本次共计算 1 个题目：每一题对 x 求 15 阶导数。
    注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数\frac{{x}^{n}}{(1 - x)} 关于 x 的 15 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{{x}^{n}}{(-x + 1)}\\\\ &\color{blue}{函数的 15 阶导数：} \\=&\frac{1307674368000{x}^{n}}{(-x + 1)^{16}} + \frac{1307674368000n{x}^{n}}{(-x + 1)^{15}x} - \frac{653837184000n{x}^{n}}{(-x + 1)^{14}x^{2}} + \frac{653837184000n^{2}{x}^{n}}{(-x + 1)^{14}x^{2}} + \frac{435891456000n{x}^{n}}{(-x + 1)^{13}x^{3}} - \frac{653837184000n^{2}{x}^{n}}{(-x + 1)^{13}x^{3}} + \frac{217945728000n^{3}{x}^{n}}{(-x + 1)^{13}x^{3}} - \frac{326918592000n{x}^{n}}{(-x + 1)^{12}x^{4}} + \frac{599350752000n^{2}{x}^{n}}{(-x + 1)^{12}x^{4}} - \frac{326918592000n^{3}{x}^{n}}{(-x + 1)^{12}x^{4}} + \frac{54486432000n^{4}{x}^{n}}{(-x + 1)^{12}x^{4}} + \frac{261534873600n{x}^{n}}{(-x + 1)^{11}x^{5}} - \frac{544864320000n^{2}{x}^{n}}{(-x + 1)^{11}x^{5}} + \frac{381405024000n^{3}{x}^{n}}{(-x + 1)^{11}x^{5}} - \frac{108972864000n^{4}{x}^{n}}{(-x + 1)^{11}x^{5}} + \frac{10897286400n^{5}{x}^{n}}{(-x + 1)^{11}x^{5}} - \frac{217945728000n{x}^{n}}{(-x + 1)^{10}x^{6}} + \frac{497642745600n^{2}{x}^{n}}{(-x + 1)^{10}x^{6}} - \frac{408648240000n^{3}{x}^{n}}{(-x + 1)^{10}x^{6}} + \frac{154378224000n^{4}{x}^{n}}{(-x + 1)^{10}x^{6}} - \frac{27243216000n^{5}{x}^{n}}{(-x + 1)^{10}x^{6}} + \frac{1816214400n^{6}{x}^{n}}{(-x + 1)^{10}x^{6}} + \frac{186810624000n{x}^{n}}{(-x + 1)^{9}x^{7}} - \frac{457686028800n^{2}{x}^{n}}{(-x + 1)^{9}x^{7}} + \frac{421361740800n^{3}{x}^{n}}{(-x + 1)^{9}x^{7}} - \frac{190702512000n^{4}{x}^{n}}{(-x + 1)^{9}x^{7}} + \frac{45405360000n^{5}{x}^{n}}{(-x + 1)^{9}x^{7}} - \frac{5448643200n^{6}{x}^{n}}{(-x + 1)^{9}x^{7}} + \frac{259459200n^{7}{x}^{n}}{(-x + 1)^{9}x^{7}} - \frac{163459296000n{x}^{n}}{(-x + 1)^{8}x^{8}} + \frac{423826603200n^{2}{x}^{n}}{(-x + 1)^{8}x^{8}} - \frac{425902276800n^{3}{x}^{n}}{(-x + 1)^{8}x^{8}} + \frac{219534915600n^{4}{x}^{n}}{(-x + 1)^{8}x^{8}} - \frac{63567504000n^{5}{x}^{n}}{(-x + 1)^{8}x^{8}} + \frac{10443232800n^{6}{x}^{n}}{(-x + 1)^{8}x^{8}} - \frac{908107200n^{7}{x}^{n}}{(-x + 1)^{8}x^{8}} + \frac{32432400n^{8}{x}^{n}}{(-x + 1)^{8}x^{8}} + \frac{145297152000n{x}^{n}}{(-x + 1)^{7}x^{9}} - \frac{394896902400n^{2}{x}^{n}}{(-x + 1)^{7}x^{9}} + \frac{425671646400n^{3}{x}^{n}}{(-x + 1)^{7}x^{9}} - \frac{242464622400n^{4}{x}^{n}}{(-x + 1)^{7}x^{9}} + \frac{80897216400n^{5}{x}^{n}}{(-x + 1)^{7}x^{9}} - \frac{16345929600n^{6}{x}^{n}}{(-x + 1)^{7}x^{9}} + \frac{1967565600n^{7}{x}^{n}}{(-x + 1)^{7}x^{9}} - \frac{129729600n^{8}{x}^{n}}{(-x + 1)^{7}x^{9}} + \frac{3603600n^{9}{x}^{n}}{(-x + 1)^{7}x^{9}} - \frac{130767436800n{x}^{n}}{(-x + 1)^{6}x^{10}} + \frac{369936927360n^{2}{x}^{n}}{(-x + 1)^{6}x^{10}} - \frac{422594172000n^{3}{x}^{n}}{(-x + 1)^{6}x^{10}} + \frac{260785324800n^{4}{x}^{n}}{(-x + 1)^{6}x^{10}} - \frac{97053957000n^{5}{x}^{n}}{(-x + 1)^{6}x^{10}} + \frac{22801058280n^{6}{x}^{n}}{(-x + 1)^{6}x^{10}} - \frac{3405402000n^{7}{x}^{n}}{(-x + 1)^{6}x^{10}} + \frac{313513200n^{8}{x}^{n}}{(-x + 1)^{6}x^{10}} - \frac{16216200n^{9}{x}^{n}}{(-x + 1)^{6}x^{10}} + \frac{360360n^{10}{x}^{n}}{(-x + 1)^{6}x^{10}} + \frac{118879488000n{x}^{n}}{(-x + 1)^{5}x^{11}} - \frac{348194246400n^{2}{x}^{n}}{(-x + 1)^{5}x^{11}} + \frac{417807149760n^{3}{x}^{n}}{(-x + 1)^{5}x^{11}} - \frac{275495220000n^{4}{x}^{n}}{(-x + 1)^{5}x^{11}} + \frac{111938626800n^{5}{x}^{n}}{(-x + 1)^{5}x^{11}} - \frac{29551321800n^{6}{x}^{n}}{(-x + 1)^{5}x^{11}} + \frac{5168643480n^{7}{x}^{n}}{(-x + 1)^{5}x^{11}} - \frac{594594000n^{8}{x}^{n}}{(-x + 1)^{5}x^{11}} + \frac{43243200n^{9}{x}^{n}}{(-x + 1)^{5}x^{11}} - \frac{1801800n^{10}{x}^{n}}{(-x + 1)^{5}x^{11}} + \frac{32760n^{11}{x}^{n}}{(-x + 1)^{5}x^{11}} - \frac{108972864000n{x}^{n}}{(-x + 1)^{4}x^{12}} + \frac{329084683200n^{2}{x}^{n}}{(-x + 1)^{4}x^{12}} - \frac{412006074480n^{3}{x}^{n}}{(-x + 1)^{4}x^{12}} + \frac{287354547480n^{4}{x}^{n}}{(-x + 1)^{4}x^{12}} - \frac{125568342900n^{5}{x}^{n}}{(-x + 1)^{4}x^{12}} + \frac{36416930550n^{6}{x}^{n}}{(-x + 1)^{4}x^{12}} - \frac{7200533340n^{7}{x}^{n}}{(-x + 1)^{4}x^{12}} + \frac{975764790n^{8}{x}^{n}}{(-x + 1)^{4}x^{12}} - \frac{89189100n^{9}{x}^{n}}{(-x + 1)^{4}x^{12}} + \frac{5255250n^{10}{x}^{n}}{(-x + 1)^{4}x^{12}} - \frac{180180n^{11}{x}^{n}}{(-x + 1)^{4}x^{12}} + \frac{2730n^{12}{x}^{n}}{(-x + 