I can't find a Clear or ClearAll command that gets rid of Derivative definitions I created. this code:
ClearAll["Global`*"];
gau[x_, v_] = (1/Sqrt[2*Pi*v])*E^-((x^2)/(2*v));
Derivative[0, n_][p][x, v] := Simplify[D[gau[z, v], {v, n}]];
ClearAll["Global`*"];
gau[x_, v_] = (1/Sqrt[2*Pi*v])*E^-((x^2)/(2*v));
Derivative[0, n_][p][x, v] := Simplify[D[gau[z, v], {v, n}]/gau[z, v]];
D[p[x, v], {v, 2}]
produces output consistent with the 1st definition, not the second.