 Solution

At a fixed cosmological time, homogeneous expansion has $\mathbf{v}=H(t)\mathbf{r}$. A comoving galaxy has $\mathbf{r}=R(t)\mathbf{x}$ with constant $\mathbf{x}$, hence $\mathbf{v}=\dot{R}\mathbf{x}=(\dot{R}/R)\mathbf{r}$. Thus $\boxed{H=\dot{R}/R}$. Peculiar velocities must be removed before applying this ideal relation to individual galaxies.