How is it derived?
Had this been ds^2 =+cdt^2+dx^2+dy^2+dz^2 it would've been the Pythagorean Theorem.
If it was ecludian space
I would say that it is not derived (except maybe as an approximation to more general spacetimes), i.e., it is arrived at by induction, not by deduction.
Well, it can derived depending on what your assumptions are. If you assume the Lorentz transform, then the Minkowski line element follows as the only possible metric that is invariant under Lorentz transforms. Similarly, if you assume the Minkowski metric, you can derive the Lorentz transform.
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