Say I have a function of $n$ variables $F(x_{1}, x_{2}, x_{3},...,x_{n})$, where $x_{1} = g_{1}(y_{1}, y_{2}, y_{3},...,y_{m})$, $x_{2} = g_{2}(y_{1}, y_{2}, y_{3},...,y_{m}),\dots, x_{n} = g_{n}(y_{1}, y_{2}, y_{3},...,y_{m})$ are also some functions of $m$ variables. How can I find the expression for
$$\frac{\partial F}{\partial x_k}$$
in terms of the variables $y_i$. I found in another answer a formula for computing partial derivatives of $F$ with respect to $y_{i}, i=1,...,n $ using the formula:
$$\frac{\partial F}{\partial y_{i}} = \frac{\partial F}{\partial x_{1}}\frac{\partial x_{1}}{\partial y_{i}} + \frac{\partial F}{\partial x_{2}}\frac{\partial x_{2}}{\partial y_{i}} + \ldots + \frac{\partial F}{\partial x_{n}}\frac{\partial x_{n}}{\partial y_{i}} = \sum_{k=1}^{n} \frac{\partial F}{\partial x_{k}}\frac{\partial x_{k}}{\partial y_{i}}$$
which I think is just an application of the total derivative followed by the chain rule, ¿am I right? However, that's not what I'm looking for as you can see.
$\partial F/\partial x_k$ just means $$\lim_{h \to 0} \frac{F(x_1,\dotsc,x_{k-1},x_k+h,x_{k+1},\dotsc)-F(x_1,\dotsc,x_{k-1},x_k,x_{k+1},\dotsc)}{h},$$ if it exists, so $g$ and $y$ don't actually enter into it. It's not clear how to talk about this partial derivative in any other way.
(This is one of the problems with partial derivative notation if it's not clear what depends on what and have multiple "layers" of functions. You can write $ D_k F $ for "the derivative of $F$ with respect to its $k$th argument" to avoid this.)