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

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