Let $d\in\mathbb N$, $U\subseteq\mathbb R^d$ be open, $\tau>0$ and $T_t$ be a $C^1$-diffeomorphism from $U$ onto $U$ for $t\in[0,\tau)$ with $T_0=\operatorname{id}_U$.
Which assumption do we need to impose in order to conclude that $$[0,\tau)\ni t\mapsto{\rm D}T_t(x)\tag1$$ is $C^1$-differentiable at $0$ and $$\left.\frac\partial{\partial t}{\rm D}T_t(x)\right|_{t=0}={\rm D}\left(\left.\frac\partial{\partial t}T_t\right|_{t=0}\right)(x)\tag2$$ for all $x\in U$?
It clearly holds if $$[0,\tau)\times U\ni(t,x)\mapsto T_t(x)\tag3$$ is $C^2$-differentiable at $(0,x)$ for all $x\in U$ by Schwarz' theorem.