What is $\frac{dXX^T}{dX}$?

Question

Given $X \in \mathbb{R}^{n \times r}$, what is $$\dfrac{dXX^T}{dX}?$$ I'm aware it is a order 4 tensor.

Answer 1

$(X+H)(X+H)^{T}=XX^{T}+\underbrace{XH^{T}+HX^{T}}_{\text{first order terms}}+HH^{T}$

Answer 2

A pair of 4th order tensors arise naturally from matrix-matrix derivatives $$\eqalign{ \omega &= \frac {\partial{{\mathbf{X^T}}}} {\partial{{\mathbf{X}}}},\,\,\,\omega_{ijkl} = \delta_{il} \delta_{jk} \cr \varepsilon &= \frac {\partial{{\mathbf{X}}}} {\partial{{\mathbf{X}}}},\,\,\,\varepsilon_{ijkl} = \delta_{ik} \delta_{jl} \cr } $$ Using these, you can write the derivative as
$$\eqalign{ {\mathbf{\frac {\partial(X\cdot X^T)} {\partial X} }} &= {\mathbf{X\cdot\omega + \varepsilon\cdot X^T}} \cr } $$