How can I get the result?

Question

I got a formula

$$ - \frac{ \frac {\partial B(\lambda, \xi)}{\partial \lambda}}{B(\lambda, \xi)} = - \Psi(\lambda) + \Psi(\lambda + \xi) $$

Where $B$ is Beta function and $\Psi$ is digamma function.

How can I get the result $- \Psi(\lambda) + \Psi(\lambda + \xi)$ ?

Answer 1

$$B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$$

$$\log B(x,y)=\log \Gamma(x)+\log \Gamma(y)-\log \Gamma(x+y)$$

$$\frac{\partial}{\partial x} \log B(x,y)= \psi(x)-\psi(x+y)$$

That follows from the definition of digamma function.

And from the chain rule:

$$\frac{\partial}{\partial x} \log B(x,y)=\frac{1}{B(x,y)} \frac{\partial}{\partial x} B(x,y)$$