 Solution

Let $d=\dim X$ and choose a local parametrizationwhose derivative has rank $d$. DefineThis is a vector subspace because it is the image of a linear map, and rank $D{\varphi }_{u_0}=d$ gives $\dim T_{p}X=d$.

If $\psi$ is another local parametrization through $p$, the transition map $h={\psi }^{-1}\circ \varphi$ is a local diffeomorphism. The chain rule givesand $Dh$ is invertible, so $\mathrm{im}D\varphi =\mathrm{im}D\psi$. Thus the definition is independent of the parametrization. Equivalently, $T_{p}X$ is the set of velocity vectors of smooth curves in $X$ through $p$. This is the tangent space from a local parametrization.