(I'll use capital letters e.g. $D$ to denote operators/matrices and small letters e.g. $d$ to name functionals)
Let's say we have the derivative operator, $D$, of real functions ($=\int _{-\infty}^{\infty} \delta '(x-x') dx'$)
We can see this as a functional $d:f\rightarrow f$ defined by $d(f)=Df$
We want to calculate the functional derivative of this at each point $f$ of its domain. Since $d$ has vector valued input and vector valued output, its derivative $d'(f)$ will be attributing a Jacobian matrix $J$ at each $f$:
$$d'(f)=J_f$$
Now, at a point $f$ in the domain, we have, from the relation between the Jacobian matrix at a point and the change in the function's output when we change the input in the direction of the vector $g(x)$:
$$\lim_{\epsilon \rightarrow 0} \frac{d(f+\epsilon g)-df}{\epsilon}=J_f g$$
where $J_f g$ is the matrix multiplication of $J_f$ and $g$
This simplifies to:
$$Dg=J_fg$$
or $$J_f=D$$
So, we have $d'(f)=D$ as the functional derivative. $d'$ is a functional which assigns the operator $D$ at each $f$. Since this is a constant functional, all the subsequent functional derivatives are zero.
So we have this Taylor series of $d$ centered at $f$:
$$d(g)=d(f)+d'(f)(g-f)+0+....$$
$$=Df+D(g-f)$$ $$=Dg$$ $$=d(g)$$
This is like the Taylor series of a linear or polynomial function turning out to be itself.
Is this work correct? Are there any non trivial applications of this?
