I have the following log-likelihood: $$l(\beta) = n \log\beta - (\beta + 1) \sum_{i = 1}^n \log(x_i)$$
the job is to find $l'(\beta)$ and show that the candidate $MLE$ is: $$\hat{\beta} =\frac{n}{\sum_{i=1}^{n}\log (x_i)} $$
so, we know that $x_i$ and $n$ are constants but then when it comes to differentiating $\sum_{i=1}^{n}\log (x_i)$ is when i get confused as in theory the differential of $\sum_{i=1}^{n}\log (x_i)$ should be ${\sum_{i=1}^{n}\frac{1}{x_i}}$ correct?
so this is what i have got to so far: $$l'(\beta) = \frac{n}{\beta}- (1+0){\sum_{i=1}^{n}\frac{1}{x_i}}$$ which simplifies to: $$l'(\beta) = \frac{n}{\beta} - {\sum_{i=1}^{n}\frac{1}{x_i}}$$
so what am I doing wrong? any help please!