I'm reading some notes on hodge theory by Charles Siegel which makes a claim on page 16 relating the space of deformations of a smooth projective hypersurface $X$ with the jacobian ideal. More specifically, let $$ Proj(\mathbb{C}[x_1,\ldots, x_n]/(f)) = Proj(S_\bullet) = X $$ then $$ H^1(X,T_X) = \frac{S_d}{\text{Jac}(f)_d} $$ where can I find a proof of this and are there any generalizations?