Physics 62 Special Relativity (23 of 43) The Lorentz Transformation Equations: Velocity (v')

In this video I will find Lorentz transformation equation in regards to velocity (v').

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[eucsstamticc]

What are you taking the derivative with respect to?

[sameedali]

How would one go about writing the Lorentz transforms for an arbitrary velocity, say with an x, y and z component. Ive been thinking that one could re-write the time equation such that it's dependance on the velocity would contain all direction components but then again one could re-write the time into 3 equations, one per each direction of motion, and use these for the transform.

[armanrahman]

As we've seen the event occurred in the S' frame wrt observer B always remains the same, it just changes when the event in S' is being observed from S stationery frame...It's so confusing to me, can you please explain