I read in the first two chapters of Urs Graf's "Introduction to Hyperfunctions and Their Integral Transforms". On the page 114 of the book, The book well defined the product of hyperfunctions in two cases. 2.5.1 Product of Upper or Lower Hyperfunctions and 2.5.2 Products in the Case of Disjoint Singular Supports. However, the book does not mention the product rule for "Products in the Case of Disjoint Singular Supports" as follows:
$$D(f\cdot g)=Df\cdot g+f\cdot Dg$$
I think this situation should also be satisfied. But it's not mentioned in the book, and all the previous situations mentioned this point, which makes me a bit skeptical. Does all the well defined product of Hyperfunction satisfy the product rule?