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Partial Derivative Properties

  1. Jul 14, 2016 #1
    1. The problem statement, all variables and given/known data
    I would just like to know if this statement is true.

    2. Relevant equations
    [tex] \frac {\partial^2 f}{\partial x^2} \frac{\partial g}{\partial x}=\frac{\partial g}{\partial x} \frac {\partial^2 f}{\partial x^2}[/tex]

    3. The attempt at a solution
    I've thought about this a bit and I haven't come to a conclusion. Thanks for the help! :smile:
     
    Last edited: Jul 14, 2016
  2. jcsd
  3. Jul 14, 2016 #2
    Well, it depends on ##f## and ##g## and not so on the partial derivative. If ##f## and ##g## are "normal" functions like ##f(x)=x^2## for example, then the statement is true. On the other hand, if they represent matrices then generally they wouldn't commute, ie. ##f\cdot g\neq g\cdot f## because ##g## and ##f## do not commute generally.
     
  4. Jul 14, 2016 #3

    Ray Vickson

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    If you set ##A = \partial g/\partial x## and ##B = \partial^2 f/\partial x^2##, you have written ##A B = B A##, which is true for any two real numbers.

    However, if what you really meant was to have
    [tex] \frac{\partial}{\partial x} \left( g \frac{\partial^2 f}{\partial x^2} \right) [/tex]
    on one side and
    [tex] \frac{\partial^2} {\partial x^2} \left( f \frac{\partial g}{\partial x} \right) [/tex]
    on the other, then that is a much different question.

    Which did you mean?
     
  5. Jul 19, 2016 #4
    I intended for the original question you had answered about [tex] AB=BA [/tex] for any real number. I was assuming that the second derivative had acted on f and the first derivative had acted on g.
     
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