Determinant of partial derivatives Suppose I have a square matrix of variables $(x_{ij})_{1 \le i,j \le n}.$ I define the operator $$\frac{d}{dX} = \mathrm{det}\Big(\frac{\partial}{\partial x_{ij}} \Big)_{1 \le i,j \le n}.$$
What is $$\frac{d}{dX} \mathrm{det}(X) \; ?$$ More generally, for real numbers $a$, what should the power rule $$\frac{d}{dX} \mathrm{det}(X)^a = ?$$ look like?
For example, when $n = 2$, we get \begin{align*} &\quad \frac{d}{dX} \mathrm{det}(X)^a \\ &= \Big( \frac{\partial^2}{\partial x_{11} \partial x_{22}} - \frac{\partial^2}{\partial x_{12} \partial x_{21}} \Big) \Big[ (x_{11} x_{22} - x_{12} x_{21})^a \Big] \\ &= \frac{\partial}{\partial x_{11}} \Big[ a x_{11} (x_{11} x_{22} - x_{12} x_{21})^{a-1} \Big] + \frac{\partial}{\partial x_{12}} \Big[a x_{12} (x_{11} x_{22} - x_{12} x_{21})^{a-1} \Big] \\ &= 2a (x_{11} x_{22} - x_{12} x_{21})^{a-1} + a (a-1) (x_{11} x_{22} - x_{12} x_{21})^{a-1} \\ &= (a+1) a \mathrm{det}(X)^{a-1}.\end{align*}
I expect the general rule to be $\frac{d}{dX} \mathrm{det}(X)^a = a(a+1)...(a+n-1) \mathrm{det}(X)^{a-1}$ but I am not sure how to prove it.
 A: This is called the Cayley identity (found by Vivanti in 1890). Here are some references:


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*Theorem 1 in Markus Fulmek, A combinatorial proof for Cayley's identity, arXiv:1309.6801v1. This gives a combinatorial proof using matchings; I cannot vouch for its readability (I have not tried reading it; in general, proofs like this can get very messy).

*The first claim in Theorem 2.1 in Sergio Caracciolo, Alan D. Sokal, Andrea Sportiello, Algebraic/combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians, arXiv:1105.6270v2. This is essentially a monograph devoted to this identity and its variants (yes, there are Cayley identities in other types), as well as various ways of proving them. There is some exciting stuff in here, but it's probably overkill if you just want to know the proof of the Cayley identity.

*Corollary A.4 in Sergio Caracciolo, Andrea Sportiello, Alan D. Sokal, Noncommutative determinants, Cauchy-Binet formulae, and Capelli-type identities. I. Generalizations of the Capelli and Turnbull identities, arXiv:0809.3516v2. This is a source I can highly recommend; I have written a list of errata for the paper (and in that list, I prove a stronger result -- Theorem A.10).
This is all related to various subjects: D-module theory (the Cayley identity is an example of a Bernstein-Sato polynomial), invariant theory, the Capelli identities (see the third reference I gave), etc.
