I'm trying to understand a paper in which the author constructs sensitivity matrices in the process of linearizing an equation. I figured that the sensitivity matrix has to be a Jacobian Matrix, however my problem is a little bit more complex (at least to me, since this kind of algebra is new to me).
I want to calculate $\frac{\partial}{\partial{p}}$$(\overline{G}\omega_b)$. In the paper it is actually $\frac{\partial}{\partial{p^T}}$ but I do not know if that makes sense and the author talks about a partial derivative with respect to p. $\omega_b$ is a 3x1 column vector and $\overline{G}$ is a nx3 matrix. $p$ is a 5n-vector of 5 other vectors (each nx1) and looks like: $p=\begin{pmatrix} b\\ \lambda \\ \mu \\ \delta_u \\ \delta_v \end{pmatrix}$. I figured the corresponding Jacobian has to be of size nx5n, however since I am actually after the derivative $\frac{\partial}{\partial{p}} \omega_b$ I figured that some kind of product-rule that creates a term $\overline{G}\frac{\partial}{\partial{p}}\omega_b$ is being used.
Here the steps that the author provides: $\frac{\partial}{\partial{p^T}}$$(\overline{G}\omega_b)$ = $\overline{G}\frac{\partial}{\partial{p}}\omega_b - [0\ 0\ 0\ C_v\ C_u]$ The latter terms looks as if it originates from the given relationship $\omega_g = \overline{G} \omega_b = W^T \omega_b - C_v(\omega_b)\delta_u-C_u(\omega_b)\delta_v$.
Can somebody explain if there is such a product rule and if yes, how it liiks like? If no, then how does the author come up with the two parts?
Edit: One additional note: $C_v$ and $C_u$ are indeed of dimension nxn, so in fact each of the two parts have dimension n x 5n.