Derivative of a Fock state

Let us suppose we have the operators $a(omega)$ and $a^dagger(omega)$ with $[a(omega),a^dagger (omega’)] = delta(omega-omega’)$. Does a derivative of the Fock state $a^dagger(omega) |0rangle$ with respect to $omega$ exist, that is $partial_{omega} a^dagger(omega) |0rangle $. Or more generally, do these expressions make senes
begin{equation}
partial_{omega} a^dagger(omega)
partial_{omega} a(omega)
end{equation}
Intuitively, I would guess that the derivative does not exist because of the orthogonality of Fock states for different $omega$. But maybe the derivative can be expressed in terms of a delta function.

My idea to find an expression for the derivative is the following. First let us rewrite the annihilation operator as
begin{equation}
a(omega) = int domega’ delta (omega’ – omega) a(omega’).
end{equation}
Differentiation of the above expression with $partial_{omega}delta (omega’ – omega) = -delta’ (omega’-omega) = frac{1}{omega’-omega} delta (omega’ – omega)$ yields
begin{equation}
partial_{omega}a(omega) = int domega’ partial_{omega}delta (omega’ – omega) a(omega’) = frac{1}{omega-omega} a(omega) .
end{equation}
So one could argue that the derivative does not exist or that is equals $infty$, is that correct?