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

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