About Divergence in polar coordinates

I’ve got a conductor in a cylinder shape that is rotating with angular velocity $omega$ around its axis, that correspond to the $z$ axis

I want to calculate the electric field and the density of electric charge.

I have no problem finding $$E = frac{m_e omega^2r}{|q_e|} u_r$$ where $m_e$ and $q_e$ are the mass and electric charge of the electron, and $u_r$ is the versor directed towards the center of the circle (and is thereofore perpendicular to the $z$ axis)

(I know that I had to use several approximation here, which are pretty standard to be honest, but that’s not the point as the result appear to be right according to the textbook)

To determin now the density of eletric charge, I would use maxwell’s equation

$$nabla cdot E = frac{rho}{epsilon_0}$$

$E$ is expressed in cylindrical coordinates, though (am I right?)

So the divergence would be

$$nabla cdot E = frac{1}{r} frac{partial rE}{partial r} = frac{2m_e omega^2}{|q_e|} = frac{rho}{epsilon_0}$$

On the book there is written though that

for a vector field that depends solely upon $r$ directed toward the
centre of the circle, the divergence is just $$nabla cdot E =
frac{dE}{dr} = frac{m_e omega^2}{|q_e|}$$

I don’t understand what he means and why I’m wrong.