Let $g:\mathbb R_+ \times \mathbb R^d \to \mathbb R$ be differentiable. Fix $t >0$. We define a map $u:(0, t) \times \mathbb R^d \to \mathbb R$ by $u(s, x) := g(t-s, x)$. The notation $\partial_s u (s, x)$ clearly means the partial derivative of $u$ w.r.t. time variable at $(s, x)$.
Could you suggest a non-ambiguous but not cumbersome notation for the partial derivative of $g$ w.r.t. time coordinate at $(t-s, x)$?
I found that Leibniz's notation is very convenient in this case, i.e., we denote the partial derivative of $g$ w.r.t. time variable at $(t-s, x)$ by $$ \frac{\partial}{\partial h} g(h, x) \bigg |_{h=t-s}. $$