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Math Help - Derivative of General Beta 2 PDF

  1. #1
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    Derivative of General Beta 2 PDF

    Need help taking the derivative of the following function w.r.t x. (this is the General Beta of the Second Kind density function):

    \frac{a}{bB(p,q)}\frac{(x/b)^{ap-1}}{(1+(x/b)^a)^{p+q}}

    B(.) is the beta function.

    I don't expect anyone to give me the answer, but would appreciate some advice on what strategy I should use, especially as regards the derivative of the beta function.

    Thanks
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  2. #2
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    Re: Derivative of General Beta 2 PDF

    This is F(x)= A\frac{f(x)}{g(x)} so use the quotient rule:

    F'= A\frac{f' g- fg'}{g^2}

    Here, f(x)= (x/b)^{ap-1}= \frac{x^{ap-1}}{b^{ap-1}} so f'= (ap-1)\frac{x^{ap-2}}{b^{ap-1}}.

    g(x)= (1+ (x/b)^a)^{p+q}

    We need the "chain rule" for that:
    g'= (p+q)(1+ (x/b)^a)^{p+q-1}\left(a\left(\frac{x^{a-1}}{b^a}\right)\right)
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