I'm trying to differentiate this function with respect to $\beta$ but there isn't much help online. If someone could explain to me why it's wrong or point me in the right direction to correct it I'd greatly appreciate it.
The summation symbol confuses me most as I don't know what happens to it exactly when differentiating.
So I'm trying to differentiate the following:
$-(\beta+1)\sum\limits_{i=1}^n \log_ex_i$
I'd guess you get something like:
$\frac{-\beta+1}{\sum\limits_{i=1}^n x_i}$
I don't have any real reason to think this and I'd guess it's wrong as I differentiate with respect to $x$ so any help really would be appreciated.
Best Answer
If $x$ and $n$ are independent of $\beta$, then
$$\frac d{d\beta}-(\beta+1)\sum_{i=1}^n\ln(x_i)=-\sum_{i=1}^n\ln(x_i)\frac d{d\beta}(\beta+1)=-\sum_{i=1}^n\ln(x_i)$$
Since we have
$$\frac d{dx}cf(x)=c\frac d{dx}f(x)$$