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$f: X\rightarrow Y$ is a diffeomorphism, then at each $x$ its derivative $df_x$ is an isomorphism of tangent spaces.could you please give me proof and insight of this result?

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Show that if $f^{-1}:Y\to X$ is your inverse map then $d(f^{-1})_{f(x)}$ is an inverse too of $df_x$.

Remark: This proves the well-definedness of the dimension of a smooth manifold. MUCH easier than to prove the well-defiedness of dimension for topological manifolds.

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