Consider a function $f\in C^{\infty}(\mathbb{R},H^{s}(\mathbb{R}^{d}))$, i.e. a function of two variables $f(t,x)$ such that for any fixed $t\in\mathbb{R}$ it holds that $f(t,\cdot)\in H^{s}(\mathbb{R}^{d})$ and such that $f(t,\cdot)$ depends smoothly on time (w.r.t. the standard Hilbert space topology on $H^{s}$).
What can we say about $\partial_{t}f$?
More precisely, my question is whether $\partial_{t}f\in C^{\infty}(\mathbb{R},H^{s}(\mathbb{R}^{d}))$, or if loose some regularity, i.e. $\partial_{t}f\in C^{\infty}(\mathbb{R},H^{s-1}(\mathbb{R}^{d}))$? Any ideas or reference about this?