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

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