The low, euclidean momentum behavior of the hadron vacuum polarization (HVP) is critical for determining, amongst other quantities, the anomalous magnetic moments of the muon. Here we present lattice QCD results for the first two derivatives of the HVP function at vanishing virtuality [1]. Computations are performed with 2+1+1 flavors of staggered quarks around the physical mass point, in volumes of linear extent larger than 6 fm, and at six values of the lattice spacing, allowing for a fully controlled continuum extrapolation.. We further consider possible uncertainties which stem from finite-volume and isospin-breaking effects. After adding to our connected contributions the disconnected terms presented in [2], we compare the resulting derivatives of the full HVP with phenomenological estimates.