There are 1 questions in this calculation: for each question, the 1 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ first\ derivative\ of\ function\ (\frac{ln(sin(x) + 1)}{4} - \frac{ln(1 - sin(x))}{4}) - \frac{1}{(2(1 + sin(x)))}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{1}{4}ln(sin(x) + 1) - \frac{1}{4}ln(-sin(x) + 1) - \frac{1}{(2sin(x) + 2)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{1}{4}ln(sin(x) + 1) - \frac{1}{4}ln(-sin(x) + 1) - \frac{1}{(2sin(x) + 2)}\right)}{dx}\\=&\frac{\frac{1}{4}(cos(x) + 0)}{(sin(x) + 1)} - \frac{\frac{1}{4}(-cos(x) + 0)}{(-sin(x) + 1)} - (\frac{-(2cos(x) + 0)}{(2sin(x) + 2)^{2}})\\=&\frac{cos(x)}{4(sin(x) + 1)} + \frac{cos(x)}{4(-sin(x) + 1)} + \frac{2cos(x)}{(2sin(x) + 2)^{2}}\\ \end{split}\end{equation} \]

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