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

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