A cohomological interpretation of derivations on graded algebras

We trace derivations through Demazure's correspondence between a finitely generated positively graded normal $k$-algebras $A$ and normal projective $k$-varieties $X$ equipped with an ample $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor $D$. We obtain a generalized Euler sequence involving a sheaf on $X$ whose space of global sections consists of all homogeneous $k$-linear derivations of $A$ and a sheaf of logarithmic derivations on $X$.