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Math Help - Differentiating sum of log

  1. #1
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    Differentiating sum of log

    Hi everyone,

    This is my first post, so please bear with me if I am posting in the wrong forum.

    I study economics and I am reading an article. I do not understand how the author gets the following conclusions:

    He takes log of this function: Differentiating sum of log-capture2.png

    And ends up with: Differentiating sum of log-capture3.png

    I thought it would be Differentiating sum of log-capture1.png instead of the log[T!/(T-t)].

    Secondly I want do differentiate logc_t^t, but I am unsure how to deal with the log sum expressions. How do you differentiate Click image for larger version. 

Name:	Capture1.PNG 
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    Wolfram alpha uses gamma functions which I honestly do not understand. The article I am reading finds the following derivative:

    Differentiating sum of log-capture4.png

    I hope someone can clearify this for me. Thank you very much!

    Jacob
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  2. #2
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    Re: Differentiating sum of log

    Hey Katmarn.

    Try expressing that term in terms of factorials and see if it is equivalent.
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  3. #3
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    Re: Differentiating sum of log

    Thank you Chiro.

    I had tried it before without luck, but as I did it again it suddently made sense

    However, I still do not know how you take the derivative of a factorial term. Could you please show me how you do it on: Differentiating sum of log-capture.png

    When I express it in terms of factorials I get: log((1+b^(1/n)*T)!/[1+b^(1/n)*(T-t-1)]!), and I have no idea how to take the derivative of that...

    Thank you!
    Last edited by Katmarn; April 29th 2013 at 03:00 AM.
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  4. #4
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    Re: Differentiating sum of log

    If you are differentiating a factorial, take a look at the Euler Gamma function and use the fundamental theorem of calculus:

    Gamma function - Wikipedia, the free encyclopedia
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  5. #5
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    Re: Differentiating sum of log

    Thank you for the answer, but I must admit I am still lost as I do not fully understand the gamma function.

    Could you, or someone else, please explain the steps the author must use to get his result?

    He takes the derivative of: Differentiating sum of log-skaermbillede-2013-04-30-kl.-14.13.18.png wrt t.

    And he ends up with: Differentiating sum of log-hej.png

    I have tried to express the sum with factorials, and got: Differentiating sum of log-factor.png

    I do not know if this is even possible as 0<beta<1. However, if it is correct i can split the log up and just focus on the bottom part when i want to take the derivative wrt. t.

    And here I am stuck...

    All help is appreciated.
    Attached Thumbnails Attached Thumbnails Differentiating sum of log-skaermbillede-2013-04-30-kl.-11.29.04.png   Differentiating sum of log-skaermbillede-2013-04-30-kl.-13.37.43.png  
    Last edited by Katmarn; April 30th 2013 at 04:20 AM.
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  6. #6
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    Re: Differentiating sum of log

    The Gamma function is the factorial function, but is extended for all real values instead of only being valid for integer values. It is not defined at 0 or the negative integers, but it is defined everywhere else.

    The connection is that n! = Gamma(n+1).
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  7. #7
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    Re: Differentiating sum of log

    Ok. Do you have the time to explain how the author ends up with the result that he does?
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  8. #8
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    Re: Differentiating sum of log

    I'll take a look at this later on so if I haven't responded tomorrow, just bump the thread.
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  9. #9
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    Re: Differentiating sum of log

    Chiro
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  10. #10
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    Re: Differentiating sum of log

    Try substituting for s values t - 1, t - 2, ..., t - t and use the fact that d/dx(u + v) = du/dx + dv/dx.

    Then collect the derivatives of all the logarithms and see if you get the same answer.
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