}^k\to X$, the map $(q,w)\mapsto (\varphi (q),d{\varphi }_{q}w)$ parametrizes $TX$ locally. Its inverse uses the base chart and derivative, and its rank is $2k$. These charts make $TX$ a $2k$-manifold.