δ Det(A) := δ (-∂s Tr ( A-s ) |s=0) = ∂s ( s Tr[(δ A ) A-s-1])|s=0
cf., for example, Burghelea, Friedlander, Kappeler, Meyer-Vietoris type formula for determinants of elliptic differential operators. J. Funct. Anal. 107 (1992), 34--65, or Kontsevich, Vishik , Geometry of determinants of elliptic operators. (of course, to define A-s one uses a spectral cut, which I did not write explicitly to simplify the notation).
Is it proven anywhere in the literature? (I do know how to prove it, but I need a reference)