  We derive the scaling of differential rotation in both slowly- and
rapidly-rotating convection zones using order of magnitude methods. Our
calculations apply across stars and fluid planets and all rotation rates, as
well as to both magnetized and purely hydrodynamic systems. We find shear
$|R\nabla\Omega|$ of order the angular frequency $\Omega$ for slowly-rotating
systems with $\Omega \ll |N|$, where $N$ is the \brvs\ frequency, and find that
it declines as a power-law in $\Omega$ for rapidly-rotating systems with
$\Omega \gg |N|$. We further calculate the meridional circulation rate and
baroclinicity and examine the magnetic field strength in the rapidly rotating
limit. Our results are in general agreement with simulations and observations
and we perform a detailed comparison with those in a companion paper.
