If $u$ has a weak derivative and $f$ is $C^1$ does $fu$ have a weak derivative (fractional Sobolev space and weak time derivatives)

Question

Let $\Omega$ be an open bounded set.

Let $s \in (0,1)$ and $H^s(\Omega) := W^{s,2}(\Omega).$ Let $f \in C^1([0,T]\times \Omega)$ and $u \in L^2(0,T;H^s(\Omega))$ with weak derivative $u' \in L^2(0,T;H^{-s}(\Omega))$. We also have that $f$ and $\nabla f$ (the spatial gradient) are uniformly bounded in time and space.

Does it follow that $(fu)'$ exists and $(fu)' \in L^2(0,T;H^{-s}(\Omega))$?

Any ideas? Maybe there is a reference for this already.