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Math Help - complicated logarithmic differentiation

  1. #1
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    complicated logarithmic differentiation

    i'm so sorry for keep asking questions but im really confused for this one.
    the problem is y=(e^(x-1)*sin^2*x)/(((x^2)+5)^2x)

    the trig function (sin^2 + 5), i don't know how to get a derivative out of that.

    help me plz!
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  2. #2
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    Hi katieeej -

    if you mean  \sin^2 x + 5 then this calls for a simple application of the chain rule for differentiation...



    ...where straight continuous line differentiate downwards with respect to x and the straight dashed line with respect to the dashed balloon expression, so that the triangular network satisfies the chain rule...



    ... though I'm not sure you meant \sin^2 x...

    Don't integrate - balloontegrate!

    Balloon Calculus: worked examples from past papers
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  3. #3
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    clarification

    the equation is

    y=

    e^(x-1)*sin^2(x)
    -------------------- division
    (x^2 +5)^2x

    i started off like
    ln y = ln(e^(x-1)sin^2 (x)) - ln (x^2 +5)^2x
    = ln(e^(x-1))+ln (sin^2 (x))- 2x ln (x^2 +5)
    = x-1 ln e +ln sin^2 (x) - 2x ln (x^2+5)
    =x-1+ln (sin^2 (x)) -2x ln (x^2 +5)

    then i don't know how to get derivatives.
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  4. #4
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    More in a moment...
    Last edited by tom@ballooncalculus; February 20th 2009 at 02:50 AM. Reason: typo
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  5. #5
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    Then the product rule...



    And then you need to apply the chain rule to \ln | \sin^2 x |...
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  6. #6
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    And using the derivative of \sin^2 x which is the same as that of \sin^2 x + 5 in the first picture...





    Don't integrate - balloontegrate!

    Balloon Calculus: worked examples from past papers
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