What shown below is a reference from the text *Differential from* by Victor Gullemin an Peter Haine. I point out that if you know the definitoin of Manifold and tangent space you can read only the part of the image beyond the theorem $4.2.10$.

So I ask effectively why the derivative of the map $g$ at the points of $X$ does not by the extension $tilde g$ any why moreover by a particular parametrization: indeed if $phi$ and $psi$ are two different local patches about $p$ then $dtilde g=d(gcircphi)big(phi^{-1}(p)big)$ and $dtilde g=d(gcircpsi)big(psi^{-1}(p)big)$ but by the chain rule

$$

dtilde g(p)=d(gcircphi)big(phi^{-1}(p)big)=d(gcircpsi)big(psi^{-1}(p)big)cdot d(psi^{-1}circphi)big(phi^{-1}(p)big)=dtilde g(p)cdot d(psi^{-1}circphi)big(phi^{-1}(p)big)

$$

and unfortunately I think that $big(phi^{-1}(p)big)neq 1$ generally. So could someone help me, please?