In II.1.3 of Shafarevich's book Basic Algebraic Geometry 1, in page 90, it is written the following:
However, I understand that the differential is given by a precomposition, so I think that it should be something similar to $$\psi(x_i)=\psi(\xi_i/\xi_0)=(d_v\pi)(\phi)(\xi_i/\xi_0)=\phi((\xi_i/\xi_0)\circ\pi)=\phi(\xi_i/\xi_0).$$ But this is probably wrong. Can someone explain in basic terms why Shafarevich gives $\psi$ in that way? Why does appear the derivative of $\xi_i/\xi_0$ but multiplied by other terms?