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

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