Is this just the product rule? I have this in my notes but I didn't think anything of it and now I'm wondering how this happens?
Edit: Im working with maximum likelihood estimation and in my notes I have that the likelihood funciton $=L(x;\theta)=\prod_{i=1}^nf(x;\theta)$ where $x$ is the variable and $\theta$ is the parameter of a probability distribution. To estimate I was told that we take the log of the likelihood function, i.e. $\ln(L)$, then take its derivative to estimate the parameter. The function I'm working with is $f(x;\theta)=(\theta +1)x^{\theta}$. So $$ L(x;\theta)=\prod_{i=1}^n(\theta +1)x_i^{\theta}=(\theta+1)^n\prod_{i=1}^nx_i^{\theta}. $$ Now $$ \ln(L(x;\theta))=n*\ln(\theta+1)+\theta \ln\left(\prod_{i=1}^nx_i\right). $$ Here's where I'm confused, I have in my notes that $$ \frac{d(\ln L)}{d\theta}=\frac{n}{\theta+1}+\sum_{i=1}^n\ln(x_i). $$ Why does the product of $x_i$ become the summation of $x_i$?