  We present lattice-QCD calculations of the hadronic form factors for the
semileptonic decays $D\to\pi\ell\nu$, $D\to K\ell\nu$, and $D_s\to K\ell\nu$.
Our calculation uses the highly improved staggered quark (HISQ) action for all
valence and sea quarks and includes $N_f=2+1+1$ MILC ensembles with lattice
spacings ranging from $a\approx0.12$ fm down to $0.042$ fm. At most lattice
spacings, an ensemble with physical-mass light quarks is included. The HISQ
action allows all the quarks to be treated with the same relativistic
light-quark action, allowing for nonperturbative renormalization using partial
conservation of the vector current. We combine our results with experimental
measurements of the differential decay rates to determine
$|V_{cd}|^{D\to\pi}=0.2238(11)^{\rm Expt}(15)^{\rm QCD}(04)^{\rm EW}(02)^{\rm
SIB}[22]^{\rm QED}$ and $|V_{cs}|^{D\to K}=0.9589(23)^{\rm Expt}(40)^{\rm
QCD}(15)^{\rm EW}(05)^{\rm SIB}[95]^{\rm QED}$ This result for $|V_{cd}|$ is
the most precise to date, with a lattice-QCD error that is, for the first time
for the semileptonic extraction, at the same level as the experimental error.
Using recent measurements from BES III, we also give the first-ever
determination of $|V_{cd}|^{D_s\to K}=0.258(15)^{\rm Expt}(01)^{\rm
QCD}[03]^{\rm QED}$ from $D_s\to K \ell\nu$. Our results also furnish new
Standard Model calculations of the lepton flavor universality ratios
$R^{D\to\pi}=0.98671(17)^{\rm QCD}[500]^{\rm QED}$, $R^{D\to
K}=0.97606(16)^{\rm QCD}[500]^{\rm QED}$, and $R^{D_s\to K}=0.98099(10)^{\rm
QCD}[500]^{\rm QED}$, which are consistent within $2\sigma$ with experimental
measurements. Our extractions of $|V_{cd}|$ and $|V_{cs}|$, when combined with
a value for $|V_{cb}|$, provide the most precise test of second-row CKM
unitarity, finding agreement with unitarity at the level of one standard
deviation.
