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Math Help - Differentiate ((x+1)(x+3)^3....(x+99)^99)/......

  1. #1
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    Differentiate ((x+1)(x+3)^3....(x+99)^99)/......

    Having problems with the following question:

    Differentiate:

    ((x+1)(x+3)^3(x+5)^5....(x+99)^99)/((x+2)^2(x+4)^4(x+6)^6...(x+100)^100)

    Any help is greatly appreciated
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  2. #2
    Senior Member MaxJasper's Avatar
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    Lightbulb Re: Differentiate ((x+1)(x+3)^3....(x+99)^99)/......

    Quote Originally Posted by ICanDivideByZero View Post
    Differentiate: ((x+1)(x+3)^3(x+5)^5....(x+99)^99)/((x+2)^2(x+4)^4(x+6)^6...(x+100)^100)
    f[\text{x}]\text{=}\frac{\prod _{k=1}^{2n-1} (x+2k-1)^{2k-1}}{\prod _{k=2}^{2n} (x+2k)^{2k}}

    Try Log[f[x]] for n=3 then differentiate:


    lnf'[x] = \frac{1}{1+x}+\frac{3}{3+x}-\frac{4}{4+x}+\frac{5}{5+x}-\frac{6}{6+x}+\frac{7}{7+x}-\frac{8}{8+x}+\frac{9}{9+x}-\frac{10}{10+x}-\frac{12}{12+x}+\text{...}

    \text{lnf[x]}'\text{=}\sum _{k=1}^n (-1)^{k+1}\frac{2k-1}{x+2k-1}

    And then use inverse Log to find diff of original f[x].

    Needs be refined and terms be checked!
    Last edited by MaxJasper; October 11th 2012 at 06:14 PM.
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