Good time of the day. In my research (I'm an electrical engineer) I'm doing a kind of quaternion analysis for a control system design. So, in my particular case, I just have a "vector part" of a voltage quaternion associated with the grid voltage waveform. And I want to find grid voltage derivative in the quaternion basis. I decided to perform partial finite-difference for each quaternion part like $q_1$, $q_2$, $q_3$ (I don't have a real part in this quaternion $q_0$). So, is it legal to call it quaternion time derivative or can you suggest a more strict way of a numeric quaternion differentiation?
$Q'=\left([q_1(k+1)-q_1(k)]/T_s,[q_2(k+1)-q_2(k)]/T_s,[q_3(k+1)-q_3(k)]/T_s\right)$, where $k$ is a sample instant and $T_s$ is a sample time.