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

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