What is wrong with the following naive derivation of the triple product rule ( https://en.wikipedia.org/wiki/Triple_product_rule)

$\frac{\partial X}{\partial Y}=\frac{\partial X}{\partial Z}\frac{\partial Z}{\partial Y}\Rightarrow\\ \left(\frac{\partial X}{\partial Z}\right)^{-1}\left(\frac{\partial Z}{\partial Y}\right)^{-1}\frac{\partial X}{\partial Y}= \frac{\partial Z}{\partial X}\frac{\partial Y}{\partial Z}\frac{\partial X}{\partial Y}=1\ne-1$

  • $\begingroup$ Your first line is wrong, you have used the chain rule for a function who's variables are not functions of another variable. $\endgroup$ – jake walsh Dec 9 '17 at 22:06

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