I'm trying to find a presentation of the ring of differential operators for a smooth affine and smooth projective variety. For example, consider the varieties \begin{align*} X = \textbf{Spec}\left( R = \frac{\mathbb{C}[x,y]}{x^2 + y^2 - 1}\right) && Y = \textbf{Proj}\left( S = \frac{\mathbb{C}[x,y,z]}{x^4 + y^4 + z^4}\right) \end{align*} In the first case, I can find the module of vector fields by computing the kahler differentials, and then computing the dual $$ T_X = \text{Hom}_R\left(\Omega_{R/\mathbb{C}},R\right) = \text{Hom}_R\left(\frac{Rdx\oplus Rdy}{xdx + ydy},R\right) = \frac{R\partial_x\oplus R\partial_y}{x\partial_x + y\partial_y} $$ and in the second case I can dualize the conormal sequence on Macaulay2.
How can I find a presentation for $D_X$ and $D_Y$? Originally I guessed that $D_X$ could be presented as $$ \frac{R[\partial_x,\partial_y]}{(x\partial_x + y\partial_y)} $$ but the operator on the bottom does not preserve the ideal.