# Derivative with respect to vector of product of two functions of the vector

I am struggling with the following derivative. Let $$\pmb{x} \in \mathbb{R}^{n}$$ be a vector, $$\pmb{y} \in \mathbb{R}^{m}$$ another vector that is a function of $$\pmb{x}$$, and $$\pmb{g}$$ and $$\pmb{h}$$ two functions returning vectors. I aim to obtain the following derivative:

$$\frac{\partial}{\partial \pmb{x}} \left( \pmb{g}(\pmb{y})^{\top} \otimes \pmb{I} \right)\pmb{h}(\pmb{x}).$$ My thoughts were that this would involve a combination of the sum and chain rule. E.g., first take the sum-rule with the left and right-hand part. But then $$\left( \pmb{g}(\pmb{y})^{\top} \otimes \pmb{I} \right)$$ takes the form of a matrix and I am quite lost. I expect that taking a differential form is needed, but I don't really understand how to do these properly yet.

## 1 Answer

Define some new matrices \eqalign{ H &= {\rm Mat}(h) \implies h = {\rm vec}(H) \cr J &= \frac{\partial h}{\partial x},\quad K = \frac{\partial g}{\partial y},\quad L = \frac{\partial y}{\partial x} \cr } Write the vector function and find its differential and gradient in terms of these new variables. \eqalign{ w &= (g^T\otimes I)h = Hg \cr dw &= (g^T\otimes I)\,dh + H\,dg \cr &= \Big((g^T\otimes I)J + HKL\Big)\,dx \cr \frac{\partial w}{\partial x} &= (g^T\otimes I)J + HKL \cr } Knowing nothing about the nature of the vector functions $$(g,h,y)$$ this is as far as we can go.

• Thank you so much for this answer and other answers you have given! I am writing a paper on maximum likelihood estimation on some multivariate models and this has all been really helpful for my work! Commented Feb 17, 2019 at 20:02