There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
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\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{(x - 2)}{({x}^{2} - 4x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{x}{(x^{2} - 4x)} - \frac{2}{(x^{2} - 4x)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{x}{(x^{2} - 4x)} - \frac{2}{(x^{2} - 4x)}\right)}{dx}\\=&(\frac{-(2x - 4)}{(x^{2} - 4x)^{2}})x + \frac{1}{(x^{2} - 4x)} - 2(\frac{-(2x - 4)}{(x^{2} - 4x)^{2}})\\=&\frac{-2x^{2}}{(x^{2} - 4x)^{2}} + \frac{8x}{(x^{2} - 4x)^{2}} + \frac{1}{(x^{2} - 4x)} - \frac{8}{(x^{2} - 4x)^{2}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2x^{2}}{(x^{2} - 4x)^{2}} + \frac{8x}{(x^{2} - 4x)^{2}} + \frac{1}{(x^{2} - 4x)} - \frac{8}{(x^{2} - 4x)^{2}}\right)}{dx}\\=&-2(\frac{-2(2x - 4)}{(x^{2} - 4x)^{3}})x^{2} - \frac{2*2x}{(x^{2} - 4x)^{2}} + 8(\frac{-2(2x - 4)}{(x^{2} - 4x)^{3}})x + \frac{8}{(x^{2} - 4x)^{2}} + (\frac{-(2x - 4)}{(x^{2} - 4x)^{2}}) - 8(\frac{-2(2x - 4)}{(x^{2} - 4x)^{3}})\\=&\frac{8x^{3}}{(x^{2} - 4x)^{3}} - \frac{48x^{2}}{(x^{2} - 4x)^{3}} - \frac{6x}{(x^{2} - 4x)^{2}} + \frac{96x}{(x^{2} - 4x)^{3}} + \frac{12}{(x^{2} - 4x)^{2}} - \frac{64}{(x^{2} - 4x)^{3}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{8x^{3}}{(x^{2} - 4x)^{3}} - \frac{48x^{2}}{(x^{2} - 4x)^{3}} - \frac{6x}{(x^{2} - 4x)^{2}} + \frac{96x}{(x^{2} - 4x)^{3}} + \frac{12}{(x^{2} - 4x)^{2}} - \frac{64}{(x^{2} - 4x)^{3}}\right)}{dx}\\=&8(\frac{-3(2x - 4)}{(x^{2} - 4x)^{4}})x^{3} + \frac{8*3x^{2}}{(x^{2} - 4x)^{3}} - 48(\frac{-3(2x - 4)}{(x^{2} - 4x)^{4}})x^{2} - \frac{48*2x}{(x^{2} - 4x)^{3}} - 6(\frac{-2(2x - 4)}{(x^{2} - 4x)^{3}})x - \frac{6}{(x^{2} - 4x)^{2}} + 96(\frac{-3(2x - 4)}{(x^{2} - 4x)^{4}})x + \frac{96}{(x^{2} - 4x)^{3}} + 12(\frac{-2(2x - 4)}{(x^{2} - 4x)^{3}}) - 64(\frac{-3(2x - 4)}{(x^{2} - 4x)^{4}})\\=&\frac{-48x^{4}}{(x^{2} - 4x)^{4}} + \frac{384x^{3}}{(x^{2} - 4x)^{4}} + \frac{48x^{2}}{(x^{2} - 4x)^{3}} - \frac{1152x^{2}}{(x^{2} - 4x)^{4}} - \frac{192x}{(x^{2} - 4x)^{3}} + \frac{1536x}{(x^{2} - 4x)^{4}} + \frac{192}{(x^{2} - 4x)^{3}} - \frac{6}{(x^{2} - 4x)^{2}} - \frac{768}{(x^{2} - 4x)^{4}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-48x^{4}}{(x^{2} - 4x)^{4}} + \frac{384x^{3}}{(x^{2} - 4x)^{4}} + \frac{48x^{2}}{(x^{2} - 4x)^{3}} - \frac{1152x^{2}}{(x^{2} - 4x)^{4}} - \frac{192x}{(x^{2} - 4x)^{3}} + \frac{1536x}{(x^{2} - 4x)^{4}} + \frac{192}{(x^{2} - 4x)^{3}} - \frac{6}{(x^{2} - 4x)^{2}} - \frac{768}{(x^{2} - 4x)^{4}}\right)}{dx}\\=&-48(\frac{-4(2x - 4)}{(x^{2} - 4x)^{5}})x^{4} - \frac{48*4x^{3}}{(x^{2} - 4x)^{4}} + 384(\frac{-4(2x - 4)}{(x^{2} - 4x)^{5}})x^{3} + \frac{384*3x^{2}}{(x^{2} - 4x)^{4}} + 48(\frac{-3(2x - 4)}{(x^{2} - 4x)^{4}})x^{2} + \frac{48*2x}{(x^{2} - 4x)^{3}} - 1152(\frac{-4(2x - 4)}{(x^{2} - 4x)^{5}})x^{2} - \frac{1152*2x}{(x^{2} - 4x)^{4}} - 192(\frac{-3(2x - 4)}{(x^{2} - 4x)^{4}})x - \frac{192}{(x^{2} - 4x)^{3}} + 1536(\frac{-4(2x - 4)}{(x^{2} - 4x)^{5}})x + \frac{1536}{(x^{2} - 4x)^{4}} + 192(\frac{-3(2x - 4)}{(x^{2} - 4x)^{4}}) - 6(\frac{-2(2x - 4)}{(x^{2} - 4x)^{3}}) - 768(\frac{-4(2x - 4)}{(x^{2} - 4x)^{5}})\\=&\frac{384x^{5}}{(x^{2} - 4x)^{5}} - \frac{3840x^{4}}{(x^{2} - 4x)^{5}} - \frac{480x^{3}}{(x^{2} - 4x)^{4}} + \frac{15360x^{3}}{(x^{2} - 4x)^{5}} - \frac{30720x^{2}}{(x^{2} - 4x)^{5}} + \frac{2880x^{2}}{(x^{2} - 4x)^{4}} + \frac{120x}{(x^{2} - 4x)^{3}} - \frac{5760x}{(x^{2} - 4x)^{4}} + \frac{30720x}{(x^{2} - 4x)^{5}} + \frac{3840}{(x^{2} - 4x)^{4}} - \frac{240}{(x^{2} - 4x)^{3}} - \frac{12288}{(x^{2} - 4x)^{5}}\\ \end{split}\end{equation} \]

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