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

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