How to make sense of quantum fields differential equations? 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?
 A: The fields of a QFT are not functions of the spatial coordinates $\boldsymbol x\in\mathbb R^n$, but operator-valued distributions (borrowing Wightman's terminology). The notion of the fields being functions of time ("sharp-time fields") can be kept in general, but their dependence on $\boldsymbol x$ is "too singular", so that they become distributions; fields need be smeared out in space.
Therefore we should write $\phi[f]$ instead of $\phi(\boldsymbol x)$, in the same way we should write $\delta[f]$ for the Dirac delta. In this sense, the commutation relations
$$
[\phi(\boldsymbol x),\pi(\boldsymbol y)]=\delta(\boldsymbol x-\boldsymbol y)
$$
should be written as
$$
[\phi(f),\pi(g)]=(f,g)
$$
for a certain scalar product $(\cdot,\cdot)$ on your space of test functions (that depends on the spin of $\phi$).
Similarly, the field equations
$$
\dot\pi(\boldsymbol x)-\Delta \phi(\boldsymbol x)+m^2\phi(\boldsymbol x)=0
$$
should actually be written as
$$
\dot\pi[f]-\phi[\Delta f]+m^2\phi[f]=0
$$
More generally, the naïve field equations
$$
\mathscr D\phi(x)=0
$$
are nothing but a short-hand notation for
$$
\phi[\mathscr Df]=0
$$
for all $f$ in the domain of $\phi$. Therefore, the derivatives acting on fields are to be understood in the sense of distributional derivatives (if $T$ is a distribution, then we define $T'[f]\equiv -T[f']$, etc.).
For free theories, the whole framework of operator-valued distributions is perfectly well understood, and one may work with all mathematical rigour one may wish. For interacting theories though, we are far from a mathematically sound theory.
For more details see for example On Relativistic Irreducible Quantum Fields Fulfilling CCR or Haag's theorem in renormalised quantum field theories. Also, anything by Wightman (e.g., PCT, Spin and Statistics, and All That).
