There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
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\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{1}{({x}^{2} - x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{1}{(x^{2} - x)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{1}{(x^{2} - x)}\right)}{dx}\\=&(\frac{-(2x - 1)}{(x^{2} - x)^{2}})\\=&\frac{-2x}{(x^{2} - x)^{2}} + \frac{1}{(x^{2} - x)^{2}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2x}{(x^{2} - x)^{2}} + \frac{1}{(x^{2} - x)^{2}}\right)}{dx}\\=&-2(\frac{-2(2x - 1)}{(x^{2} - x)^{3}})x - \frac{2}{(x^{2} - x)^{2}} + (\frac{-2(2x - 1)}{(x^{2} - x)^{3}})\\=&\frac{8x^{2}}{(x^{2} - x)^{3}} - \frac{8x}{(x^{2} - x)^{3}} - \frac{2}{(x^{2} - x)^{2}} + \frac{2}{(x^{2} - x)^{3}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{8x^{2}}{(x^{2} - x)^{3}} - \frac{8x}{(x^{2} - x)^{3}} - \frac{2}{(x^{2} - x)^{2}} + \frac{2}{(x^{2} - x)^{3}}\right)}{dx}\\=&8(\frac{-3(2x - 1)}{(x^{2} - x)^{4}})x^{2} + \frac{8*2x}{(x^{2} - x)^{3}} - 8(\frac{-3(2x - 1)}{(x^{2} - x)^{4}})x - \frac{8}{(x^{2} - x)^{3}} - 2(\frac{-2(2x - 1)}{(x^{2} - x)^{3}}) + 2(\frac{-3(2x - 1)}{(x^{2} - x)^{4}})\\=&\frac{-48x^{3}}{(x^{2} - x)^{4}} + \frac{72x^{2}}{(x^{2} - x)^{4}} + \frac{24x}{(x^{2} - x)^{3}} - \frac{36x}{(x^{2} - x)^{4}} - \frac{12}{(x^{2} - x)^{3}} + \frac{6}{(x^{2} - x)^{4}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-48x^{3}}{(x^{2} - x)^{4}} + \frac{72x^{2}}{(x^{2} - x)^{4}} + \frac{24x}{(x^{2} - x)^{3}} - \frac{36x}{(x^{2} - x)^{4}} - \frac{12}{(x^{2} - x)^{3}} + \frac{6}{(x^{2} - x)^{4}}\right)}{dx}\\=&-48(\frac{-4(2x - 1)}{(x^{2} - x)^{5}})x^{3} - \frac{48*3x^{2}}{(x^{2} - x)^{4}} + 72(\frac{-4(2x - 1)}{(x^{2} - x)^{5}})x^{2} + \frac{72*2x}{(x^{2} - x)^{4}} + 24(\frac{-3(2x - 1)}{(x^{2} - x)^{4}})x + \frac{24}{(x^{2} - x)^{3}} - 36(\frac{-4(2x - 1)}{(x^{2} - x)^{5}})x - \frac{36}{(x^{2} - x)^{4}} - 12(\frac{-3(2x - 1)}{(x^{2} - x)^{4}}) + 6(\frac{-4(2x - 1)}{(x^{2} - x)^{5}})\\=&\frac{384x^{4}}{(x^{2} - x)^{5}} - \frac{768x^{3}}{(x^{2} - x)^{5}} - \frac{288x^{2}}{(x^{2} - x)^{4}} + \frac{576x^{2}}{(x^{2} - x)^{5}} + \frac{288x}{(x^{2} - x)^{4}} - \frac{192x}{(x^{2} - x)^{5}} - \frac{72}{(x^{2} - x)^{4}} + \frac{24}{(x^{2} - x)^{3}} + \frac{24}{(x^{2} - x)^{5}}\\ \end{split}\end{equation} \]

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