The derivation of things like the Schwarzchild metric in relativity is found by solving [tex]R_{ab}=0[/tex] for a static, spherically symmetric space-time with [tex]T_{ab}=0[/tex]. It's essentially solving the Einstein Field Equations for certain conditions (as all black hole metric's are).
Deriving the existence of the notion of a metric is much more indepth. Finding the Schwarzchild metric is already assuming all the machinary of (Pseudo)Riemannian manifolds etc. Actually developing all that machinary from more basic ideas like norms and tangent spaces is much more involved.
As others have said, what precisely are you referring to, because the answer would differ a lot!