Derivation
An informal derivation follows. Suppose that f(x, y, z) = 0. Write z as a function of x and y. Thus the total derivative dz is
Suppose that we move along a curve with dz = 0, where the curve is parameterized by x. Thus y can be written in terms of x, so on this curve
Therefore the equation for dz = 0 becomes
Since this must be true for all dx, rearranging terms gives
Dividing by the derivatives on the right hand side gives the triple product rule
Note that this proof makes many implicit assumptions regarding the existence of partial derivatives, the existence of the exact differential dz, the ability to construct a curve in some neighborhood with dz = 0, and the nonzero value of partial derivatives and their reciprocals. A formal proof based on mathematical analysis would eliminate these potential ambiguities.
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