The triple product rule in multivariable calculus is widely used. Can a quadruple product rule equation be written for an equation f(x,y,z,z2)=0?

Actually, there is general product rule for $n$-tuple. It is almost the direct consequence of implicit function theorem. Assume $n$ variables satisfies $F(x_1,\ldots,x_n)=0$, we have $$\frac{\partial x_i}{\partial x_j}=-{\partial F/\partial x_j\over\partial F/\partial x_i}$$ Then multiply all fractions $$\frac{\partial x_1}{\partial x_2}\cdots\frac{\partial x_{n-1}}{\partial x_n}\frac{\partial x_n}{\partial x_1}=(-1)^n{\partial F/\partial x_2\over\partial F/\partial x_1}\cdots{\partial F/\partial x_n\over\partial F/\partial x_{n-1}}{\partial F/\partial x_1\over\partial F/\partial x_n}=(-1)^n$$

This is what would make sense to me: $(fghi)'=f'ghi+fg'hi+fgh'i+fghi'$ You can keep adding $n$ functions to this rule.

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    I think you misunderstood what the question is asking about. Look at the link given by anon in his comment. – Willie Wong Nov 18 '13 at 13:40

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