Laplacian cylindrical coordinates

[LagrangeEuler]

Laplacian in cylindrical coordinates is defined by

$$\Delta=\frac{\partial^2}{\partial \rho^2}+\frac{1}{\rho}\frac{\partial}{\partial \rho}+\frac{1}{\rho^2}\frac{\partial^2}{\partial \varphi^2}+\frac{\partial^2}{\partial z^2}$$
I am confused. I I have spherical symmetric function f(r) then
$$\Delta f(r)=\frac{d^2}{dr^2}f(r)+\frac{2}{r}\frac{d}{dr}f(r)$$
If I worked on function $f(r)$ with Laplacian in cylindrical coordinates. I suppose that $$f(r)=f(\rho)$$ but then factor $2$ is problem.

 

[Delta2]

I think it is $f(r)=f(\sqrt{\rho^2+z^2})$.

 

[LagrangeEuler]

Of course. Thanks.