There are 1 questions in this calculation: for each question, the 3 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ third\ derivative\ of\ function\ \frac{x({x}^{2} + 3)}{(3{x}^{2} + 1)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{x^{3}}{(3x^{2} + 1)} + \frac{3x}{(3x^{2} + 1)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{x^{3}}{(3x^{2} + 1)} + \frac{3x}{(3x^{2} + 1)}\right)}{dx}\\=&(\frac{-(3*2x + 0)}{(3x^{2} + 1)^{2}})x^{3} + \frac{3x^{2}}{(3x^{2} + 1)} + 3(\frac{-(3*2x + 0)}{(3x^{2} + 1)^{2}})x + \frac{3}{(3x^{2} + 1)}\\=&\frac{-6x^{4}}{(3x^{2} + 1)^{2}} + \frac{3x^{2}}{(3x^{2} + 1)} - \frac{18x^{2}}{(3x^{2} + 1)^{2}} + \frac{3}{(3x^{2} + 1)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-6x^{4}}{(3x^{2} + 1)^{2}} + \frac{3x^{2}}{(3x^{2} + 1)} - \frac{18x^{2}}{(3x^{2} + 1)^{2}} + \frac{3}{(3x^{2} + 1)}\right)}{dx}\\=&-6(\frac{-2(3*2x + 0)}{(3x^{2} + 1)^{3}})x^{4} - \frac{6*4x^{3}}{(3x^{2} + 1)^{2}} + 3(\frac{-(3*2x + 0)}{(3x^{2} + 1)^{2}})x^{2} + \frac{3*2x}{(3x^{2} + 1)} - 18(\frac{-2(3*2x + 0)}{(3x^{2} + 1)^{3}})x^{2} - \frac{18*2x}{(3x^{2} + 1)^{2}} + 3(\frac{-(3*2x + 0)}{(3x^{2} + 1)^{2}})\\=&\frac{72x^{5}}{(3x^{2} + 1)^{3}} - \frac{42x^{3}}{(3x^{2} + 1)^{2}} + \frac{6x}{(3x^{2} + 1)} + \frac{216x^{3}}{(3x^{2} + 1)^{3}} - \frac{54x}{(3x^{2} + 1)^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{72x^{5}}{(3x^{2} + 1)^{3}} - \frac{42x^{3}}{(3x^{2} + 1)^{2}} + \frac{6x}{(3x^{2} + 1)} + \frac{216x^{3}}{(3x^{2} + 1)^{3}} - \frac{54x}{(3x^{2} + 1)^{2}}\right)}{dx}\\=&72(\frac{-3(3*2x + 0)}{(3x^{2} + 1)^{4}})x^{5} + \frac{72*5x^{4}}{(3x^{2} + 1)^{3}} - 42(\frac{-2(3*2x + 0)}{(3x^{2} + 1)^{3}})x^{3} - \frac{42*3x^{2}}{(3x^{2} + 1)^{2}} + 6(\frac{-(3*2x + 0)}{(3x^{2} + 1)^{2}})x + \frac{6}{(3x^{2} + 1)} + 216(\frac{-3(3*2x + 0)}{(3x^{2} + 1)^{4}})x^{3} + \frac{216*3x^{2}}{(3x^{2} + 1)^{3}} - 54(\frac{-2(3*2x + 0)}{(3x^{2} + 1)^{3}})x - \frac{54}{(3x^{2} + 1)^{2}}\\=&\frac{-1296x^{6}}{(3x^{2} + 1)^{4}} + \frac{864x^{4}}{(3x^{2} + 1)^{3}} - \frac{162x^{2}}{(3x^{2} + 1)^{2}} - \frac{3888x^{4}}{(3x^{2} + 1)^{4}} + \frac{1296x^{2}}{(3x^{2} + 1)^{3}} - \frac{54}{(3x^{2} + 1)^{2}} + \frac{6}{(3x^{2} + 1)}\\ \end{split}\end{equation} \]

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