The aim is to find the maximum likelihood estimator for theta. $f(x)$ is given and we can assume that $1\le x\le-1$. I have completed the steps seen in the image, however I am having difficulty differentiating the logarithmic summation.
$$f(x) = \frac 1 2 (1+\theta x)$$ $$ L(\theta \mid x_1,\ldots,x_n) = \frac 1 {2^n}\prod_{i=1}^n(1+\theta x_i)$$ \begin{align} \ell & = \ln\left[ \frac 1 {2^n}\prod_{i=1}^n(1+\theta x_i) \right] \\[10pt] & = \ln\frac1{2^n} + \ln\prod_{i=1}^n (1+\theta x_i) \\[10pt] & = n\ln\frac 1 2 + \sum_{i=1}^n \ln(1+\theta x_i) \end{align} $$ \frac{\partial\ell}{\partial\theta} = 0 + \frac{\partial\sum_{i=1}^n \ln(1+\theta x_i)}{\partial\theta} $$
Thanks in advance!
You want linearity of differentiation and the chain rule:
$$ \begin{align} \frac{\partial \sum \ln (1 + \theta x_i)}{\partial \theta} &= \sum \frac{1}{1 + \theta x_i} \frac{\partial (1 + \theta x_i)}{\partial \theta} \\ &= \sum \frac{x_i}{1 + \theta x_i} = 0 \end{align} $$
I can't see a solution to that though... I would use the Newton-Rhapson method.