Following Shafarevich Reid - Basic algebraic geometry 1, the differential at $p$ of a polynomial $F(T_1,\dots,T_n)$ is the linear part of the Taylor expansion at $x$ (p. 87), so $$ (dF=)d_xF=\sum \frac{\partial F}{\partial T_i}(T_i-p_i).$$ This is an object of $\Theta_x^*$, the dual of the tangent space.
Later, on p. 190, it is defined as a 1-form something that is generated, at some open set $U$, by differentials of polynomials as $k[U]$-submodule of the module of maps from $x\in X$ to $\Theta_x^*$.
For example, in $\mathbb{A}^n$ with coordinates $y_1,\dots,y_n$, $$ d_p y_i = y_i - p_i$$ is a 1-form. We still have this if we consider the curve $y^2=x^3+x$. We can deduce that $ 2ydy= (3x^2+1)dx $ as a relation in $\Theta^*$: at any point $(p,q)$ in the curve we should have $$ 2q(y-q)=(3p^2+1)(x-p),$$ precisely the equation of the tangent space, so it is true in $\Theta_{(p,q)}^*$ (right?).
My question is in the projective cases and the relations between the coverings (I guess).
(i) $\mathbb{P}^1$ with the covering $\mathbb{A}^1_0\cup \mathbb{A}^1_1$ and variables $u,t$. We have the relation $u=t^{-1}$ in the intersection $\mathbb{A}^1_0\cap \mathbb{A}^1_1$. How do we end up with $du=-dt/t^2$? If this were true, then by definition $$ u-p = \frac{-(t-1/p)}{1/p^2} $$ at the point $[1:p]$, which seems not true.
(ii)Another example I am struggling with is the following relation $\phi=\psi=\chi$, how do we deduce that with the definition $d_{point}y=y-point_y$?
