Are you familiar with the chain rule for partial derivatives? Applying it should give you the formulas.
In deriving the reduced mass two-body Hamiltonian from Bethe and Salpeter's book I encountered the attached text. I am unsure of the details behind how they arrive at the first two equations of (5.2). I am looking for the explicit steps in reaching the formulas directly following 'we have...' if anyone can help out