I am looking into an inertial navigation filter implementation and I would like to take advantage of a doppler velocity log (DVL) for the velocity measurements. This requires me to take the Jacobian of the measurement function, suppose my state vector is $$ x = \begin{bmatrix}q &p &v \end{bmatrix} $$ for quaternion orientation, position, and velocity. The error state vector is thus $$ \delta x = \begin{bmatrix} \delta\rho &\delta p & \delta v\end{bmatrix} $$ where $\delta\rho$ is the attitude error about the three cartesian axes.
The measurement function is thus $$ y = \hat{R}_{n}^{b} v^{n} $$ where $R_{n}^{b}=(R_{b}^{n})^{T}$ is the rotation matrix resulting from the quaternion $q$. This is the case because the DVL measures velocity with respect to body frame and the estimate is given with respect to NED frame. I am in doubt about how to take the derivative $$ \frac{\partial h}{\partial \delta x} $$ where $h=\hat{R}_{n}^{b} v^{n}$
I understand the derivative can be broken into the chain rule $$ \frac{\partial h}{\partial \delta x} = \frac{\partial h}{\partial x}\frac{\partial x}{\partial\delta x} $$ but I am not sure how to begin evaluating this derivative?