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Math Help - Logarithmic Differentiation

  1. #1
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    Logarithmic Differentiation

    Hi I am having trouble solving a couple of problems involving logarithmic differentiation.

    1.Find if


    2.If , find .

    3.Let

    Determine the derivative at the point .

    4. If , find .

    Any tips on how to do these? Thanks.
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  2. #2
    MHF Contributor harish21's Avatar
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    Quote Originally Posted by KK88 View Post
    Hi I am having trouble solving a couple of problems involving logarithmic differentiation.

    1.Find if


    2.If , find .

    3.Let

    Determine the derivative at the point .

    4. If , find .

    Any tips on how to do these? Thanks.
    For the first question, use the chain rule by solving:

    \frac{\mbox{d log(u)}}{\mbox{du}} \frac{\mbox{du}}{\mbox{dx}} where u = \sqrt{\frac{4x+8}{5x+7}}

    and differentiate..

    For further assistance, show your work on this and the other problems on where you are getting stuck.
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  3. #3
    Senior Member AllanCuz's Avatar
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    Quote Originally Posted by KK88 View Post
    Hi I am having trouble solving a couple of problems involving logarithmic differentiation.





    1.Find if






    2.If , find .






    3.Let







    Determine the derivative at the point .



    4. If , find .



    Any tips on how to do these? Thanks.

    2:

    The first part of this is easy, but the derivative of  x^x is not, so let us go through that

     y = x^x

     lny = xlnx

     \frac{1}{y} y` = lnx + 1

     y` = x^x lnx + x^x

    Thus,

     F(x) = 4sinx + 3x^x

     F`(x) = 4cosx + 3x^x( lnx + 1)

    3:

    For  y= ln(x^2 + y^2)

     y` = \frac{1}{x^2 + y^2} (x^2+y^2)` = \frac{1}{x^2 + y^2} (2x + 2y y` )

    Bring y prime over to one side and factor it out,

     y`[ 1 - \frac{2y}{x^2 + y^2} ] = \frac{1}{x^2 + y^2} (2x)

     y` = \frac { \frac{2x}{x^2 + y^2} }{ 1 - \frac{2y}{x^2 + y^2} }

    Sub in the point (1,0) to find the value.

    4:

    is a repeat of 2
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  4. #4
    Super Member 11rdc11's Avatar
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    Quote Originally Posted by KK88 View Post
    Hi I am having trouble solving a couple of problems involving logarithmic differentiation.






    1.Find if







    2.If , find .







    3.Let








    Determine the derivative at the point .



    4. If , find .



    Any tips on how to do these? Thanks.


    Here is abother way to approach the first problem. Use your log rules to seperate the the experession.

     y = \frac{1}{2}\bigg(\ln{(4x+8)} - \ln{(5x-7)}\bigg)

    y' = \frac{1}{2}\bigg(\frac{4}{4x+8}-\frac{5}{5x-7}\bigg)

    now just simplify
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