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I want to calculate the derivative of quaternion product. Say $p$ and $q$ are unit quaternions. And I want to calculate $\frac{\partial p\bigotimes q}{\partial q}$.

From one reference

Quaternion kinematics for the error-state Kalman filter

I found that I can calculate this by converting p to a left-quaternion-product matrix: $p\bigotimes q=[p]_{R}q$ Then, the partial derivative is just $[p]_{R}$.

My question is that what if I calculate this by the definition of partial derivative like the following $$\frac{\partial p\bigotimes q}{\partial q} =\frac{p\bigotimes q\bigotimes dq\ominus p\bigotimes q}{dq} =\frac{q^{*}\bigotimes p^{*}\bigotimes p\bigotimes q\bigotimes dq}{dq} =\frac{dq}{dq} =I.$$

What is the problem of this derivation? Also I would like to ask what is the common way to calculate quaternion derivative. Just treat them as normal 4 by 1 vector and calculate vector derivative?

Thanks

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