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Differentiation - Product Rule

  1. Aug 24, 2011 #1
    1. The problem statement, all variables and given/known data
    In general I havn't had problems using the differentiation rules until I came on this question, I'm probably doing something stupid any help is handy. Plugged it into an online differentiation solver and it comes up with (x^2-1)/(x^2) which I am getting nowhere near to in my attempts.

    Differentiate y = ((1/x)-1)*(1-x)


    2. The attempt at a solution

    Using product rule

    f(x)= x^-1 - 1
    f'(x)= -x^-2
    g(x)= 1-x
    g'(x)= 1

    f(x)*g'(x) = (x^-1 -1)(1) = x^-1 - 1
    g(x)*f'(x) = (1-x)(-x^-2) = -x^-2 + x^-1

    Adding them together = -x^-2 + 2x^-1 - 1

    Simplifying slightly = -1 / (-x^2 + 2x)

    And that's where I'm stuck, really not sure what I'm doing wrong here if its in the differentiation or messing up simplifying etc with algebra (been a long time since I've done math)

    Help much appreciated
    Thanks
     
    Last edited: Aug 24, 2011
  2. jcsd
  3. Aug 24, 2011 #2

    Hootenanny

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    Welcome to Physics Forums.
    A little slip here, which unfortunatly carries through: [itex]g^\prime(x)=-1[/itex].
     
  4. Aug 24, 2011 #3
    Ah yep silly mistake. Just gave it another go, still not getting the answer I'm expecting though, can anyone confirm if the answer I think I should be getting (x^2 - 1)/(x^2) is correct or not as I might be chasing something that is wrong to begin with (although its much more likely it's me doing something wrong).

    Attempt 2

    f(x) = x^-1 - 1
    f'(x) = -x^-2
    g(x) = (1 - x)
    g'(x) = -1

    f(x)*g'(x) = -x^-1 + 1
    g(x)*f'(x) = -x^-2 + x

    Adding together

    -x^-2 - x^-1 + x + 1

    Simplifying

    (x + 1)/(-x^2 - x)
     
  5. Aug 24, 2011 #4

    Hootenanny

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    Yes, that is correct.
    These final two lines are incorrect.

    Do you have to use the product rule? If not, you may find it easier to multiply out the brackets before differentiating.
     
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