Partial Derivatives and Operator Commutivity

Question

I have an operator $$L\psi=\frac{1}{r^2}\partial_z^2\psi+\frac{1}{r}\partial_r(\frac{1}{r}\partial_r\psi)$$

I am interested in taking $\partial_rL\psi$ and $\partial_zL\psi$. Do the partial derivatives and the $L$ operator commute? For instance is $\partial_aL\psi$ (where $a$ represents both $r, z$ as well as any other dependent variable of $\psi$) equivalent to $L\partial_a\psi$ just as $\partial_a\partial_b\psi=\partial_b\partial_a\psi$?

Answer 1

See @Dr.MV 's answer above in his comment.

$\partial_zL\psi=L\partial_z\psi$ but $\partial_rL\psi\ne L\partial_r\psi$ because of the $1/r$ in L.