Deceleration parameter from the Hubble derivative ($q=-1-\dot{H}/H^2$)

Differentiating $H=\dot{a}/a$ gives $\dot{H}=\ddot{a}/a-H^2$, which proves the displayed identity. A falling Hubble parameter does not by itself imply deceleration: linear growth $a\propto t$ has $H=1/t$ but $q=0$.