I am looking for a correct way to find a derivative of a scalar function over multiple vectors. Let
$$\begin{aligned} f : \mathbb{R}^m \times \mathbb{R}^n &\to \mathbb{R}, \\ (\vec{x}, \vec{y}) &\mapsto c\end{aligned}$$
Is there any way to formalize partial and exact derivatives on this function or functions of several vectors?
I've searched around, and I'm not even sure if there is a mathematical way to denote functions of this type in a clear, concise way.
Would a general matrix derivative be appropriate here in the event that $m \ne n$?