Let $f\colon M\rightarrow N$ be a $C^\infty$ function between $C^\infty$ manifolds. Show that $$df_x: TM_x\rightarrow TN_{f(x)}$$ defined by $$df_x([c_0]) := [f \circ c_0]$$ is a well-defined map. Where $[c_0]$ is an equivalence class.
As a side-question; Is it possible to do this without charts? I think my course doesn't use charts