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.

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