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Math Help - More differentiation help

  1. #1
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    More differentiation help

    Find in a simplified form, the derivative of the function
    \frac{2+\ln (1+x)^2}{2-\ln (1-x)^2}

    Used the quotient rule:
    \frac{(2-2\ln (1-x))(\frac{2(1)}{(1+x)})-(2+2\ln (1+x))(\frac{-2(-1)}{(1-x)})}{(2-2\ln (1-x))^2}
    \frac{\frac{4-4\ln(1-x)}{1+x}-\frac{4+4\ln (1+x)}{1-x}}{(2-2\ln(1-x))^2}
    \frac{\frac{4(1-x)(1-\ln (1-x))-4(1+x)(1+\ln (1+x))}{1-x^2}}{(2-2\ln (1-x))^2}
    \frac{4(1-x)-4(1-x)(\ln (1-x))-4(1+x)-4(1+x)(\ln (1+x))}{(1-x^2)(2-2\ln (1-x))^2}
    \frac{4(x-1)(\ln (1-x))-4(1+x)(\ln (1+x))-8x}{(1-x^2)(2-2\ln (1-x))^2}
    The answer is supposed to be
    \frac{(x-1)\ln (1-x)-(1+x)\ln (1+x)-2x}{(1-x^2)(2-2\ln (1-x))^2}
    so I have a factor 4. Where have i missed out?
    Thanks
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  2. #2
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    Quote Originally Posted by arze View Post
    Find in a simplified form, the derivative of the function
    \frac{2+\ln (1+x)^2}{2-\ln (1-x)^2}

    Used the quotient rule:
    \frac{(2-2\ln (1-x))(\frac{2(1)}{(1+x)})-(2+2\ln (1+x))(\frac{-2(-1)}{(1-x)})}{(2-2\ln (1-x))^2}
    \frac{\frac{4-4\ln(1-x)}{1+x}-\frac{4+4\ln (1+x)}{1-x}}{(2-2\ln(1-x))^2}
    \frac{\frac{4(1-x)(1-\ln (1-x))-4(1+x)(1+\ln (1+x))}{1-x^2}}{(2-2\ln (1-x))^2}
    \frac{4(1-x)-4(1-x)(\ln (1-x))-4(1+x)-4(1+x)(\ln (1+x))}{(1-x^2)(2-2\ln (1-x))^2}
    \frac{4(x-1)(\ln (1-x))-4(1+x)(\ln (1+x))-8x}{(1-x^2)(2-2\ln (1-x))^2} (**)
    The answer is supposed to be
    \frac{(x-1)\ln (1-x)-(1+x)\ln (1+x)-2x}{(1-x^2)(2-2\ln (1-x))^2}
    so I have a factor 4. Where have i missed out?
    Thanks
    Who says that's the correct answer? Once you cancel out the 4's from your final answer (** above), then Maple agrees with you!
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  3. #3
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    I don't exactly get what you mean. I have the factor 4, but i can't just eliminate it.
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  4. #4
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    Quote Originally Posted by arze View Post
    \frac{4(x-1)(\ln (1-x))-4(1+x)(\ln (1+x))-8x}{(1-x^2)(2-2\ln (1-x))^2}
    From your answer, notice the factor of 4 in the denominator

    (1-x^2)(2-2\ln (1-x))^2 = (1-x^2) 2^2 (1 - \ln(1-x))^2 = 4 (1-x^2) (1 - \ln(1-x))^2
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  5. #5
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    so do you mean that the answer given is wrong?
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  6. #6
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    Quote Originally Posted by arze View Post
    so do you mean that the answer given is wrong?
    I believe so. I gave this problem to Maple (a symbolic mathematics package) and like I said, it agrees with you.
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  7. #7
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    oh, that's what you meant by maple, i was wondering what you were talking about. haha
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