Why Do Partial Derivatives Not Always Multiply to One?

[unscientific]

Homework Statement

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.

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?

 

[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..

 

[Ray Vickson]

Homework Statement

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.

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?

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

 

[unscientific]

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

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

 

[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
$\frac{\partial P}{\partial x}\frac{\partial x}{\partial P}≠1$