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The problem given is this. If the variables $P, V$, and $T$ are related by the equation $PV = nRT$, where $n$ and $R$ are constants, simplify the expression

$$\frac{\partial V}{\partial T} \cdot \frac{\partial T}{\partial P} \cdot \frac{\partial P}{\partial V}$$

After doing the calculations, we see that we get $-1$ as an answer. My question: why can't we just apply the chain rule to "cancel out" the numerators/denominators and get 1 as an answer?

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    Terminological note: this is called [the cyclic chain rule or Euler's chain relation](http://en.wikipedia.org/wiki/Triple_product_rule).2011-09-28

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