Struggling to grasp derivation for the Body Jacobian from Modern Robotics

Hey everyone,

I'm following the book Modern Robotics and have made it to chapter 5.1.2 Body Jacobian, but I'm struggling to understand the derivation for the body Jacobian.

When the book derives T for the space Jacobian it follows the chain rule which places the matrix for the screw to the left of the exponential rotation, complying with the commutative properties of matrices:

However, when the book derives T for the body Jacobian the screw is to the right of the exponential rotation:

Surely though this goes against the chain rule? Plus if the screw matrix needs to be on the right for the rest of the derivation to work:

I feel I have missed something but I can't find any resources to explain why this is the case here. Could someone please explain why the body Jacobian is different here, and why when we derive the exponential rotation the screw matrix is on the right side of the rotation?

A member of Reddit has resolved this issue. In the same book, chapter 3, 3.44, it proves that Ae^(At) is equal to e^(At)A. Makes sense when you calculate it yourself as you can treat the t as a scalar throughout the exponential function.