Basic qns on partial derivatives

1. Jul 27, 2012

unscientific

1. The problem statement, all variables and given/known data

1. Is (∂P/∂x)(∂x/∂P) = 1?

I realized that's not true, but i'm not sure why.

2. Say we have an equation PV = T*exp(VT)

The question wanted to find (∂P/∂V), (∂V/∂T) and (∂T/∂P) and show that product of all 3 = -1.

3. The attempt at a solution

I tried moving the variables about then differentiate but I got all the wrong answers, for example:

V = (T/P) * exp (VT)

then to find (∂V/∂T) with P constant, i did product rule.. which gave me wrong answers

T = PV * exp(-VT)

then to find (∂T/∂P) with V constant, i use product rule again..which completely gave me the wrong answers..

So I thought that you're not allowed to move the variables around?

Strangely I got (∂P/∂V) correct despite moving the variables around... coincidence?

2. Jul 27, 2012

unscientific

update: i realized its the same! but this way it took much much longer than the solution which they simply took the "ln" throughout to simplify..

3. Jul 27, 2012

Ray Vickson

Show what you did: you said " ... then to find (∂V/∂T) with P constant, i did product rule.. which gave me wrong answers...". What did you get, and how do you know the answer is wrong?

RGV

4. Jul 27, 2012

unscientific

I worked it out finally. It comes out the same as the solutions, so i'm relieved. But can anyone answer the first qn?

5. Jul 28, 2012

Hassan2

I don't have a clear understanding of partial derivative but the following hint may help you:

Suppose P is a function of x and y, we can write

$\frac{\partial P}{\partial P}=1$

using chain rule:
$\frac{\partial P}{\partial x}\frac{\partial x}{\partial P}+\frac{\partial P}{\partial y}\frac{\partial y}{\partial P}=1$

Due to the second term
$\frac{\partial P}{\partial x}\frac{\partial x}{\partial P}≠1$