There are 1 questions in this calculation: for each question, the 1 derivative of x is calculated.
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\[ \begin{equation}\begin{split}[1/1]Find\ the\ first\ derivative\ of\ function\ e^{{\frac{1}{x}}^{2}}arctan(\frac{({x}^{2} + x + 1)}{((x + 1)(x - 2))})\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = e^{\frac{1}{x^{2}}}arctan(\frac{x^{2}}{(x^{2} - x - 2)} + \frac{x}{(x^{2} - x - 2)} + \frac{1}{(x^{2} - x - 2)})\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( e^{\frac{1}{x^{2}}}arctan(\frac{x^{2}}{(x^{2} - x - 2)} + \frac{x}{(x^{2} - x - 2)} + \frac{1}{(x^{2} - x - 2)})\right)}{dx}\\=&\frac{e^{\frac{1}{x^{2}}}*-2arctan(\frac{x^{2}}{(x^{2} - x - 2)} + \frac{x}{(x^{2} - x - 2)} + \frac{1}{(x^{2} - x - 2)})}{x^{3}} + e^{\frac{1}{x^{2}}}(\frac{((\frac{-(2x - 1 + 0)}{(x^{2} - x - 2)^{2}})x^{2} + \frac{2x}{(x^{2} - x - 2)} + (\frac{-(2x - 1 + 0)}{(x^{2} - x - 2)^{2}})x + \frac{1}{(x^{2} - x - 2)} + (\frac{-(2x - 1 + 0)}{(x^{2} - x - 2)^{2}}))}{(1 + (\frac{x^{2}}{(x^{2} - x - 2)} + \frac{x}{(x^{2} - x - 2)} + \frac{1}{(x^{2} - x - 2)})^{2})})\\=&\frac{-2e^{\frac{1}{x^{2}}}arctan(\frac{x^{2}}{(x^{2} - x - 2)} + \frac{x}{(x^{2} - x - 2)} + \frac{1}{(x^{2} - x - 2)})}{x^{3}} - \frac{2x^{3}e^{\frac{1}{x^{2}}}}{(x^{2} - x - 2)^{2}(\frac{x^{4}}{(x^{2} - x - 2)^{2}} + \frac{2x^{3}}{(x^{2} - x - 2)^{2}} + \frac{3x^{2}}{(x^{2} - x - 2)^{2}} + \frac{2x}{(x^{2} - x - 2)^{2}} + \frac{1}{(x^{2} - x - 2)^{2}} + 1)} - \frac{x^{2}e^{\frac{1}{x^{2}}}}{(x^{2} - x - 2)^{2}(\frac{x^{4}}{(x^{2} - x - 2)^{2}} + \frac{2x^{3}}{(x^{2} - x - 2)^{2}} + \frac{3x^{2}}{(x^{2} - x - 2)^{2}} + \frac{2x}{(x^{2} - x - 2)^{2}} + \frac{1}{(x^{2} - x - 2)^{2}} + 1)} + \frac{2xe^{\frac{1}{x^{2}}}}{(x^{2} - x - 2)(\frac{x^{4}}{(x^{2} - x - 2)^{2}} + \frac{2x^{3}}{(x^{2} - x - 2)^{2}} + \frac{3x^{2}}{(x^{2} - x - 2)^{2}} + \frac{2x}{(x^{2} - x - 2)^{2}} + \frac{1}{(x^{2} - x - 2)^{2}} + 1)} - \frac{xe^{\frac{1}{x^{2}}}}{(x^{2} - x - 2)^{2}(\frac{x^{4}}{(x^{2} - x - 2)^{2}} + \frac{2x^{3}}{(x^{2} - x - 2)^{2}} + \frac{3x^{2}}{(x^{2} - x - 2)^{2}} + \frac{2x}{(x^{2} - x - 2)^{2}} + \frac{1}{(x^{2} - x - 2)^{2}} + 1)} + \frac{e^{\frac{1}{x^{2}}}}{(x^{2} - x - 2)(\frac{x^{4}}{(x^{2} - x - 2)^{2}} + \frac{2x^{3}}{(x^{2} - x - 2)^{2}} + \frac{3x^{2}}{(x^{2} - x - 2)^{2}} + \frac{2x}{(x^{2} - x - 2)^{2}} + \frac{1}{(x^{2} - x - 2)^{2}} + 1)} + \frac{e^{\frac{1}{x^{2}}}}{(x^{2} - x - 2)^{2}(\frac{x^{4}}{(x^{2} - x - 2)^{2}} + \frac{2x^{3}}{(x^{2} - x - 2)^{2}} + \frac{3x^{2}}{(x^{2} - x - 2)^{2}} + \frac{2x}{(x^{2} - x - 2)^{2}} + \frac{1}{(x^{2} - x - 2)^{2}} + 1)}\\ \end{split}\end{equation} \]

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