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Basic qns on partial derivatives

  1. Jul 27, 2012 #1
    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. jcsd
  3. Jul 27, 2012 #2
    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..
     
  4. Jul 27, 2012 #3

    Ray Vickson

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    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
     
  5. Jul 27, 2012 #4
    I worked it out finally. It comes out the same as the solutions, so i'm relieved. But can anyone answer the first qn?
     
  6. Jul 28, 2012 #5
    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

    [itex]\frac{\partial P}{\partial P}=1[/itex]

    using chain rule:
    [itex]\frac{\partial P}{\partial x}\frac{\partial x}{\partial P}+\frac{\partial P}{\partial y}\frac{\partial y}{\partial P}=1[/itex]

    Due to the second term
    [itex]\frac{\partial P}{\partial x}\frac{\partial x}{\partial P}≠1[/itex]
     
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