There are 1 questions in this calculation: for each question, the 2 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ second\ derivative\ of\ function\ arctan(\frac{2x}{(2 - {x}^{2})})\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = arctan(\frac{2x}{(-x^{2} + 2)})\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( arctan(\frac{2x}{(-x^{2} + 2)})\right)}{dx}\\=&(\frac{(2(\frac{-(-2x + 0)}{(-x^{2} + 2)^{2}})x + \frac{2}{(-x^{2} + 2)})}{(1 + (\frac{2x}{(-x^{2} + 2)})^{2})})\\=&\frac{4x^{2}}{(-x^{2} + 2)^{2}(\frac{4x^{2}}{(-x^{2} + 2)^{2}} + 1)} + \frac{2}{(-x^{2} + 2)(\frac{4x^{2}}{(-x^{2} + 2)^{2}} + 1)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{4x^{2}}{(-x^{2} + 2)^{2}(\frac{4x^{2}}{(-x^{2} + 2)^{2}} + 1)} + \frac{2}{(-x^{2} + 2)(\frac{4x^{2}}{(-x^{2} + 2)^{2}} + 1)}\right)}{dx}\\=&\frac{4(\frac{-2(-2x + 0)}{(-x^{2} + 2)^{3}})x^{2}}{(\frac{4x^{2}}{(-x^{2} + 2)^{2}} + 1)} + \frac{4(\frac{-(4(\frac{-2(-2x + 0)}{(-x^{2} + 2)^{3}})x^{2} + \frac{4*2x}{(-x^{2} + 2)^{2}} + 0)}{(\frac{4x^{2}}{(-x^{2} + 2)^{2}} + 1)^{2}})x^{2}}{(-x^{2} + 2)^{2}} + \frac{4*2x}{(-x^{2} + 2)^{2}(\frac{4x^{2}}{(-x^{2} + 2)^{2}} + 1)} + \frac{2(\frac{-(-2x + 0)}{(-x^{2} + 2)^{2}})}{(\frac{4x^{2}}{(-x^{2} + 2)^{2}} + 1)} + \frac{2(\frac{-(4(\frac{-2(-2x + 0)}{(-x^{2} + 2)^{3}})x^{2} + \frac{4*2x}{(-x^{2} + 2)^{2}} + 0)}{(\frac{4x^{2}}{(-x^{2} + 2)^{2}} + 1)^{2}})}{(-x^{2} + 2)}\\=&\frac{16x^{3}}{(-x^{2} + 2)^{3}(\frac{4x^{2}}{(-x^{2} + 2)^{2}} + 1)} - \frac{64x^{5}}{(-x^{2} + 2)^{5}(\frac{4x^{2}}{(-x^{2} + 2)^{2}} + 1)^{2}} - \frac{32x^{3}}{(-x^{2} + 2)^{4}(\frac{4x^{2}}{(-x^{2} + 2)^{2}} + 1)^{2}} + \frac{8x}{(\frac{4x^{2}}{(-x^{2} + 2)^{2}} + 1)(-x^{2} + 2)^{2}} + \frac{4x}{(-x^{2} + 2)^{2}(\frac{4x^{2}}{(-x^{2} + 2)^{2}} + 1)} - \frac{32x^{3}}{(\frac{4x^{2}}{(-x^{2} + 2)^{2}} + 1)^{2}(-x^{2} + 2)^{4}} - \frac{16x}{(\frac{4x^{2}}{(-x^{2} + 2)^{2}} + 1)^{2}(-x^{2} + 2)^{3}}\\ \end{split}\end{equation} \]

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