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Math Help - Differentiating Big Multiple

  1. #1
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    Differentiating Big Multiple

    Hi I need urgent help for this expression as I have no idea how to get its derivative:


    f(x)=(x^2+1)^100(x^3-7)^2(x-1)(x^{10}+x+3)^{-1}

    I have to use the ln function but I can only come up to here:

    \frac{d}{dx}ln(f(x))=100\frac{2x}{x^2+1}+2\frac{3x  ^2}{x^3-7}+\frac{1}{x-1}-\frac{10x^9+1}{x^{10}+x+3}

    Specifically I don't know how to get rid of the natural log function on the left side of the equation. I would appreciate if anyone could tell me how?
    Last edited by Kataangel; October 14th 2009 at 08:59 AM.
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  2. #2
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    Quote Originally Posted by Kataangel View Post
    Hi I need urgent help for this expression as I have no idea how to get its derivative:


    f(x)=(x^2+1)^100(x^3-7)^2(x-1)(x^{10}+x+3)^{-1}

    I have to use the ln function but I can only come up to here:

    \frac{d}{dx}ln(f(x))=100\frac{2x}{x^2+1}+2\frac{3x  ^2}{x^3-7}+\frac{1}{x-1}-\frac{10x^9+1}{x^{10}+x+3}

    Specifically I don't know how to get rid of the natural log function on the left side of the equation. I would appreciate if anyone could tell me how?
    \frac{d}{dx}ln(f(x))= \frac{1}{f(x)}f'(x)
     therefore \quad \frac{1}{f(x)}f'(x)==\frac{200x}{x^2+1}+\frac{6x^2  }{x^3-7}+\frac{1}{x-1}-\frac{10x^9+1}{x^{10}+x+3}
     therefore \quad f'(x)=f(x)\times (\frac{200x}{x^2+1}+\frac{6x^2}{x^3-7}+\frac{1}{x-1}-\frac{10x^9+1}{x^{10}+x+3})
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  3. #3
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    Hello, Kataangel!

    Hi, I need urgent help for this expression as I have no idea how to get its derivative.
    . .
    Why do you say that? You obviously know how to use logs and to differentiate!

    f(x)=(x^2+1)^{100}(x^3-7)^2(x-1)(x^{10}+x+3)^{-1}

    I have to use the ln function but I can only come up to here:

    \frac{d}{dx}ln(f(x))=100\frac{2x}{x^2+1}+2\frac{3x  ^2}{x^3-7}+\frac{1}{x-1}-\frac{10x^9+1}{x^{10}+x+3}

    Specifically I don't know how to get rid of the natural log function on the left side of the equation.
    I would appreciate if anyone could tell me how.
    Write it like this: . {\color{blue}y} \;=\;(x^2+1)^{100}(x^3-7)^2(x-1)(x^{10}+x+3)^{-1}

    Take logs: . \ln(y) \;=\;100\ln(x^2+1) + 2\ln(x^3-7) + \ln(x-1) - \ln(x^{10}+x+3)

    Diffrerentiate: . \frac{1}{y}\cdot\frac{dy}{dx} \;=\;\frac{200x}{x^2+1} + \frac{6x^2}{x^3-7} + \frac{1}{x-1} - \frac{10x^9 + 1}{x^{10}+x+3}

    Then: . \frac{dy}{dx} \;=\;y\bigg[\frac{200x}{x^2+1} + \frac{6x^2}{x^3-7} + \frac{1}{x-1} - \frac{10x^9 + 1}{x^{10}+x+3}\bigg]

    . . . . . \frac {dy}{dx}\;=\;(x^2+1)^{100}(x^3-7)^2(x-1)(x^{10}+x+3)^{-1}\bigg[\frac{200x}{x^2+1} + \frac{6x^2}{x^3-7} + \frac{1}{x-1} - \frac{10x^9 + 1}{x^{10}+x+3}\bigg]


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