Suppose I have a function that takes from sets $X \subset \mathbb{R}$, $Y \subset\mathbb{R}$ to $\mathbb{R}$
$$d: X \times Y \to \mathbb{R}$$
The elements are identified by $d(x,y) = xy \in \mathbb{R}$
Now I make a change, and treat $x,y$ as quantities that depends on time $t$
So $d(x(t),y(t)) = x(t)y(t)$
How should change the original definition of the function $$d: X \times Y \to \mathbb{R}?$$
I am thinking of something like $$d: (X \times \mathbb{R}) \times (Y \times \mathbb{R}) \to \mathbb{R}$$
Update: Case 2: Suppose instead I have:
$X \subset \mathbb{R}^n$, $Y \subset\mathbb{R}^n$ to $\mathbb{R}$
$$d: X \times Y \to \mathbb{R}$$
The elements are identified by $d(x,y) = x^Ty \in \mathbb{R}$
Similarly, I make a change and treat $x,y$ as quantities that depends on time $t$
So $d(x(t),y(t)) = x(t)^Ty(t)$
How should change the original definition of the function $d$?
The whole thing actually becomes a function $d':\Bbb R\to \Bbb R$ (don't call two different functions $d$, just don't). The resoning is simple: You input a real number ($t$), and the function outputs a real number (the value of $d$ at $(x(t),y(t))$).
If you want to retain the information that this function actually goes via $X\times Y$, then you do that by writing it as a composition: $$ \Bbb R\overset{x\,\times\, y}{\longrightarrow}X\times Y\overset{d}{\longrightarrow}\Bbb R $$