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...Partial differentiation for functions of more than one variable f x y take area as an example a xy increase xx in yx constant yy simultaneous aa the limits da dx dy total differential real single value function two independent variables fx fxy lim paul percival chem spring derivative relations consider z so zz dz derivatives can be taken any order zx taking inverse xz to find third zy yz xyz chain rule and thermodynamics from generalized equation state closed system fp vt six written pt pv tp vvt p pvtpvt but given three inverses e g tv pp v pvt there are only basic properties matter by convention these chosen coefficient thermal expansion isobaric...