Differentiability of inner product involving Bochner functions

Question

Suppose we are given $f\in L^2(0, T; H^1)$ and that there exists $\xi\in L^2(0, T; H^1)$ such that, for any $g\in L^2(0,T; H^1)$ with $\dot{g}\in L^2(0, T; H^1)$, it holds that $$t\mapsto (\nabla f(t), \nabla g(t))_{L^2}$$ is differentiable (in the usual week sense) with $$\dfrac{d}{dt} (\nabla f, \nabla g)_{L^2} = (\nabla \xi, \nabla g)_{L^2} + (\nabla f, \nabla \dot{g})_{L^2}.$$ Is it possible to conclude that then $f$ must have a weak time derivative and $\dot{f}=\xi$ (up to a constant, maybe)? I have been working on a similar problem to this for some time and have not been able to make progress. I can do it if I replace the $L^2$ inner product of the gradients by the $L^2$ inner product of the functions. Any idea is appreciated!