Cotangent Euler sequence in homogeneous coordinates ($0\to {\Omega }_{{\mathbb{P}}^n}^1\to \mathcal{O}(-1{)}^{\oplus (n+1)}\to \mathcal{O}\to 0$)

For a degree-zero rational function regular on an open subset of projective space, each derivative $\partial f/\partial X_i$ is a regular homogeneous rational function of degree $-1$. These derivatives define a derivation with values in $\mathcal{O}(-1{)}^{\oplus (n+1)}$, hence a map from the Kähler differential sheaf. The last map is $(g_i)\mapsto {\sum }_i{X}_i{g}_i$. On $X_i\ne 0$, put $y_j=X_j/X_i$ and use frame $X_i^{-1}$. The derivative of $y_j$ has coordinate vector with $1$ at $j$, $-y_j$ at $i$ and zero elsewhere. Those vectors form a free basis of the kernel of $(b_i)\mapsto b_i+{\sum }_{j\ne i}{y}_j{b}_j$, proving exactness directly in every characteristic.

Dualize the Euler sequence on complex projective space and tensor by the hyperplane line bundle to obtain $0\to {\Omega }^1(1)\to {\mathcal{O}}^{n+1}\to \mathcal{O}(1)\to 0$. On global sections the last map sends $(c_i)$ to ${\sum }_i{c}_i{X}_i$, an isomorphism onto the linear forms. Left exactness of sheaf cohomology therefore gives the asserted vanishing. No higher-cohomology assumption is needed for this final kernel calculation.