There are 1 questions in this calculation: for each question, the 3 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ third\ derivative\ of\ function\ \frac{({x}^{2} + 1)}{({x}^{3} + 1)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{x^{2}}{(x^{3} + 1)} + \frac{1}{(x^{3} + 1)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{x^{2}}{(x^{3} + 1)} + \frac{1}{(x^{3} + 1)}\right)}{dx}\\=&(\frac{-(3x^{2} + 0)}{(x^{3} + 1)^{2}})x^{2} + \frac{2x}{(x^{3} + 1)} + (\frac{-(3x^{2} + 0)}{(x^{3} + 1)^{2}})\\=&\frac{-3x^{4}}{(x^{3} + 1)^{2}} + \frac{2x}{(x^{3} + 1)} - \frac{3x^{2}}{(x^{3} + 1)^{2}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-3x^{4}}{(x^{3} + 1)^{2}} + \frac{2x}{(x^{3} + 1)} - \frac{3x^{2}}{(x^{3} + 1)^{2}}\right)}{dx}\\=&-3(\frac{-2(3x^{2} + 0)}{(x^{3} + 1)^{3}})x^{4} - \frac{3*4x^{3}}{(x^{3} + 1)^{2}} + 2(\frac{-(3x^{2} + 0)}{(x^{3} + 1)^{2}})x + \frac{2}{(x^{3} + 1)} - 3(\frac{-2(3x^{2} + 0)}{(x^{3} + 1)^{3}})x^{2} - \frac{3*2x}{(x^{3} + 1)^{2}}\\=&\frac{18x^{6}}{(x^{3} + 1)^{3}} - \frac{18x^{3}}{(x^{3} + 1)^{2}} + \frac{18x^{4}}{(x^{3} + 1)^{3}} - \frac{6x}{(x^{3} + 1)^{2}} + \frac{2}{(x^{3} + 1)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{18x^{6}}{(x^{3} + 1)^{3}} - \frac{18x^{3}}{(x^{3} + 1)^{2}} + \frac{18x^{4}}{(x^{3} + 1)^{3}} - \frac{6x}{(x^{3} + 1)^{2}} + \frac{2}{(x^{3} + 1)}\right)}{dx}\\=&18(\frac{-3(3x^{2} + 0)}{(x^{3} + 1)^{4}})x^{6} + \frac{18*6x^{5}}{(x^{3} + 1)^{3}} - 18(\frac{-2(3x^{2} + 0)}{(x^{3} + 1)^{3}})x^{3} - \frac{18*3x^{2}}{(x^{3} + 1)^{2}} + 18(\frac{-3(3x^{2} + 0)}{(x^{3} + 1)^{4}})x^{4} + \frac{18*4x^{3}}{(x^{3} + 1)^{3}} - 6(\frac{-2(3x^{2} + 0)}{(x^{3} + 1)^{3}})x - \frac{6}{(x^{3} + 1)^{2}} + 2(\frac{-(3x^{2} + 0)}{(x^{3} + 1)^{2}})\\=&\frac{-162x^{8}}{(x^{3} + 1)^{4}} + \frac{216x^{5}}{(x^{3} + 1)^{3}} - \frac{60x^{2}}{(x^{3} + 1)^{2}} - \frac{162x^{6}}{(x^{3} + 1)^{4}} + \frac{108x^{3}}{(x^{3} + 1)^{3}} - \frac{6}{(x^{3} + 1)^{2}}\\ \end{split}\end{equation} \]

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