There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
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\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{(1)}{(1 + x + xx + xxx)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{1}{(x^{3} + x^{2} + x + 1)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{1}{(x^{3} + x^{2} + x + 1)}\right)}{dx}\\=&(\frac{-(3x^{2} + 2x + 1 + 0)}{(x^{3} + x^{2} + x + 1)^{2}})\\=&\frac{-3x^{2}}{(x^{3} + x^{2} + x + 1)^{2}} - \frac{2x}{(x^{3} + x^{2} + x + 1)^{2}} - \frac{1}{(x^{3} + x^{2} + x + 1)^{2}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-3x^{2}}{(x^{3} + x^{2} + x + 1)^{2}} - \frac{2x}{(x^{3} + x^{2} + x + 1)^{2}} - \frac{1}{(x^{3} + x^{2} + x + 1)^{2}}\right)}{dx}\\=&-3(\frac{-2(3x^{2} + 2x + 1 + 0)}{(x^{3} + x^{2} + x + 1)^{3}})x^{2} - \frac{3*2x}{(x^{3} + x^{2} + x + 1)^{2}} - 2(\frac{-2(3x^{2} + 2x + 1 + 0)}{(x^{3} + x^{2} + x + 1)^{3}})x - \frac{2}{(x^{3} + x^{2} + x + 1)^{2}} - (\frac{-2(3x^{2} + 2x + 1 + 0)}{(x^{3} + x^{2} + x + 1)^{3}})\\=&\frac{18x^{4}}{(x^{3} + x^{2} + x + 1)^{3}} + \frac{24x^{3}}{(x^{3} + x^{2} + x + 1)^{3}} - \frac{6x}{(x^{3} + x^{2} + x + 1)^{2}} + \frac{20x^{2}}{(x^{3} + x^{2} + x + 1)^{3}} + \frac{8x}{(x^{3} + x^{2} + x + 1)^{3}} - \frac{2}{(x^{3} + x^{2} + x + 1)^{2}} + \frac{2}{(x^{3} + x^{2} + x + 1)^{3}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{18x^{4}}{(x^{3} + x^{2} + x + 1)^{3}} + \frac{24x^{3}}{(x^{3} + x^{2} + x + 1)^{3}} - \frac{6x}{(x^{3} + x^{2} + x + 1)^{2}} + \frac{20x^{2}}{(x^{3} + x^{2} + x + 1)^{3}} + \frac{8x}{(x^{3} + x^{2} + x + 1)^{3}} - \frac{2}{(x^{3} + x^{2} + x + 1)^{2}} + \frac{2}{(x^{3} + x^{2} + x + 1)^{3}}\right)}{dx}\\=&18(\frac{-3(3x^{2} + 2x + 1 + 0)}{(x^{3} + x^{2} + x + 1)^{4}})x^{4} + \frac{18*4x^{3}}{(x^{3} + x^{2} + x + 1)^{3}} + 24(\frac{-3(3x^{2} + 2x + 1 + 0)}{(x^{3} + x^{2} + x + 1)^{4}})x^{3} + \frac{24*3x^{2}}{(x^{3} + x^{2} + x + 1)^{3}} - 6(\frac{-2(3x^{2} + 2x + 1 + 0)}{(x^{3} + x^{2} + x + 1)^{3}})x - \frac{6}{(x^{3} + x^{2} + x + 1)^{2}} + 20(\frac{-3(3x^{2} + 2x + 1 + 0)}{(x^{3} + x^{2} + x + 1)^{4}})x^{2} + \frac{20*2x}{(x^{3} + x^{2} + x + 1)^{3}} + 8(\frac{-3(3x^{2} + 2x + 1 + 0)}{(x^{3} + x^{2} + x + 1)^{4}})x + \frac{8}{(x^{3} + x^{2} + x + 1)^{3}} - 2(\frac{-2(3x^{2} + 2x + 1 + 0)}{(x^{3} + x^{2} + x + 1)^{3}}) + 2(\frac{-3(3x^{2} + 2x + 1 + 0)}{(x^{3} + x^{2} + x + 1)^{4}})\\=&\frac{-162x^{6}}{(x^{3} + x^{2} + x + 1)^{4}} - \frac{324x^{5}}{(x^{3} + x^{2} + x + 1)^{4}} + \frac{108x^{3}}{(x^{3} + x^{2} + x + 1)^{3}} - \frac{378x^{4}}{(x^{3} + x^{2} + x + 1)^{4}} - \frac{264x^{3}}{(x^{3} + x^{2} + x + 1)^{4}} + \frac{108x^{2}}{(x^{3} + x^{2} + x + 1)^{3}} + \frac{60x}{(x^{3} + x^{2} + x + 1)^{3}} - \frac{126x^{2}}{(x^{3} + x^{2} + x + 1)^{4}} - \frac{36x}{(x^{3} + x^{2} + x + 1)^{4}} + \frac{12}{(x^{3} + x^{2} + x + 