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

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