# Differentiation with Log and Summation operators

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.

If $x$ and $n$ are independent of $\beta$, then
$$\frac d{d\beta}-(\beta+1)\sum_{i=1}^n\ln(x_i)=-\sum_{i=1}^n\ln(x_i)\frac d{d\beta}(\beta+1)=-\sum_{i=1}^n\ln(x_i)$$
$$\frac d{dx}cf(x)=c\frac d{dx}f(x)$$
• Thank you :) For future reference, if you were to differentiate with respect to $x$ how would you go about doing that? – Evan Jan 16 '17 at 2:20
• @N.K Well, $x_i$ would just be a function, and so,$$\frac d{dt}\sum_{i=1}^n\ln(x_i(t))=\sum_{i=1}^n\frac d{dt}\ln(x_i(t))$$and I imagine you can take the rest? – Simply Beautiful Art Jan 16 '17 at 12:54