There are 1 questions in this calculation: for each question, the 3 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ third\ derivative\ of\ function\ \frac{(x + 3)}{((x - 3)(x - 2))}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{x}{(x^{2} - 5x + 6)} + \frac{3}{(x^{2} - 5x + 6)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{x}{(x^{2} - 5x + 6)} + \frac{3}{(x^{2} - 5x + 6)}\right)}{dx}\\=&(\frac{-(2x - 5 + 0)}{(x^{2} - 5x + 6)^{2}})x + \frac{1}{(x^{2} - 5x + 6)} + 3(\frac{-(2x - 5 + 0)}{(x^{2} - 5x + 6)^{2}})\\=&\frac{-2x^{2}}{(x^{2} - 5x + 6)^{2}} - \frac{x}{(x^{2} - 5x + 6)^{2}} + \frac{1}{(x^{2} - 5x + 6)} + \frac{15}{(x^{2} - 5x + 6)^{2}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2x^{2}}{(x^{2} - 5x + 6)^{2}} - \frac{x}{(x^{2} - 5x + 6)^{2}} + \frac{1}{(x^{2} - 5x + 6)} + \frac{15}{(x^{2} - 5x + 6)^{2}}\right)}{dx}\\=&-2(\frac{-2(2x - 5 + 0)}{(x^{2} - 5x + 6)^{3}})x^{2} - \frac{2*2x}{(x^{2} - 5x + 6)^{2}} - (\frac{-2(2x - 5 + 0)}{(x^{2} - 5x + 6)^{3}})x - \frac{1}{(x^{2} - 5x + 6)^{2}} + (\frac{-(2x - 5 + 0)}{(x^{2} - 5x + 6)^{2}}) + 15(\frac{-2(2x - 5 + 0)}{(x^{2} - 5x + 6)^{3}})\\=&\frac{8x^{3}}{(x^{2} - 5x + 6)^{3}} - \frac{16x^{2}}{(x^{2} - 5x + 6)^{3}} - \frac{6x}{(x^{2} - 5x + 6)^{2}} - \frac{70x}{(x^{2} - 5x + 6)^{3}} + \frac{4}{(x^{2} - 5x + 6)^{2}} + \frac{150}{(x^{2} - 5x + 6)^{3}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{8x^{3}}{(x^{2} - 5x + 6)^{3}} - \frac{16x^{2}}{(x^{2} - 5x + 6)^{3}} - \frac{6x}{(x^{2} - 5x + 6)^{2}} - \frac{70x}{(x^{2} - 5x + 6)^{3}} + \frac{4}{(x^{2} - 5x + 6)^{2}} + \frac{150}{(x^{2} - 5x + 6)^{3}}\right)}{dx}\\=&8(\frac{-3(2x - 5 + 0)}{(x^{2} - 5x + 6)^{4}})x^{3} + \frac{8*3x^{2}}{(x^{2} - 5x + 6)^{3}} - 16(\frac{-3(2x - 5 + 0)}{(x^{2} - 5x + 6)^{4}})x^{2} - \frac{16*2x}{(x^{2} - 5x + 6)^{3}} - 6(\frac{-2(2x - 5 + 0)}{(x^{2} - 5x + 6)^{3}})x - \frac{6}{(x^{2} - 5x + 6)^{2}} - 70(\frac{-3(2x - 5 + 0)}{(x^{2} - 5x + 6)^{4}})x - \frac{70}{(x^{2} - 5x + 6)^{3}} + 4(\frac{-2(2x - 5 + 0)}{(x^{2} - 5x + 6)^{3}}) + 150(\frac{-3(2x - 5 + 0)}{(x^{2} - 5x + 6)^{4}})\\=&\frac{-48x^{4}}{(x^{2} - 5x + 6)^{4}} + \frac{216x^{3}}{(x^{2} - 5x + 6)^{4}} + \frac{48x^{2}}{(x^{2} - 5x + 6)^{3}} + \frac{180x^{2}}{(x^{2} - 5x + 6)^{4}} - \frac{108x}{(x^{2} - 5x + 6)^{3}} - \frac{1950x}{(x^{2} - 5x + 6)^{4}} - \frac{30}{(x^{2} - 5x + 6)^{3}} - \frac{6}{(x^{2} - 5x + 6)^{2}} + \frac{2250}{(x^{2} - 5x + 6)^{4}}\\ \end{split}\end{equation} \]

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