Multivariate analogue of triple product rule

I was doing some thermodynamic calculations and I found the need for a multivariable analogue of the triple product rule. Basically I had a set of $m$ functions $z_i(y, x_1, x_2, ldots, x_m)$, and I needed to calculate partial derivatives of changes in $x$ resulting from a change in $y$ where all $z_i$ values are simultaneously held constant, i.e., I want to calculate:

$$ left(frac{partial x_1}{partial y}right)_{z_1, z_2, ldots, z_m}, ldots, left(frac{partial x_m}{partial y}right)_{z_1, z_2, ldots, z_m} . $$

As a reminder, the regular triple product rule would tell us the following, which unfortunately is not helpful in the multivariable case:

$$ left(frac{partial x}{partial y}right)_{z} = – frac{left(frac{partial z}{partial y}right)_{x}}{left(frac{partial z}{partial x}right)_{y}} qquad qquad text{when $m=1$}. $$

To compactify, let’s use linear algebra notation and write our problem as us knowing $mathbf z (y, mathbf x)$, and wanting to find:

$$ left(frac{partial mathbf x}{partial y}right)_{mathbf z} = ~ ??? $$

I believe I have the answer to this (which I will post as an Answer), but I am unsure as I cannot find it in standard places. This raises a follow-up question for me, which is whether this has a name?

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Example

I came across the need for the above rule when trying to work out heat capacity in the grand canonical ensemble, in the case of a multi-component system (multiple chemical potentials $mu_1, ldots, mu_m$ and multiple particle counts $N_1, ldots, N_m$). In this ensemble our natural variables are temperature $T$ and $boldsymbol mu$, but heat capacity is a constant-$mathbf N$ derivative. We have:

begin{align}
C_V & = T left(frac{partial S}{partial T}right)_{V, mathbf N}

& = T left(frac{partial S}{partial T}right)_{V, boldsymbol mu} + T { left(frac{partial S}{partial boldsymbol mu}right)_{V, T} }^T left(frac{partial boldsymbol mu}{partial T}right)_{V, mathbf N} qquad text {(using regular multidimensional chain rule)}
end{align}

So the question is then how to get $ left(frac{partial boldsymbol mu}{partial T}right)_{V, mathbf N} $, i.e., here $mathbf x = boldsymbol mu$ and $mathbf z = mathbf N $. Besides grand canonical ensemble, similar examples would appear if we are considering ensembles involving vectorial quantities, such as magnetization or angular momentum.