I have question about comment in Lee's Introduction to Smooth Manifolds - page 51.
Given smooth map $F:M\to N$ between smooth manifolds $M$ and $N$ we say that the total derivative of $F$ at $p\in M$ (given chart $(U,\varphi)$ around $p$ and $(V,\psi)$ around $F(p)$) is given by $D(\psi\circ F\circ \varphi^{-1})(\varphi(p))$. The comment in the book is that total derivative is chart independent. If there is another chart $(U',\varphi')$ around $p$ then we should have
$D(\psi\circ F\circ\varphi'^{-1})(\varphi'(p))=D(\psi\circ F\circ \varphi^{-1}\circ \varphi \circ \varphi'^{-1})(\varphi'(p))=$ $D(\psi\circ F\circ\varphi^{-1})(\varphi(p))\cdot D(\varphi \circ\varphi'^{-1})(\varphi (p)).$
If total derivative is chart independent we should have $D(\varphi\circ \varphi'^{-1})(\varphi'(p))=\mathbb{Id}$, which doesn't have to be the case.
Whats wrong in my reasoning?