Dear Stefan Boersen,
You claim that a mathematical procedure that is correct for mathematical reasons is also correct for physical reasons. If one has an equation with a '=' in the middle then one can alway substract or add a number left and right from the '=' sign. If one wants to differentiate (taking a parameter p), left and right of the '=' sign , then that would be correct for mathematical reasons but also correct for physical reasons. If a scientist states that one cannot differentiate this (dE = F dx) equation taking the time t, then one simply writes it all out. If a scientist states that the product rule ( d(gh) = hdg + gdh ) is not valid for dE = F dx, then one simply writes it all out. Please do this for me.
Regards ir.Simon Boersen
A small change in distance dx = x1 - x0, a small change in time is dt = t1 - t0
I done it and , you are looking happy with the two pictures.