What is the derivative of matrix vector product $(A^Tx)$ with respect to A?

Question

What is the derivative of a vector with respect to a matrix? Specifically, $\frac{d(A^Tx)}{dA} = ? $, where $ A \in R^{n \times m}$ and $x \in R^n$.

Answer 1

Note that $f:A\rightarrow A^Tx$ is linear. Then the derivative is $Df_A:H\in M_{n,m}\rightarrow H^Tx\in\mathbb{R}^m$.

In other words, if $(e_i)$ is the canonical basis of $\mathbb{R}^m$, then $\dfrac{\partial{f}}{\partial{a_{i,j}}}=x_ie_j$