There are 1 questions in this calculation: for each question, the 3 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ third\ derivative\ of\ function\ \frac{((1 - x){e}^{x})}{(1 + {x}^{2})}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{-x{e}^{x}}{(x^{2} + 1)} + \frac{{e}^{x}}{(x^{2} + 1)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{-x{e}^{x}}{(x^{2} + 1)} + \frac{{e}^{x}}{(x^{2} + 1)}\right)}{dx}\\=&-(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x{e}^{x} - \frac{{e}^{x}}{(x^{2} + 1)} - \frac{x({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))}{(x^{2} + 1)} + (\frac{-(2x + 0)}{(x^{2} + 1)^{2}}){e}^{x} + \frac{({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))}{(x^{2} + 1)}\\=&\frac{2x^{2}{e}^{x}}{(x^{2} + 1)^{2}} - \frac{2x{e}^{x}}{(x^{2} + 1)^{2}} - \frac{x{e}^{x}}{(x^{2} + 1)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{2x^{2}{e}^{x}}{(x^{2} + 1)^{2}} - \frac{2x{e}^{x}}{(x^{2} + 1)^{2}} - \frac{x{e}^{x}}{(x^{2} + 1)}\right)}{dx}\\=&2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x^{2}{e}^{x} + \frac{2*2x{e}^{x}}{(x^{2} + 1)^{2}} + \frac{2x^{2}({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))}{(x^{2} + 1)^{2}} - 2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x{e}^{x} - \frac{2{e}^{x}}{(x^{2} + 1)^{2}} - \frac{2x({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))}{(x^{2} + 1)^{2}} - (\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x{e}^{x} - \frac{{e}^{x}}{(x^{2} + 1)} - \frac{x({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))}{(x^{2} + 1)}\\=&\frac{-8x^{3}{e}^{x}}{(x^{2} + 1)^{3}} + \frac{2x{e}^{x}}{(x^{2} + 1)^{2}} + \frac{4x^{2}{e}^{x}}{(x^{2} + 1)^{2}} + \frac{8x^{2}{e}^{x}}{(x^{2} + 1)^{3}} - \frac{2{e}^{x}}{(x^{2} + 1)^{2}} - \frac{{e}^{x}}{(x^{2} + 1)} - \frac{x{e}^{x}}{(x^{2} + 1)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-8x^{3}{e}^{x}}{(x^{2} + 1)^{3}} + \frac{2x{e}^{x}}{(x^{2} + 1)^{2}} + \frac{4x^{2}{e}^{x}}{(x^{2} + 1)^{2}} + \frac{8x^{2}{e}^{x}}{(x^{2} + 1)^{3}} - \frac{2{e}^{x}}{(x^{2} + 1)^{2}} - \frac{{e}^{x}}{(x^{2} + 1)} - \frac{x{e}^{x}}{(x^{2} + 1)}\right)}{dx}\\=&-8(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{3}{e}^{x} - \frac{8*3x^{2}{e}^{x}}{(x^{2} + 1)^{3}} - \frac{8x^{3}({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))}{(x^{2} + 1)^{3}} + 2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x{e}^{x} + \frac{2{e}^{x}}{(x^{2} + 1)^{2}} + \frac{2x({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))}{(x^{2} + 1)^{2}} + 4(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x^{2}{e}^{x} + \frac{4*2x{e}^{x}}{(x^{2} + 1)^{2}} + \frac{4x^{2}({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))}{(x^{2} + 1)^{2}} + 8(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{2}{e}^{x} + \frac{8*2x{e}^{x}}{(x^{2} + 1)^{3}} + \frac{8x^{2}({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))}{(x^{2} + 1)^{3}} - 2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}}){e}^{x} - \frac{2({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))}{(x^{2} + 1)^{2}} - (\frac{-(2x + 0)}{(x^{2} + 1)^{2}}){e}^{x} - \frac{({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))}{(x^{2} + 1)} - (\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x{e}^{x} - \frac{{e}^{x}}{(x^{2} + 1)} - \frac{x({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))}{(x^{2} + 1)}\\=&\frac{48x^{4}{e}^{x}}{(x^{2} + 1)^{4}} - \frac{24x^{2}{e}^{x}}{(x^{2} + 1)^{3}} - \frac{24x^{3}{e}^{x}}{(x^{2} + 1)^{3}} + \frac{12x{e}^{x}}{(x^{2} + 1)^{2}} + \frac{6x^{2}{e}^{x}}{(x^{2} + 1)^{2}} - \frac{48x^{3}{e}^{x}}{(x^{2} + 1)^{4}} + \frac{24x{e}^{x}}{(x^{2} + 1)^{3}} - \frac{2{e}^{x}}{(x^{2} + 1)} - \frac{x{e}^{x}}{(x^{2} + 1)}\\ \end{split}\end{equation} \]

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