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Partial Derivative Product with variables as functions

  1. Nov 21, 2012 #1
    1. The problem statement, all variables and given/known data

    I'm trying to understand how a certain substitution can be made with regards to taking the partial derivative of a function product when the variable I am differentiating by is a function itself.


    2. Relevant equations

    (∂/∂p) (v(p)p(x,t)) = v(p) + (∂v/∂p)p


    3. The attempt at a solution

    My first thought was to use the product rule, but that seems to fall down with the function p(x,t) there. Am I missing something obvious here?
     
    Last edited: Nov 21, 2012
  2. jcsd
  3. Nov 21, 2012 #2

    HallsofIvy

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    Because the derivative is with respect to p, the fact that p is a function of x and y is irrelevent. With f(p)= pv(p), by the product rule, df/dp= v+ p dv/dp. Notice that these are ordinary derivatives, not partial derivatives, because f depends on the single variable, p.
     
  4. Nov 21, 2012 #3
    So if I'm understanding this correctly, all the derivatives should be ordinary rather than partial in this case?
     
  5. Nov 21, 2012 #4

    HallsofIvy

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  6. Nov 21, 2012 #5
    Ok, thank you for your help.
     
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