There are 1 questions in this calculation: for each question, the 1 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ first\ derivative\ of\ function\ \frac{2ln(\frac{({x}^{\frac{1}{2}} + 1)}{(x - {x}^{\frac{1}{2}} + 1)})}{3}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{2}{3}ln(\frac{x^{\frac{1}{2}}}{(x - x^{\frac{1}{2}} + 1)} + \frac{1}{(x - x^{\frac{1}{2}} + 1)})\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{2}{3}ln(\frac{x^{\frac{1}{2}}}{(x - x^{\frac{1}{2}} + 1)} + \frac{1}{(x - x^{\frac{1}{2}} + 1)})\right)}{dx}\\=&\frac{\frac{2}{3}((\frac{-(1 - \frac{\frac{1}{2}}{x^{\frac{1}{2}}} + 0)}{(x - x^{\frac{1}{2}} + 1)^{2}})x^{\frac{1}{2}} + \frac{\frac{1}{2}}{(x - x^{\frac{1}{2}} + 1)x^{\frac{1}{2}}} + (\frac{-(1 - \frac{\frac{1}{2}}{x^{\frac{1}{2}}} + 0)}{(x - x^{\frac{1}{2}} + 1)^{2}}))}{(\frac{x^{\frac{1}{2}}}{(x - x^{\frac{1}{2}} + 1)} + \frac{1}{(x - x^{\frac{1}{2}} + 1)})}\\=&\frac{-2x^{\frac{1}{2}}}{3(x - x^{\frac{1}{2}} + 1)^{2}(\frac{x^{\frac{1}{2}}}{(x - x^{\frac{1}{2}} + 1)} + \frac{1}{(x - x^{\frac{1}{2}} + 1)})} + \frac{1}{3(x - x^{\frac{1}{2}} + 1)(\frac{x^{\frac{1}{2}}}{(x - x^{\frac{1}{2}} + 1)} + \frac{1}{(x - x^{\frac{1}{2}} + 1)})x^{\frac{1}{2}}} + \frac{1}{3(x - x^{\frac{1}{2}} + 1)^{2}(\frac{x^{\frac{1}{2}}}{(x - x^{\frac{1}{2}} + 1)} + \frac{1}{(x - x^{\frac{1}{2}} + 1)})x^{\frac{1}{2}}} - \frac{1}{3(x - x^{\frac{1}{2}} + 1)^{2}(\frac{x^{\frac{1}{2}}}{(x - x^{\frac{1}{2}} + 1)} + \frac{1}{(x - x^{\frac{1}{2}} + 1)})}\\ \end{split}\end{equation} \]

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