A quantum field is an operator valued function, that is, a function $\varphi(x)$ defined on spacetime which assigns operators on a Hilbert space to each event $x$. In a more rigorous approach a quantum field could be defined as an operator valued distribution on spacetime.
Anyway, it is quite common that these quantum fields obey differential equations, like the Klein-Gordon equation $$(\Box +m^2)\varphi=0$$ and Dirac's equation $$(i\gamma^\mu \partial_\mu - m)\psi=0.$$
In that sense we need to understand what is the derivative of a quantum field. In this seems a little complicated.
Of course one can say: "a quantum field takes values in a Hilbert space so you can use the Frechet derivative", but it is not even clear what Hilbert space it is that the quantum fields takes values on. Also, as is clear from Quantum Mechanics, most of the operators we deal in QM are unbounded and hence discontinuous. I believe this would have a great impact on how should we deal with things like derivatives.
So, in order for quantum fields to satisfy differential equations, how is the correct way to define and understand the derivative of a quantum field? How can we make sense of quantum field differential equations?