In defining the tangent space of a smooth manifold at a point $p$, one can consider a local coordinate $\phi: U \to \phi(U)$ with $p \in U$ and define the tangent space as the tangent space to $\phi(p)$ in $\phi(U)$ which is an open subset of $\{\mathbb R}^n$. However this definition would depend on the choice of $\phi$, which is undesirable. Now, the idea is to use the natural isomorphisms between tangent spaces thus defined (the derivatives of the change of coordinate maps) to identify these spaces.

In defining the tangent space of a smooth manifold at a point $p$, one can consider a local coordinate $\phi: U \to \phi(U)$ with $p \in U$ and define the tangent space as the tangent space to $\phi(p)$ in $\phi(U)$ which is an open subset of ${\mathbb R}^n$. However this definition would depend on the choice of $\phi$, which is undesirable. Now, the idea is to use the natural isomorphisms between tangent spaces thus defined (the derivatives of the change of coordinate maps) to identify these spaces.