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

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