# Is there a quadruple product rule?

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?

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What is the triple product rule? –  Qiaochu Yuan Apr 23 '12 at 3:46
–  anon Apr 23 '12 at 3:47
If you follow the argument given in that wikipedia page, and if I understamnd correctly what you want, you can easily see what the quadruple product rule is. –  Mariano Suárez-Alvarez Apr 23 '12 at 3:53

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.