I am currently working through Guillemin & Pollack (slowly) and I've run into a bump quite early on. So far, we've recalled that maps from open sets to $\mathbb{R}^n$ are "well behaved". Their derivatives are linear maps between the copies of affine space that they are defined on and they satisfy the chain rule. We wish to extend derivatives to smooth maps between manifolds so that they act like normal derivatives when the manifolds are affine space, and so that they also satisfy the chain rule. The approach is this: Let $f:X\to Y$ be a smooth map of smooth $k$ and $l$ manifolds, respectively, with $x\mapsto y$. Let $\phi:(U,0)\to (X,x)$ parametrize $X$ around $x$ and let $\psi:(V,0)\to (Y,y)$ parametrize $Y$ around $y$ with $U\subset \mathbb{R}^k$ and $V\subset\mathbb{R}^l$ open sets. To get a commutative diagram, shrink $U$ and let $h$ complete the diagram, $h=\psi^{-1}\circ f\circ\phi$.
(To "shrink" properly, I believe we let $U'=U\cap\phi^{-1}(f^{-1}(\psi(V)))$)
So then we make the definition $df_x=d\psi_0\circ dh_0\circ d\phi_0^{-1}$. I'm having a hard time seeing why this definition is independent of the choice of parametrization. I believe it, but I can't lock it down set theoretically if you know what I mean. The book advises that we use a similar argument to showing that the tangent space of $X$ at $x$ is well defined. I've tried introducing alternative parametrizations $\eta$ and $\rho$ to parallel $\phi$ and $\psi$, but my diagrams are confusing and I am not sure which clever composition to make and then subsequently take a derivative of. Any help would be much appreciated. Thanks!
EDIT: Some more details on my progress. Suppose $\rho:V'\to Y$ is an alternative parametrization of $Y$ at $y$. Without loss of generality, let $\rho(V')=\psi(V)$. Then we have $h'=\rho^{-1}\circ f\circ \phi$ and the possibly different derivative map $df_x'=d\rho_0\circ dh_0'\circ d\phi_0^{-1}$. Then notice that $\rho\circ h'=\psi\circ h=f\circ\phi$ is a map from an open set in affince space to affine space. Then the chain rule applies and we get that $df_x'=d\rho_0\circ dh_0'\circ d\phi_0^{-1}=d\psi_0\circ dh_0\circ d\phi_0^{-1}=df_x$ as desired. I can get a symmetric result if we swap out $\phi$ for $\eta$, also. But I can't figure out how to change them both at once.