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My experience is that a derivation is usually called some procedure which will show you that it is reasonable to believe some theorem is true and bother to prove it. In that it is not P=>Q but rather something along the lines Q=>A, P=>A, so maybe P=>Q
So you first derive a result which will give you an idea that it is true, and then you mathematically prove it.
In many lectures, mostly physics, only derivations are presented
