What is the derivative of a unit vector?
The derivative of any vector whether it is unit or not is simply the derivative of each component in the vector. If you have some vector valued function r(t) for example which you divide by its magnitude to obtain a unit vector, the derivative is simply a vector :(derivative of the x component, the derivative of the y component)/IIr(t)||
If the unit vector is just a number ( given) then obviously the derivative is 0.
In summary, to get a unit vector divide the vector by its magnitude. To find the derivative, take the derivative of each component of the vector separately. That will give you the new vector. This works for functions of more than two dimensions as well.