  The determination of $\vert V_{us}\vert$ from kaon semileptonic decays
requires the value of the form factor $f_+(q^2=0)$, which can be calculated
precisely on the lattice. We provide the one-loop partially quenched staggered
chiral perturbation theory expressions that may be employed to analyze
staggered simulations of $f_+(q^2)$ with three light flavors. We consider both
the case of a mixed action, where the valence and sea sectors have different
staggered actions, and the standard case where these actions are the same. The
momentum transfer $q^2$ of the form factor is allowed to have an arbitrary
value. We give results for the generic situation where the $u$, $d$, and $s$
quark masses are all different, $N_f=1+1+1$, and for the isospin limit,
$N_f=2+1$. The expression we obtain for $f_+(q^2)$ is independent of the mass
of the (valence) spectator quark. In the limit of vanishing lattice spacing,
our results reduce to the one-loop continuum partially quenched expression for
$f_+(q^2)$, which has not previously been reported in the literature for the
$N_f=1+1+1$ case. Our expressions have already been used in staggered lattice
analyses of $f_+(0)$, and should prove useful in future calculations as well.
