I have tried to get $\frac{\partial (ab^T)}{\partial a}$ and $\frac{\partial (ab^T)}{\partial b}$ or $d(ab^T)$, where $a,b \in \mathbb{R}^{n \times 1}$.
But all I searched on the Internet are $\frac{\partial (a^Tb)}{\partial a}$ and $\frac{\partial (a^Tb)}{\partial b}$.
So how to solve $\frac{\partial (ab^T)}{\partial a}$ and $\frac{\partial (ab^T)}{\partial b}$ ?
Thanks.
$\def\R#1{\mathbb R^{#1}}\def\p#1#2{\frac{\partial #1}{\partial #2}}$The gradient of a vector (or transpose vector) with respect to its own components is $$\p{a}{a_k}=e_k\qquad\quad\p{b^T}{b_k}=e_k^T$$ where $e_k$ is the $k^{th}$ cartesian basis vector for $\R{n}$
Use the above to compute component-wise gradients $$\eqalign{ \p{(ab^T)}{a_k} &= e_kb^T\qquad\quad \p{(ab^T)}{b_k} &= ae_k^T }$$ Note that both $e_kb^T{\;\rm and\;}ae_k^T\;$ are themselves $\,\R{n\times n}$ matrices, hence the full gradients are third-order tensors which do not lend themselves to standard matrix/vector notation.