There are 1 questions in this calculation: for each question, the 1 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ first\ derivative\ of\ function\ (\frac{-1}{4})ln(\frac{(tan(\frac{x}{2}) - 2)}{(tan(\frac{x}{2}) - 2)})\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{-1}{4}ln(\frac{tan(\frac{1}{2}x)}{(tan(\frac{1}{2}x) - 2)} - \frac{2}{(tan(\frac{1}{2}x) - 2)})\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{-1}{4}ln(\frac{tan(\frac{1}{2}x)}{(tan(\frac{1}{2}x) - 2)} - \frac{2}{(tan(\frac{1}{2}x) - 2)})\right)}{dx}\\=&\frac{\frac{-1}{4}((\frac{-(sec^{2}(\frac{1}{2}x)(\frac{1}{2}) + 0)}{(tan(\frac{1}{2}x) - 2)^{2}})tan(\frac{1}{2}x) + \frac{sec^{2}(\frac{1}{2}x)(\frac{1}{2})}{(tan(\frac{1}{2}x) - 2)} - 2(\frac{-(sec^{2}(\frac{1}{2}x)(\frac{1}{2}) + 0)}{(tan(\frac{1}{2}x) - 2)^{2}}))}{(\frac{tan(\frac{1}{2}x)}{(tan(\frac{1}{2}x) - 2)} - \frac{2}{(tan(\frac{1}{2}x) - 2)})}\\=&\frac{tan(\frac{1}{2}x)sec^{2}(\frac{1}{2}x)}{8(\frac{tan(\frac{1}{2}x)}{(tan(\frac{1}{2}x) - 2)} - \frac{2}{(tan(\frac{1}{2}x) - 2)})(tan(\frac{1}{2}x) - 2)^{2}} - \frac{sec^{2}(\frac{1}{2}x)}{8(tan(\frac{1}{2}x) - 2)(\frac{tan(\frac{1}{2}x)}{(tan(\frac{1}{2}x) - 2)} - \frac{2}{(tan(\frac{1}{2}x) - 2)})} - \frac{sec^{2}(\frac{1}{2}x)}{4(\frac{tan(\frac{1}{2}x)}{(tan(\frac{1}{2}x) - 2)} - \frac{2}{(tan(\frac{1}{2}x) - 2)})(tan(\frac{1}{2}x) - 2)^{2}}\\ \end{split}\end{equation} \]

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