1)^{3}} - \frac{6}{(x^{3} + x^{2} + x + 1)^{2}} - \frac{6}{(x^{3} + x^{2} + x + 1)^{4}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-162x^{6}}{(x^{3} + x^{2} + x + 1)^{4}} - \frac{324x^{5}}{(x^{3} + x^{2} + x + 1)^{4}} + \frac{108x^{3}}{(x^{3} + x^{2} + x + 1)^{3}} - \frac{378x^{4}}{(x^{3} + x^{2} + x + 1)^{4}} - \frac{264x^{3}}{(x^{3} + x^{2} + x + 1)^{4}} + \frac{108x^{2}}{(x^{3} + x^{2} + x + 1)^{3}} + \frac{60x}{(x^{3} + x^{2} + x + 1)^{3}} - \frac{126x^{2}}{(x^{3} + x^{2} + x + 1)^{4}} - \frac{36x}{(x^{3} + x^{2} + x + 1)^{4}} + \frac{12}{(x^{3} + x^{2} + x + 1)^{3}} - \frac{6}{(x^{3} + x^{2} + x + 1)^{2}} - \frac{6}{(x^{3} + x^{2} + x + 1)^{4}}\right)}{dx}\\=&-162(\frac{-4(3x^{2} + 2x + 1 + 0)}{(x^{3} + x^{2} + x + 1)^{5}})x^{6} - \frac{162*6x^{5}}{(x^{3} + x^{2} + x + 1)^{4}} - 324(\frac{-4(3x^{2} + 2x + 1 + 0)}{(x^{3} + x^{2} + x + 1)^{5}})x^{5} - \frac{324*5x^{4}}{(x^{3} + x^{2} + x + 1)^{4}} + 108(\frac{-3(3x^{2} + 2x + 1 + 0)}{(x^{3} + x^{2} + x + 1)^{4}})x^{3} + \frac{108*3x^{2}}{(x^{3} + x^{2} + x + 1)^{3}} - 378(\frac{-4(3x^{2} + 2x + 1 + 0)}{(x^{3} + x^{2} + x + 1)^{5}})x^{4} - \frac{378*4x^{3}}{(x^{3} + x^{2} + x + 1)^{4}} - 264(\frac{-4(3x^{2} + 2x + 1 + 0)}{(x^{3} + x^{2} + x + 1)^{5}})x^{3} - \frac{264*3x^{2}}{(x^{3} + x^{2} + x + 1)^{4}} + 108(\frac{-3(3x^{2} + 2x + 1 + 0)}{(x^{3} + x^{2} + x + 1)^{4}})x^{2} + \frac{108*2x}{(x^{3} + x^{2} + x + 1)^{3}} + 60(\frac{-3(3x^{2} + 2x + 1 + 0)}{(x^{3} + x^{2} + x + 1)^{4}})x + \frac{60}{(x^{3} + x^{2} + x + 1)^{3}} - 126(\frac{-4(3x^{2} + 2x + 1 + 0)}{(x^{3} + x^{2} + x + 1)^{5}})x^{2} - \frac{126*2x}{(x^{3} + x^{2} + x + 1)^{4}} - 36(\frac{-4(3x^{2} + 2x + 1 + 0)}{(x^{3} + x^{2} + x + 1)^{5}})x - \frac{36}{(x^{3} + x^{2} + x + 1)^{4}} + 12(\frac{-3(3x^{2} + 2x + 1 + 0)}{(x^{3} + x^{2} + x + 1)^{4}}) - 6(\frac{-2(3x^{2} + 2x + 1 + 0)}{(x^{3} + x^{2} + x + 1)^{3}}) - 6(\frac{-4(3x^{2} + 2x + 1 + 0)}{(x^{3} + x^{2} + x + 1)^{5}})\\=&\frac{1944x^{8}}{(x^{3} + x^{2} + x + 1)^{5}} + \frac{5184x^{7}}{(x^{3} + x^{2} + x + 1)^{5}} - \frac{1944x^{5}}{(x^{3} + x^{2} + x + 1)^{4}} + \frac{7776x^{6}}{(x^{3} + x^{2} + x + 1)^{5}} + \frac{7488x^{5}}{(x^{3} + x^{2} + x + 1)^{5}} - \frac{3240x^{4}}{(x^{3} + x^{2} + x + 1)^{4}} + \frac{360x^{2}}{(x^{3} + x^{2} + x + 1)^{3}} + \frac{5136x^{4}}{(x^{3} + x^{2} + x + 1)^{5}} - \frac{3024x^{3}}{(x^{3} + x^{2} + x + 1)^{4}} + \frac{2496x^{3}}{(x^{3} + x^{2} + x + 1)^{5}} + \frac{864x^{2}}{(x^{3} + x^{2} + x + 1)^{5}} - \frac{1584x^{2}}{(x^{3} + x^{2} + x + 1)^{4}} + \frac{240x}{(x^{3} + x^{2} + x + 1)^{3}} - \frac{504x}{(x^{3} + x^{2} + x + 1)^{4}} + \frac{192x}{(x^{3} + x^{2} + x + 1)^{5}} - \frac{72}{(x^{3} + x^{2} + x + 1)^{4}} + \frac{72}{(x^{3} + x^{2} + x + 1)^{3}} + \frac{24}{(x^{3} + x^{2} + x + 1)^{5}}\\ \end{split}\end{equation} \]

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