I am trying to take the derivative of a vector with respect to a vector but I am stuck and I do not know how to proceed using matrix notation or any other notation.
The equation I am trying to derive is the following
$$ \frac{\partial}{\partial\boldsymbol{\eta}}\left(\boldsymbol{H}\boldsymbol{A}\left(\boldsymbol{\eta}\right)\boldsymbol{c}\right) $$
where $\boldsymbol{H}$ is a constant $[9\times3]$ rectangular matrix, $\boldsymbol{A}$ is a NON constant $[3\times3]$ square matrix and $\boldsymbol{c}$ is a constant $[3\times1]$ vector. Clearly the product $\boldsymbol{H}\boldsymbol{A}\left(\boldsymbol{\eta}\right)\boldsymbol{c}$ is a vector.
I know the functional dependence of $\boldsymbol{A}\left(\boldsymbol{\eta}\right)$ upon $\boldsymbol{\eta}$, therefore I am able to evaluate some sort of derivative of $\boldsymbol{A}$, but being it the derivative of a matrix with respect to a vector I am not really sure how to proceed besides trying the brute force approach, i.e. by writing explicitly the elements of the vector $\boldsymbol{H}\boldsymbol{A}\left(\boldsymbol{\eta}\right)\boldsymbol{c}$ and trying to derive these.
EDIT
I want to actually compute the derivative point by point. More specifically I am looking for a way to rewrite the derivative of the matrix with respect to the vector in order to obtain a more useful expression.
Moreover, I add some details on $\boldsymbol{A}$, which can be written as: $$\boldsymbol{A}(\boldsymbol{\eta})=\boldsymbol{B}\boldsymbol{R}(\boldsymbol{q}(\boldsymbol{\eta}))$$ where $\boldsymbol{B}$ is another $\left[3\times3\right]$ matrix and $R$ is a rotation matrix written through quaternions which depend on the parameter $\boldsymbol{\eta}$. I have a way to compute $\frac{\partial\boldsymbol{q}}{\partial\boldsymbol{\eta}}$ but I don't really know how to use it.