  In the 2022 study, together with Paul McFadden and Kostas Skenderis, I
analyzed tree-level 3- and 4-point Witten diagrams (amplitudes) of scalar
operators in anti-de Sitter space in momentum space. This paper constitutes its
extension to Witten diagrams with bulk interactions involving spacetime
derivatives. In $d = 3$ boundary dimensions the Witten diagrams involving
conformally coupled and massless scalars can be evaluated in closed form. Such
cases are of interest in holographic cosmology and correspond to dual operators
of conformal dimensions $\Delta = 2$ and $3$ respectively. I present explicit
formulae for all such amplitudes and provide a Mathematica package serving as
the repository of all the results. I discuss renormalization issues and show
that, contrary to the expectation, even finite correlators may acquire
non-trivial renormalization effects.
