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Math Help - Logarithm.... Too long, need assistance

  1. #1
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    Unhappy Logarithm.... Too long, need assistance

    Find \frac{dy}{dx} by logarithmic Differentiation:
    y = \frac{x^2(x-1)^2(x+2)^3}{(x-4)^5}

    Answer is \frac{x(x-1)(x+2)^2}{(x - 4)^6} (2x^3 - 30x^2 - 6x +16)
    EDIT Wahhhh very sorry for the mistake!!!
    Last edited by ^_^Engineer_Adam^_^; January 26th 2007 at 03:51 AM.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by ^_^Engineer_Adam^_^ View Post
    Find \frac{dy}{dx} by logarithmic Differentiation:
    y = \frac{x^2(x-1)^2(x+2)^3}{(x-4)^5}

    Answer is \frac{x(x-1)(x+2)^2}{(x - 4)^6} (2x^3 - 30x^2 - 6x +16)
    EDIT Wahhhh very sorry for the mistake!!!
    As in the other problem, take logs:

    \ln(y) = 2\ln(x)+2\ln(x-1)+3\ln(x+2)-5\ln(x-4)

    so:

    \frac{1}{y}\,\frac{dy}{dx} = \frac{2}{x}+\frac{2}{x-1}+\frac{3}{x+2}-\frac{5}{x-4}

    \frac{dy}{dx} = \frac{x^2(x-1)^2(x+2)^3}{(x-4)^5} \frac{2(x^3-15x^2-3x+8)}{x(x-1)(x+2)(x-4)}

    \frac{dy}{dx} = \frac{x(x-1)(x+2)^2}{(x-4)^6} {2(x^3-15x^2-3x+8)}

    RonL
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  3. #3
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    How did you get
    2(x^3 - 15x^2 - 3x + 8)?
    Sorry because thats the part where i got stuck

    This is where ive started:
    2(x-1)(x+2)(x-4) + 2(x)(x+2)(x-4) + 3(x)(x-1)(x-4) - 5(x)(x - 1)(x +2)
    I tried about 10 times but i still haven't got it right
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by ^_^Engineer_Adam^_^ View Post
    How did you get
    2(x^3 - 15x^2 - 3x + 8)?
    Sorry because thats the part where i got stuck

    This is where ive started:
    2(x-1)(x+2)(x-4) + 2(x)(x+2)(x-4) + 3(x)(x-1)(x-4) - 5(x)(x - 1)(x +2)
    I tried about 10 times but i still haven't got it right
    I'm tempted to say its just algebra, but I don't trust myself to get that right,
    so I used machine assistance to simplify it.

    But I will have a go:

    2(x-1)(x+2)(x-4)=2x^3-6x^2-12x+16

    2(x)(x+2)(x-4)=2x^3-4x^2-16x

    3(x)(x-1)(x-4)=3x^3-15x^2+12x

    -5(x)(x - 1)(x +2)=-5x^3-5x^2+10x

    So adding these we get:

    2(x-1)(x+2)(x-4) + 2(x)(x+2)(x-4) + 3(x)(x-1)(x-4) - 5(x)(x - 1)(x +2)

    ..... =2x^3-30x^2-6x+16

    Which is what we wanted.

    RonL
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