How to calculate the derivative of the angular momentum vector $ dvec L = d(hat I vec omega)?$

My last question, but also the most important one How to calculate the derivative of the angular momentum vector?
$$ dvec L = d(hat I vec omega)$$
I’m especially interested in derivative tensor to vector for the law of conservation of angular momentum for a rotating rigid body.
$$ d(hat I vec omega)=0$$

I would like to note that the change of the moment of inertia $d hat I$ must result in change vector $vec L$ and there must be a second element $d vec omega$ that compensates for this change that the angular momentum vector remains unchanged.

I have a solution that is strange but has not been verified by me yet and for now I will not give my answer. So maybe someone will show me how to calculate this derivative.

Don’t kill innocent questions, answer them.