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I would like to know if there is any physical significance associated with the derivative of a quantity with respect to proper time or is it just a mathematical trick. Since proper time is measured in its "rest" frame of a moving particle, it seems to me that particle is not going through any dynamics and therefore time derivatives should be zero in the rest frame. I understand that we use derivatives with respect to proper time to keep things Lorentz invariant....but that sounds more like a mathematical requirement rather than something of physical significance. Example: what is the physical significance of four-velocity. I know four velocity is tangent to the worldline but I find it hard to remember through physical intuition. I always have to go through a book to find its definition.

Kindly excuse my ignorance.

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The derivative with respect to proper time is the derivative with respect to time of the instantaneously comoving inertial frame.

This does not mean the particle is at rest. That's why I had to have the word instantaneously in there.

As for four velocity, that's the unit tangent vector to the world line. It tells you what direction in spacetime something is going.

Just like a unit velocity vector tells you the direction a particle is going in Newtonian mechanics.

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