Sorry. Final sentence should read: The second of these cancels your Term 1 the third your Term 2 and the first would cancel your Term 5 if you had the correct relation in the Weyl algebra namely [d,x]=1[d,x]=1 rather than [x,d]=1[x,d]=1.

I apologise for being scary; it is not intentional. Terms 3 and 4 should be $-((dx)(f))(m)=(-d(xf))(m)=-d((xf)(m))+(xf)(dm)$ and this is $-f(m)-xd(f(m))+(xf)(dm)$. The second of these cancels your Term 1 the third your Term 2 and the first would cancel your Term 1 if you had the correct relation in the Weyl algebra namely $[d,x]=1$ rather than $[x,d]=1$.

Go to under what operation? You now seem to be asking for much more than your original question.

Doesn't the claim in the first paragraph depend on the formula. The reference arxiv.org/abs/dg-ga/9702008 claims that $(\frac{d}{dx}(f))(m)=\frac{d}{dx}(f(m)) - f(\frac{d}{dx}m)$ works. Proposition 1.2.9 of this book math.columbia.edu/~scautis/dmodules/hottaetal.pdf agrees

There are lots of questions here. What exactly do you mean by the $\mathcal{O}_X$-dual of a left $\mathcal{D}_X$-module $\mathcal{M}$? Do you mean the left $\mathcal{D}_X$-module $\mathrm{Hom}_{\mathcal{O}_X}(\mathcal{M},\mathcal{O}_X)$? What do you mean by 'construct the category of $D$-modules in a canonical way? What is non-canonical about the usual construction? I think you must mean more by 'construct the category' than I understand.

Why does it seem that the problem can be reduced? Have you read it somewhere? If so, where?

Thanks again. In fact this has made me realise that my question wasn't quite stated correctly. I really wanted to assume $Y$ is smooth rather than $X$. I think this is what I need for my claim that $\iota^\ast \mathcal{T}_X\to \mathcal{N}_{Y/X}$ is surjective. I'll edit the question. If you want to make appropriate changes/delete comments about smoothness then I'll do the same.