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Quicky on derivative of absolute value in exponential

  1. May 10, 2008 #1
    Hey folks,

    I'm looking for a little guidance in solving the derivative y'(x)of the following function containing an absolute in the exponent:


    I'm pretty sure its not as simple as

    [tex]y'(x)=a e^{a|x|}[/tex]

    Any suggestions??
  2. jcsd
  3. May 10, 2008 #2
    i think the problem is that [tex] |x| [/tex] is not differential in zero so, [tex]e^{a|x|}[/tex] is not differential in zerp, so if you want to calculate the differential somewhere else, then just do in the two cases. Then you get for [tex]x<0[/tex]

    [tex]\partial_x e^{a|x|} = \partial_x e^{-ax} = -ae^{-ax} [/tex]

    for [tex]x>0[/tex] you get

    [tex]\partial_x e^{a|x|} = \partial_x e^{ax} = ae^{ax} [/tex]

    combining these could be

    [tex]\partial_x e^{a|x|} = sign(x) a e^{a|x|} = \frac{x}{|x|} a e^{a|x|}[/tex]

    but remember that it is not defined in 0.
  4. May 10, 2008 #3

    so the [tex]\frac{x}{|x|}[/tex] is really just a neat way of setting the coefficient to [tex]\pm 1[/tex], depending on where x is.

    Thats cool. :)

    Thanks mranderson, very helpful.
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