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Second derivative notation

  1. May 4, 2009 #1
    I often see the second derivative written down like this:

    [tex]\frac{d^2y}{dx^2}[/tex]

    Although it seems more logical to me to write

    [tex]\frac{d^2y}{d^2x^2}[/tex]

    Or

    [tex]\frac{d^2y}{(dx)^2}[/tex]

    Since it represents

    [tex]\frac{d}{dx} \frac{dy}{dx} [/tex]

    Is there any logic behind this or is it just a shortcut notation to omit the square in d², or brackets in the denominator?
     
  2. jcsd
  3. May 4, 2009 #2

    tiny-tim

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    Hi ImAnEngineer! :smile:

    It's because it's short for (d/dx)2(y) …

    for example, you might write (d/dx)2(x3 + sinx), or indeed (d/dx)28(x3 + sinx) …

    and (d/dx)n is naturally written without brackets as dn/dxn

    the x3 + sinx stays as it is. :wink:
     
  4. May 4, 2009 #3
    I think that if the notation had d2x2 then people may be tempted to do silly things like cancel the d2 and the x2 and get really confused :) As it is, there's only a slight bit of confusion in areas such as this :)
     
  5. May 4, 2009 #4
    Is it?

    I would say:

    [tex]\left(\frac{d}{dx}\right)^n=\frac{d^n}{(dx)^n}=\frac{d^n}{d^nx^n}[/tex]

    Because (ab)²=a²b² and not ab²

    So is it just a shortcut notation to leave out the ² in the denominator?
     
  6. May 4, 2009 #5
    Semantically, d2x2 may imply that the differential operator is being applied to x twice, which is not the case in (dx)2. Ie., it is like mistaking (sin x)2 for sin2x2.
    In the case of writing dx2, it is just treating dx as a single entity, not as d(x2).
     
  7. May 5, 2009 #6
    It is the same in differential geometry and relativity, where line element (metric) is written as ds^2 instead of (ds)^2. It save some works in writing I suppose...
     
  8. May 5, 2009 #7

    tiny-tim

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    Yup! :biggrin:
     
  9. May 5, 2009 #8
    Aah OK! This makes sense, that really helps.

    Thanks everyone! :smile:
     
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