There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{({x}^{3} - 2)}{(2{(x - 1)}^{2})}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{\frac{1}{2}x^{3}}{(x - 1)^{2}} - \frac{1}{(x - 1)^{2}}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{\frac{1}{2}x^{3}}{(x - 1)^{2}} - \frac{1}{(x - 1)^{2}}\right)}{dx}\\=&\frac{1}{2}(\frac{-2(1 + 0)}{(x - 1)^{3}})x^{3} + \frac{\frac{1}{2}*3x^{2}}{(x - 1)^{2}} - (\frac{-2(1 + 0)}{(x - 1)^{3}})\\=&\frac{-x^{3}}{(x - 1)^{3}} + \frac{3x^{2}}{2(x - 1)^{2}} + \frac{2}{(x - 1)^{3}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-x^{3}}{(x - 1)^{3}} + \frac{3x^{2}}{2(x - 1)^{2}} + \frac{2}{(x - 1)^{3}}\right)}{dx}\\=&-(\frac{-3(1 + 0)}{(x - 1)^{4}})x^{3} - \frac{3x^{2}}{(x - 1)^{3}} + \frac{3(\frac{-2(1 + 0)}{(x - 1)^{3}})x^{2}}{2} + \frac{3*2x}{2(x - 1)^{2}} + 2(\frac{-3(1 + 0)}{(x - 1)^{4}})\\=&\frac{3x^{3}}{(x - 1)^{4}} - \frac{6x^{2}}{(x - 1)^{3}} + \frac{3x}{(x - 1)^{2}} - \frac{6}{(x - 1)^{4}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{3x^{3}}{(x - 1)^{4}} - \frac{6x^{2}}{(x - 1)^{3}} + \frac{3x}{(x - 1)^{2}} - \frac{6}{(x - 1)^{4}}\right)}{dx}\\=&3(\frac{-4(1 + 0)}{(x - 1)^{5}})x^{3} + \frac{3*3x^{2}}{(x - 1)^{4}} - 6(\frac{-3(1 + 0)}{(x - 1)^{4}})x^{2} - \frac{6*2x}{(x - 1)^{3}} + 3(\frac{-2(1 + 0)}{(x - 1)^{3}})x + \frac{3}{(x - 1)^{2}} - 6(\frac{-4(1 + 0)}{(x - 1)^{5}})\\=&\frac{-12x^{3}}{(x - 1)^{5}} + \frac{27x^{2}}{(x - 1)^{4}} - \frac{18x}{(x - 1)^{3}} + \frac{24}{(x - 1)^{5}} + \frac{3}{(x - 1)^{2}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-12x^{3}}{(x - 1)^{5}} + \frac{27x^{2}}{(x - 1)^{4}} - \frac{18x}{(x - 1)^{3}} + \frac{24}{(x - 1)^{5}} + \frac{3}{(x - 1)^{2}}\right)}{dx}\\=&-12(\frac{-5(1 + 0)}{(x - 1)^{6}})x^{3} - \frac{12*3x^{2}}{(x - 1)^{5}} + 27(\frac{-4(1 + 0)}{(x - 1)^{5}})x^{2} + \frac{27*2x}{(x - 1)^{4}} - 18(\frac{-3(1 + 0)}{(x - 1)^{4}})x - \frac{18}{(x - 1)^{3}} + 24(\frac{-5(1 + 0)}{(x - 1)^{6}}) + 3(\frac{-2(1 + 0)}{(x - 1)^{3}})\\=&\frac{60x^{3}}{(x - 1)^{6}} - \frac{144x^{2}}{(x - 1)^{5}} + \frac{108x}{(x - 1)^{4}} - \frac{24}{(x - 1)^{3}} - \frac{120}{(x - 1)^{6}}\\ \end{split}\end{equation} \]

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