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In a partial differentiation lecture , it is written that $$\frac{\partial x}{\partial y} \ne \frac{1}{\partial y/\partial x} $$ if the relation between $x$ and $y$ is explicit . Can you give me an example? I am not convinced because for example: $$y=2x^3 + 2 z^4$$ $$\frac{\partial y}{\partial x}=6x^2$$ then if we differentiate the equation partially with respect to $y$ : $$1=6x^2\frac{\partial x}{\partial y}$$ so $$\frac{\partial x}{\partial y}=\frac{1}{6x^2}=\frac{1}{\partial y/\partial x} $$

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  • $\begingroup$ Good answers to this question here math.stackexchange.com/questions/530733/… $\endgroup$ – Kevin Sep 26 '17 at 11:22
  • $\begingroup$ I add here a reference in which you can be interested, where is mentioned different cases for functions of one variable. Maybe you want to read it to think yours comparison with the multivariable case for partial differential equations. That is C. J. Brookes, Functions satisfying a certain type of reciprocity of derivatives, The American Mathematical MONTHLY, 74(5): 578–580, (1967). Good luck. $\endgroup$ – user243301 Sep 26 '17 at 11:31
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    $\begingroup$ The $\neq$ statement in the lecture is a warning that the two sides are not always equal, not an assertion that the two sides are always unequal. In this particular case you got lucky; or I would say rather you were unlucky, because what you really needed was to find one of the many counterexamples to the notion that the two sides should be equal. $\endgroup$ – David K Sep 26 '17 at 12:11

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