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Thread: Derivatives involving absolute values

  1. #1
    Sep 2009

    Derivatives involving absolute values


    I am slowly moving through the calculus portion of a math textbook my girlfriend lent me and am stuck on a few problems involving finding the first derivative of equations involving absolute values.
    I will use one of these as an example.

    Find the derivative of: y = \frac{ln|x|}{x^4}

    Normally, if there was no absolute value, I would rewrite the equation as  (lnx)(x^{-4}) . Then I would use the product rule to get  (\frac{1}{x})(x^{-4}) + (-4x^{-5})(lnx) . Likely ending up writing it as  \frac {x}{x^4} + \frac {-4lnx}{x^5} .
    *I did this to show that I know how to do it with no absolute value*

    I know what an absolute value is, I am unsure of what to do with it when trying to derive a function that has one.
    I searched google and found a couple websites that tried to explain it, but I could not grasp what they were saying. I was hoping that someone could, using my example, explain (in english) how to go about doing this.
    Thanks in advanced!
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  2. #2
    MHF Contributor alexmahone's Avatar
    Oct 2008
    The reason the absolute value is given in the question is simply because the logarithm of a negative number does not exist. So, you should write your answer as:


    \frac {1}{x^5}-\frac {4}{x^5}ln|x|

    Hope that helps!
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