1)^{4}x^{12}} + \frac{100590336000n{x}^{n}}{(-x + 1)^{3}x^{13}} - \frac{312153004800n^{2}{x}^{n}}{(-x + 1)^{3}x^{13}} + \frac{405627505920n^{3}{x}^{n}}{(-x + 1)^{3}x^{13}} - \frac{296943126480n^{4}{x}^{n}}{(-x + 1)^{3}x^{13}} + \frac{138013435560n^{5}{x}^{n}}{(-x + 1)^{3}x^{13}} - \frac{43274731500n^{6}{x}^{n}}{(-x + 1)^{3}x^{13}} + \frac{9447948510n^{7}{x}^{n}}{(-x + 1)^{3}x^{13}} - \frac{1454593140n^{8}{x}^{n}}{(-x + 1)^{3}x^{13}} + \frac{157387230n^{9}{x}^{n}}{(-x + 1)^{3}x^{13}} - \frac{11711700n^{10}{x}^{n}}{(-x + 1)^{3}x^{13}} + \frac{570570n^{11}{x}^{n}}{(-x + 1)^{3}x^{13}} - \frac{16380n^{12}{x}^{n}}{(-x + 1)^{3}x^{13}} + \frac{210n^{13}{x}^{n}}{(-x + 1)^{3}x^{13}} - \frac{93405312000n{x}^{n}}{(-x + 1)^{2}x^{14}} + \frac{297041385600n^{2}{x}^{n}}{(-x + 1)^{2}x^{14}} - \frac{398950755840n^{3}{x}^{n}}{(-x + 1)^{2}x^{14}} + \frac{304706296440n^{4}{x}^{n}}{(-x + 1)^{2}x^{14}} - \frac{149365556340n^{5}{x}^{n}}{(-x + 1)^{2}x^{14}} + \frac{50041781790n^{6}{x}^{n}}{(-x + 1)^{2}x^{14}} - \frac{11864147295n^{7}{x}^{n}}{(-x + 1)^{2}x^{14}} + \frac{2025547095n^{8}{x}^{n}}{(-x + 1)^{2}x^{14}} - \frac{250044795n^{9}{x}^{n}}{(-x + 1)^{2}x^{14}} + \frac{22117095n^{10}{x}^{n}}{(-x + 1)^{2}x^{14}} - \frac{1366365n^{11}{x}^{n}}{(-x + 1)^{2}x^{14}} + \frac{55965n^{12}{x}^{n}}{(-x + 1)^{2}x^{14}} - \frac{1365n^{13}{x}^{n}}{(-x + 1)^{2}x^{14}} + \frac{15n^{14}{x}^{n}}{(-x + 1)^{2}x^{14}} + \frac{87178291200n{x}^{n}}{(-x + 1)x^{15}} - \frac{283465647360n^{2}{x}^{n}}{(-x + 1)x^{15}} + \frac{392156797824n^{3}{x}^{n}}{(-x + 1)x^{15}} - \frac{310989260400n^{4}{x}^{n}}{(-x + 1)x^{15}} + \frac{159721605680n^{5}{x}^{n}}{(-x + 1)x^{15}} - \frac{56663366760n^{6}{x}^{n}}{(-x + 1)x^{15}} + \frac{14409322928n^{7}{x}^{n}}{(-x + 1)x^{15}} - \frac{2681453775n^{8}{x}^{n}}{(-x + 1)x^{15}} + \frac{368411615n^{9}{x}^{n}}{(-x + 1)x^{15}} - \frac{37312275n^{10}{x}^{n}}{(-x + 1)x^{15}} + \frac{2749747n^{11}{x}^{n}}{(-x + 1)x^{15}} - \frac{143325n^{12}{x}^{n}}{(-x + 1)x^{15}} + \frac{5005n^{13}{x}^{n}}{(-x + 1)x^{15}} - \frac{105n^{14}{x}^{n}}{(-x + 1)x^{15}} + \frac{n^{15}{x}^{n}}{(-x + 1)x^{15}}\\ \end{split}\end{equation} \]

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