I am trying to find the following derivative:
$$\frac{\partial}{\partial x} \left( Axb^TC \right)$$
where $A, C \in \mathbb R^{n \times n}$ and $x, b \in \mathbb R^{n \times 1}$. $A,C,b$ are independent of $x$. So far, I was not able to find any identities that can help me here.
It's easiest to use index notation and work with the individual components. $$\eqalign{ Y_{im} &= A_{ij}(x_{j})b_{k}C_{km} \\ G_{imp} =\; \frac{\partial Y_{im}}{\partial x_p} &= A_{ij}\big(\delta_{jp}\big)b_{k}C_{km} &= A_{ip}b_{k}C_{km} \\ }$$ Since $G$ requires three free indexes, it is called a third-order tensor.