A vector in a rotating frame. Find the rate of change of this vector ('particular step derivation')

Question

I am doing a project with the fundamental background not in my major. I am reading the following lecture:

How to get the green part?

Can anyone show me the detailed derivation or provide the note for reference?

(There might be similar questions; however I only want the specific derivation or explanation of some steps not the last result.)

Answer 1

• The first green part is the Rodrigue's rotation formula.
• The second green part is a small angle approximation for $\delta \theta$.