# To ensure whether chain rule is applicable

In my thermodynamics book, I came across the following equation:

$$0=\left[ 1- \left( \dfrac{\partial V(p,T)}{\partial p}\right) \left( \dfrac{\partial p(V,T)}{\partial V}\right) \right] dV +.......$$

Now, is chain rule applicable to this term so that the product of the two partial derivatives is 1 and the whole term is 0?

My book doesn't apply chain rule.

• Could you add more details on the nature of your problem? – Kiarash Aug 10 '17 at 17:54
• I only want to know whether we can apply chain rule to the above partial derivative. – N.G.Tyson Aug 10 '17 at 18:09
• Yeah, I think that you're right. The same variable is being held constant in both of those partial derivatives (i.e., the temperature T), so those two partial derivatives should be reciprocals of each other. Their product should equal 1 and so that entire dV term should be zero. – Samuel Weir Aug 10 '17 at 19:48
• I found the relevant equation in my copy of "Statistical Mechanics" by Franz Mohling. Here's the Google book link ( books.google.com/books/about/… ) and if you type "9.4a" in the "From inside this book" search window, then equation 9.4a will be displayed, which is the equation to use here. – Samuel Weir Aug 10 '17 at 20:09