# Partial derivatives of Gas Law

In James Stewart's Calculus exercise 82 page 891 asks you to show that:

$\frac{\partial P}{\partial V}\frac{\partial V}{\partial T}\frac{\partial T}{\partial P} = -1$

I can do this by noting that $V = \frac{nRT}{P}$ so that:

$\frac{\partial V}{\partial T}$ = $\frac{\partial}{\partial T}\left(\frac{nRT}{P}\right)$ = $\frac{nR}{P}$

and then doing likewise to find the other terms.

But this confuses me because I have treated P as a constant like n and R while it is actually a function of T and V. If I make it a function of T and V I get stuck and anyway the thing gets a lot more complicated than the exercise intends. So how can I just treat P as a constant when I know its not?

Thanks for any help, Peter

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## Answers and Replies

When you take a partial derivative of a function of several variables, everything except the variable you are taking the derivative of are considered a constant. All I think the exercise is doing is asking you to take the three partials and multiply them together recognizing that PV in the denominator equals nRT so they cancel leaving -1.