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The chain rule

  1. Mar 20, 2009 #1
    1. The problem statement, all variables and given/known data
    Hi all.

    I have a time-dependent variable called x. I wish to differentiate x2 with respect to the time t, and this is what I have done:

    [tex]
    \frac{d}{dt}x^2 = \frac{d}{dx}x^2\frac{dx}{dt} = \frac{d}{dx}x^2\dot x.
    [/tex]

    where the dot over x denotes differentiation with respect to time t. Now my question is, must I differentiate [itex]\dot x[/itex] also with respect to x, since it is standing on its right side?
     
  2. jcsd
  3. Mar 20, 2009 #2

    danago

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    Gold Member

    Nah you dont. It is a little bit ambiguous how you have written it, but in this context it is just d(x^2)/dx MULTIPLED by x dot.
     
    Last edited: Mar 20, 2009
  4. Mar 20, 2009 #3
    How is the proper way to write it then?

    Thanks for replying.
     
  5. Mar 20, 2009 #4

    danago

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    Gold Member

    Its not that how you have written it is wrong, it is just that it could possibly be misinterpreted. I personally would have witten it with the x dot on the other side of the derivative operator just so there will be no confusion:

    [tex]

    \dot x \frac{d}{dx} x^2

    [/tex]
     
  6. Mar 20, 2009 #5

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Better would be
    [tex]\dot{x}\frac{dx^2}{dx}[/tex]
    or just
    [tex]\left(\frac{dx}{dt}\right)\left(\frac{dx^2}{dx}\right)[/tex]

    Now, what is
    [tex]\frac{d x^2}{dx}[/tex]?
     
  7. Mar 20, 2009 #6
    Ahh, now you are just teasing me :smile:

    Thanks to all for helping.
     
  8. Mar 20, 2009 #7

    Mark44

    Staff: Mentor

    I don't think Halls was teasing - you shouldn't leave it as dx^2/dx. I think you understand, but I'm not 100% certain.
     
  9. Mar 20, 2009 #8
    The reason why I didn't evaluate it in my first post was because I wasn't sure if I had to differentiate [itex]\dot x[/itex] as well, but now I would of course just write 2x instead.
     
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