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

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