I'd like to derivate a dual quaternion
\begin{align} \hat{q}&=(1 + \frac{1}{2}\epsilon\vec{t})q \end{align}
where
\begin{align} q &= e^\vec{w} , \\\vec{w}&=(0, w_1,w_2,w_3)^t \end{align} is a quaternion in exponential map representation that describes a rotation like: \begin{align} q &= (\cos(\frac{1}{2}\theta), \vec{u}*\sin(\frac{1}{2}\theta)) \end{align}
I'd like to get \begin{align} \frac{\partial \hat{q}}{\partial \vec{w}} &= (1 + \frac{1}{2}\epsilon\vec{t}) \frac{\partial q}{\partial \vec{w}} &(1)\\ \frac{\partial q}{\partial \vec{w}}&=(\frac{\partial q}{\partial w_1},\frac{\partial q}{\partial w_2},\frac{\partial q}{\partial w_3}) \end{align} where the derivation in the last line consists of 12 partial derivations and returns a 4x3 matrix.
Getting to this point I'm not sure about how to finish the multiplication in (1), which consists of a dual quaternion and a 4x3 matrix.
Or is there another common way to derivate a dual quaternion?
Please could anyone solve this? Thank you for any hints!