I'm studying the functional derivatives on Mahler's book "Statistical Multisource-Multitarget Information Fusion". I have some difficulties in understand a chain rule for functional derivatives.
Given a functional $F[h]$, Mahler defines the functional derivative of $F$ in a point $x\in\mathbb{R}^n$ as follows \begin{equation}\frac{\partial F}{\partial x}[h] \triangleq \lim_{\epsilon\to 0}\frac{F[h+\epsilon \delta_x]-F[h]}{\epsilon}\tag{1}\end{equation} where $\delta_x$ is the Dirac delta density concentrated at $x$. In the same, given a functional transformation $T[h]$, which is a test function resulting to the transformation of the starting test function $h$, he defines pointwise the functional derivative for $T$ as follows \begin{equation}\frac{\partial T}{\partial x}[h](w)\triangleq \lim_{\epsilon\to 0}\frac{T[h+\epsilon \delta_x](w)-T[h](w)}{\epsilon} \qquad \forall w \in \mathbb{R}^n\tag{2}\end{equation}
Given $(1)$, $(2)$, the chain rule that I don't understand is the following \begin{equation}\frac{\partial}{\partial x} F[T[h]]= \int \frac{\partial T}{\partial x}[h](w) \cdot \frac{\partial F}{\partial w} [T[h]] \text{ d}w\end{equation}
Question
I cannot see the difference between the notations $\frac{\partial}{\partial x} F[T[h]]$ and $\frac{\partial F}{\partial w} [T[h]]$. This two terms, I think, need to be different because if they are equal then the chain rule express a relation of the type \begin{equation}f(x)=\int g(x,w) f(w) \text{ d}w\end{equation} which it seems to me without sense (it looks like to be satisfied only for $f=0$)