How does a derivation differ from an experiment or a theory?
A derivation does not require an experiment, if it can be DERIVED mathematically from an accepted Law or Theory. It can also be used to experimentally test a theory.
E = mc^2 is a derivation from the Special Theory of Relativity. It has been verified experimentally. The bending of light by gravity and the unusual orbital behavior of Mercury have neen derived from the General Theory of Relavity and experimantally verified.
The law of conservation of momentum can be derived from Newton's second and third laws of motion. It does not require experimental verification to be proven.
Some laws (like Snell's Law of optics), can be derived from the wave theory of light.
I see why you care confused by all this.