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

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