本次共计算 1 个题目：每一题对 x 求 4 阶导数。
    注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数xsin(x) - \frac{(1 + xx)}{(1 + x + xx)} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = xsin(x) - \frac{x^{2}}{(x^{2} + x + 1)} - \frac{1}{(x^{2} + x + 1)}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( xsin(x) - \frac{x^{2}}{(x^{2} + x + 1)} - \frac{1}{(x^{2} + x + 1)}\right)}{dx}\\=&sin(x) + xcos(x) - (\frac{-(2x + 1 + 0)}{(x^{2} + x + 1)^{2}})x^{2} - \frac{2x}{(x^{2} + x + 1)} - (\frac{-(2x + 1 + 0)}{(x^{2} + x + 1)^{2}})\\=&sin(x) + xcos(x) + \frac{2x^{3}}{(x^{2} + x + 1)^{2}} + \frac{x^{2}}{(x^{2} + x + 1)^{2}} - \frac{2x}{(x^{2} + x + 1)} + \frac{2x}{(x^{2} + x + 1)^{2}} + \frac{1}{(x^{2} + x + 1)^{2}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( sin(x) + xcos(x) + \frac{2x^{3}}{(x^{2} + x + 1)^{2}} + \frac{x^{2}}{(x^{2} + x + 1)^{2}} - \frac{2x}{(x^{2} + x + 1)} + \frac{2x}{(x^{2} + x + 1)^{2}} + \frac{1}{(x^{2} + x + 1)^{2}}\right)}{dx}\\=&cos(x) + cos(x) + x*-sin(x) + 2(\frac{-2(2x + 1 + 0)}{(x^{2} + x + 1)^{3}})x^{3} + \frac{2*3x^{2}}{(x^{2} + x + 1)^{2}} + (\frac{-2(2x + 1 + 0)}{(x^{2} + x + 1)^{3}})x^{2} + \frac{2x}{(x^{2} + x + 1)^{2}} - 2(\frac{-(2x + 1 + 0)}{(x^{2} + x + 1)^{2}})x - \frac{2}{(x^{2} + x + 1)} + 2(\frac{-2(2x + 1 + 0)}{(x^{2} + x + 1)^{3}})x + \frac{2}{(x^{2} + x + 1)^{2}} + (\frac{-2(2x + 1 + 0)}{(x^{2} + x + 1)^{3}})\\=&2cos(x) - xsin(x) - \frac{8x^{4}}{(x^{2} + x + 1)^{3}} - \frac{8x^{3}}{(x^{2} + x + 1)^{3}} + \frac{10x^{2}}{(x^{2} + x + 1)^{2}} - \frac{10x^{2}}{(x^{2} + x + 1)^{3}} + \frac{4x}{(x^{2} + x + 1)^{2}} - \frac{8x}{(x^{2} + x + 1)^{3}} + \frac{2}{(x^{2} + x + 1)^{2}} - \frac{2}{(x^{2} + x + 1)} - \frac{2}{(x^{2} + x + 1)^{3}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( 2cos(x) - xsin(x) - \frac{8x^{4}}{(x^{2} + x + 1)^{3}} - \frac{8x^{3}}{(x^{2} + x + 1)^{3}} + \frac{10x^{2}}{(x^{2} + x + 1)^{2}} - \frac{10x^{2}}{(x^{2} + x + 1)^{3}} + \frac{4x}{(x^{2} + x + 1)^{2}} - \frac{8x}{(x^{2} + x + 1)^{3}} + \frac{2}{(x^{2} + x + 1)^{2}} - \frac{2}{(x^{2} + x + 1)} - \frac{2}{(x^{2} + x + 1)^{3}}\right)}{dx}\\=&2*-sin(x) - sin(x) - xcos(x) - 8(\frac{-3(2x + 1 + 0)}{(x^{2} + x + 1)^{4}})x^{4} - \frac{8*4x^{3}}{(x^{2} + x + 1)^{3}} - 8(\frac{-3(2x + 1 + 0)}{(x^{2} + x + 1)^{4}})x^{3} - \frac{8*3x^{2}}{(x^{2} + x + 1)^{3}} + 10(\frac{-2(2x + 1 + 0)}{(x^{2} + x + 1)^{3}})x^{2} + \frac{10*2x}{(x^{2} + x + 1)^{2}} - 10(\frac{-3(2x + 1 + 0)}{(x^{2} + x + 1)^{4}})x^{2} - \frac{10*2x}{(x^{2} + x + 1)^{3}} + 4(\frac{-2(2x + 1 + 0)}{(x^{2} + x + 1)^{3}})x + \frac{4}{(x^{2} + x + 1)^{2}} - 8(\frac{-3(2x + 1 + 0)}{(x^{2} + x + 1)^{4}})x - \frac{8}{(x^{2} + x + 1)^{3}} + 2(\frac{-2(2x + 1 + 0)}{(x^{2} + x + 1)^{3}}) - 2(\frac{-(2x + 1 + 0)}{(x^{2} + x + 1)^{2}}) - 2(\frac{-3(2x + 1 + 0)}{(x^{2} + x + 1)^{4}})\\=&-3sin(x) - xcos(x) + \frac{48x^{5}}{(x^{2} + x + 1)^{4}} + \frac{72x^{4}}{(x^{2} + x + 1)^{4}} - \frac{72x^{3}}{(x^{2} + x + 1)^{3}} + \frac{84x^{3}}{(x^{2} + x + 1)^{4}} + \frac{78x^{2}}{(x^{2} + x + 1)^{4}} - \frac{60x^{2}}{(x^{2} + x + 1)^{3}} + \frac{24x}{(x^{2} + x + 1)^{2}} - \frac{36x}{(x^{2} + x + 1)^{3}} + \frac{36x}{(x^{2} + x + 1)^{4}} - \frac{12}{(x^{2} + x + 1)^{3}} + \frac{6}{(x^{2} + x + 1)^{2}} + \frac{6}{(x^{2} + x + 1)^{4}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( -3sin(x) - xcos(x) + \frac{48x^{5}}{(x^{2} + x + 1)^{4}} + \frac{72x^{4}}{(x^{2} + x + 1)^{4}} - \frac{72x^{3}}{(x^{2} + x + 1)^{3}} + \frac{84x^{3}}{(x^{2} + x + 1)^{4}} + \frac{78x^{2}}{(x^{2} + x + 1)^{4}} - \frac{60x^{2}}{(x^{2} + x + 1)^{3}} + \frac{24x}{(x^{2} + x + 1)^{2}} - \frac{36x}{(x^{2} + x + 1)^{3}} + \frac{36x}{(x^{2} + x + 1)^{4}} - \frac{12}{(x^{2} + x + 1)^{3}} + \frac{6}{(x^{2} + x + 1)^{2}} + \frac{6}{(x^{2} + x + 1)^{4}}\right)}{dx}\\=&-3cos(x) - cos(x) - x*-sin(x) + 48(\frac{-4(2x + 1 + 0)}{(x^{2} + x + 1)^{5}})x^{5} + \frac{48*5x^{4}}{(x^{2} + x + 1)^{4}} + 72(\frac{-4(2x + 1 + 0)}{(x^{2} + x + 1)^{5}})x^{4} + \frac{72*4x^{3}}{(x^{2} + x + 1)^{4}} - 72(\frac{-3(2x + 1 + 0)}{(x^{2} + x + 1)^{4}})x^{3} - \frac{72*3x^{2}}{(x^{2} + x + 1)^{3}} + 84(\frac{-4(2x + 1 + 0)}{(x^{2} + x + 1)^{5}})x^{3} + \frac{84*3x^{2}}{(x^{2} + x + 1)^{4}} + 78(\frac{-4(2x + 1 + 0)}{(x^{2} + x + 1)^{5}})x^{2} + \frac{78*2x}{(x^{2} + x + 1)^{4}} - 60(\frac{-3(2x + 1 + 0)}{(x^{2} + x + 1)^{4}})x^{2} - \frac{60*2x}{(x^{2} + x + 1)^{3}} + 24(\frac{-2(2x + 1 + 0)}{(x^{2} + x + 1)^{3}})x + \frac{24}{(x^{2} + x + 1)^{2}} - 36(\frac{-3(2x + 1 + 0)}{(x^{2} + x + 1)^{4}})x - \frac{36}{(x^{2} + x + 1)^{3}} + 36(\frac{-4(2x + 1 + 0)}{(x^{2} + x + 1)^{5}})x + \frac{36}{(x^{2} + x + 1)^{4}} - 12(\frac{-3(2x + 1 + 0)}{(x^{2} + x + 1)^{4}}) + 6(\frac{-2(2x + 1 + 0)}{(x^{2} + x + 1)^{3}}) + 6(\frac{-4(2x + 1 + 0)}{(x^{2} + x + 1)^{5}})\\=&-4cos(x) + xsin(x) - \frac{384x^{6}}{(x^{2} + x + 1)^{5}} - \frac{768x^{5}}{(x^{2} + x + 1)^{5}} + \frac{672x^{4}}{(x^{2} + x + 1)^{4}} - \frac{960x^{4}}{(x^{2} + x + 1)^{5}} - \frac{960x^{3}}{(x^{2} + x + 1)^{5}} + \frac{864x^{3}}{(x^{2} + x + 1)^{4}} - \frac{312x^{2}}{(x^{2} + x + 1)^{3}} - \frac{600x^{2}}{(x^{2} + x + 1)^{5}} + \frac{648x^{2}}{(x^{2} + x + 1)^{4}} + \frac{336x}{(x^{2} + x + 1)^{4}} - \frac{192x}{(x^{2} + x + 1)^{3}} - \frac{192x}{(x^{2} + x + 1)^{5}} - \frac{48}{(x^{2} + x + 1)^{3}} + \frac{72}{(x^{2} + x + 1)^{4}} + \frac{24}{(x^{2} + x + 1)^{2}} - \frac{24}{(x^{2} + x + 1)^{5}}\\ \end{split}\end{equation} \]

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