Handbook of derivative AdS amplitudes

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.