How to derive Non-normalized quaternion with respect to time?

• I
Roni BM
I know that for normalized quaternion, $$\hat{q}$$, the derivative is given by $$\frac{d\hat{q}}{dt}=\frac{1}{2}\hat{q}\cdot \omega$$ where $$\cdot$$ denotes the quaternion multiplication.

I want to calculate the time derivative of a non-normalized quaternion q.

I tried to calculate the derivative by using the chain rule, $$\dot{q}=\left|q\right|\dot{\hat{q}}+\hat{q}\frac{d\left|q\right|}{dt}$$ and I got a very complicated term. I wonder if I am having a wrong approach and if there is a known formula?