Partial with respect to a function

Question

I suspect this is either an elementary issue I have forgotten how to address since my long ago undergraduate vector calculus, or a nonsensical question. I apologize in advance either way.

There is a well explored model where the literature is full of partials with respect to a variable $$\frac{\partial}{\partial x_1} \vec{f}(\vec{x})$$ but in my reformulation of this model $\vec{x}$ is a function of other variables $\vec{y}$ so I have $$\frac{\partial}{\partial x_1} \vec{g}\big(\vec{x}(\vec{y}),\vec{y}\big).$$ How do you take a partial with respect to a function?

EDIT:

Since each $\frac{\partial x_i}{\partial y_j}$ is well defined, is this as simple as abusing the chain rule, $$\frac{\partial g}{\partial x} =\frac{\partial g}{\partial y}\frac{\partial y}{\partial x} =\frac{\partial g}{\partial y}\frac{1}{\frac{\partial x}{\partial y}}$$ ?

Answer 1

You don't. In the usual notation, $\frac\partial{\partial x_i}g$ just means taking the derivative of $g$ with respect its $i$-th entry. With this in mind, $\frac\partial{\partial x_i}g(\vec x(\vec y),\vec y)$ means evaluating the function $\frac\partial{\partial x_i}g$ at the point $(\vec x(\vec y),\vec y)$.

Edited:

Say $g$ is a function $g:\mathbb R^n\times\mathbb R^m\to \mathbb R$. Then $\frac{\partial g}{\partial x_i}$ (if it exists) stands for the function $\mathbb R^n\times\mathbb R^m\to \mathbb R$ defined by $$\frac{\partial g}{\partial x_1}(a,b)=\lim_{h\to 0}\frac{g(a+he_1,b)-g(a,b)}{h}.$$ for every $a\in\mathbb R^n$, $b\in\mathbb R^m$. Then you can plug in the point $(a,b)=(\vec x(\vec y),\vec y)$.