Is there an equivalent of partial differentiation for functions taking multiple vectors as input?
I mean the following.
If we have a function $f(x,y)$, then a partial derivative is denoted as $\frac{\partial}{\partial x}f(x,y)$. So far so good.
But what if we have a function $f(\mathbf{x},\mathbf{y})$ (or written with scalar variables $f(x_1, x_2,y_1, y_2)$) and we want to get a vector $\Big(\frac{\partial}{\partial x_1}f(x_1, x_2,y_1, y_2), \frac{\partial}{\partial x_2}f(x_1, x_2,y_1, y_2)\Big)^\top$?
Can I write $\frac{\partial}{\partial \mathbf{x}}f(\mathbf{x},\mathbf{y})$, or maybe $\nabla_\mathbf{x} f(\mathbf{x},\mathbf{y}) $? Is there some standard notation or will I have to make up one?