Let $A: H^s(\Omega) \to H^1(\Omega)$ be a bounded linear map. Let $u \in H^s(\Omega)$. Let $f:\mathbb{R} \to \mathbb{R}$ be nonlinear and Lipschitz such that $f(u) \in H^s(\Omega)$.

Is it possible to write the weak gradient $$\nabla(A(f(u))$$ in terms of the derivative of $f$, the (possibly Frechet) derivative of $A$ and (possibly?) $\nabla (Au)$? Basically I am looking for a chain rule for such kinds of expressions but preferably with a gradient term which is linear in $u$.

Motivation: the motivation is that this gradient term comes up in a PDE, and I wish to linearise the PDE and look for a fixed point. So to take another example, instead of considering $\nabla g(u)$ I would write it as $g'(u)\nabla u$, then linearise: $g'(v)\nabla u$ then look for a fixed point of the map $v \mapsto u$ where $u$ is the solution of an equation with the coefficient $g'(v)$.

This is related to Nemytskii maps but it is not quite the same.