TITLE: What is the derivative of a vector with respect to its transpose?
QUESTION [5 upvotes]: I've already looked at Vector derivative w.r.t its transpose $\frac{d(Ax)}{d(x^T)}$, but I wasn't able to find the direct answer to my question in that question. What is the value of $$\frac{d}{dx} x^T\text{ ?}$$
My initial intuition is that it is $1$, but I'm not exactly sure of why that would be so.

REPLY [5 votes]: The answer is relatively straightforward. Without going into the detail, let me tell the answer first: $I_{N\times N}$. 
The derivative of a vector valued function with respect to a vector is called Jacobian $J$
\begin{equation}
\textbf{J}:=\left(\frac{d \textbf{f}(\textbf{x})}{d\textbf{x}}\right)_{ij} = \frac{d\textbf{f}_i(\textbf{x})}{d\textbf{x}_j}. 
\end{equation}
For the question of interest, $\textbf{f}(\textbf{x})\rightarrow\textbf{x}^T$. The remaining task is use the property: 
\begin{equation}
\frac{d\textbf{x}_i}{d\textbf{x}_j}=\delta_{ij}=I_{N\times N}.
\end{equation}