Let $M$ be a manifold and $\varphi, \psi$ two charts. Now for $\varphi=(x^1,...,x^n), \psi=(y^1,...,y^n)$ I know that it holds
$d (\varphi \circ \psi^{-1})= (\dfrac{\partial x^i}{\partial y^1} \vert ... \vert \dfrac{\partial x^i}{\partial y^n})$
but I don't know how to calculate this.
I can write $d (\varphi \circ \psi^{-1})=d \varphi \cdot d \psi^{-1} = d \varphi \cdot (d \psi)^{-1} = (\dfrac{\partial x^i}{\partial x^1} \vert... \vert \dfrac{\partial x^i}{\partial x^n} ) \cdot (\dfrac{\partial y^i}{\partial y^1}\vert...\vert \dfrac{\partial y^i}{\partial y^n})^{-1}$ , correct?
But how to continue?
Assume that the two coordinate systems have a common open portion $U\subset M$, where they are both valid. In general this $U$ consists of "abstract" points $p$. (These points may have $N\gg n$ coordinates in some ambient carrying space.) In order to do analysis or differential geometry on $U$ (resp., $M$) we use various systems $$\phi: \>U\to {\mathbb R}^n,\quad p\mapsto x=(x_1,\ldots, x_n)$$ or $$\psi: \>U\to {\mathbb R}^n,\quad p\mapsto y=(y_1,\ldots, y_n)\ ,$$ whereby the $n$ denotes the dimension of the manifold $M$. When dealing with such systems you may assume that the function $$f:=\phi\circ\psi^{-1}:\qquad{\mathbb R}^n\to{\mathbb R}^n, \qquad y\mapsto x$$ with the precise domain and range $$f:\quad\psi(U)\to\phi(U), \qquad (y_1,\ldots, y_n)\mapsto(x_1,\ldots, x_n)$$ is given to you "at any time", i.e., as an $n$-tuple of expressions $$x_i=f_i(y_1,\ldots,y_n)\qquad(1\leq i\leq n)\ ,$$ which you may differentiate partially when needed.