Calculate the adjoint of a derivative of the function

Question

If

$\partial_{x}\psi = \psi_{x} + \psi \partial_{x}$,

and when we apply $^{\dagger}$ on both sides, then

$(\partial_{x}\psi = \psi_{x} + \psi \partial_{x})^{\dagger},$

$-\psi^{\dagger}\partial_{x} = \psi_{x}^{\dagger}-\partial_{x}\psi^{\dagger}, \quad \partial_{x}^{\dagger}=-\partial_{x}.$

Is it right?