Shouldn't the chain rule be applied instead of the product rule in linear operators? If I have the linear operators $A(x)$ and $B(x)$, and i want to calculate the derivative of $A(B(x))$, shouldn't it be:
$$A(B(x))'= A'(B(x))B'(x)$$
Instead of $(AB(x))'= A'(x)B(x) + A(x)B'(x)$ ?
This may not make much sense, but if you think about this linear operators as matrices, when we have $AB\vec{x} $, should't we undestand it as "$A(B(x))$", instead of "$A(x)B(x)$"? After all A is acting on $x$ transformed by "B", not in $x$ itself.
I am making this stupid question because I am trying to study QM through Functional analysis and operator theory, but when I came to this example I thought about something that I hadn't thought before and now I am in doubt, sorry for the silly question.
It came from this check @littleO :
"Let $A(x)$ be an operator (It probably means a linear operator,but you know how physicists are!) dependant on a continuous variable x, define its variable by:
$$\frac{dA(x)}{dx} \equiv  A'(x) = \lim_{\varepsilon \rightarrow 0} \frac{A(x+\varepsilon)-A(x)}{\varepsilon}$$
If $A$ has an inverse show that
$$\frac{dA^{-1}}{dx}= - A^{-1}A'A^{-1}$$
Show also that
$$\frac{dAB}{dx}= A'B+ AB'$$ "
 A: It depends on what you are taking the derivative with respect to!
Suppose you have two different operators $A_t$ and $B_t$, which depend on time.  At each moment, each operator is a function from your Hilbert space to itself, and so has a derivative with respect to the quantum state.  In that case, you do apply the chain rule: $$\frac{\partial}{\partial x}(A_tB_tx)=\frac{\partial A_t}{\partial x}(B_tx)\frac{\partial B_t}{\partial x}(x)$$
On the other hand, since $\{A_t\}_t$ and $\{B_t\}_t$ are each a function from time to the set of operators on your Hilbert space, it too has a derivative with respect to time.  Since $x$ is constant in time and composition coincides with multiplication for linear operators, we can apply the product rule: $$\frac{\partial}{\partial t}(A_tB_tx)=\left(\frac{\partial A_t}{\partial t}B_t+A_t\frac{\partial B_t}{\partial t}\right)x$$
(I've ignored issues with whether these derivatives exist and are bounded, in what sense they exist (Gateau/Frechet), how to handle if x might vary in time too, etc.  But you mentioned you came to this from physics, so you probably don't need to know/can guess how to handle those.)
