# Math Help - Derivative of General Beta 2 PDF

1. ## 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

2. ## 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:
$g'= (p+q)(1+ (x/b)^a)^{p+q-1}\left(a\left(\frac{x^{a-1}}{b^a}\right)\right)$