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

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