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。
\[ \begin{equation}\begin{split}【1/1】求函数\frac{{x}^{x}ln(x)x}{x} + lg(x) + {e}^{x} - sin(x) 关于 x 的 15 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = {x}^{x}ln(x) + lg(x) + {e}^{x} - sin(x)\\\\ &\color{blue}{函数的 15 阶导数：} \\=&{x}^{x}ln^{15}(x)ln(x) + 15{x}^{x}ln^{14}(x)ln(x) + \frac{105{x}^{x}ln^{13}(x)ln(x)}{x} + \frac{1365{x}^{x}ln^{12}(x)ln(x)}{x} + 105{x}^{x}ln^{13}(x)ln(x) + 455{x}^{x}ln^{12}(x)ln(x) - \frac{455{x}^{x}ln^{12}(x)ln(x)}{x^{2}} + \frac{15015{x}^{x}ln^{10}(x)ln(x)}{x^{2}} + \frac{8190{x}^{x}ln^{11}(x)ln(x)}{x} + \frac{30030{x}^{x}ln^{10}(x)ln(x)}{x} + \frac{2730{x}^{x}ln^{11}(x)ln(x)}{x^{3}} + 1365{x}^{x}ln^{11}(x)ln(x) + 3003{x}^{x}ln^{10}(x)ln(x) - \frac{75075{x}^{x}ln^{9}(x)ln(x)}{x^{3}} - \frac{225225{x}^{x}ln^{8}(x)ln(x)}{x^{3}} + 5005{x}^{x}ln^{9}(x)ln(x) + 6435{x}^{x}ln^{8}(x)ln(x) + \frac{125125{x}^{x}ln^{9}(x)ln(x)}{x^{2}} + \frac{450450{x}^{x}ln^{8}(x)ln(x)}{x^{2}} - \frac{18018{x}^{x}ln^{10}(x)ln(x)}{x^{4}} + \frac{20020{x}^{x}ln^{9}(x)ln(x)}{x^{4}} + \frac{75075{x}^{x}ln^{9}(x)ln(x)}{x} + \frac{135135{x}^{x}ln^{8}(x)ln(x)}{x} + \frac{315315{x}^{x}ln^{8}(x)ln(x)}{x^{4}} - \frac{1156155{x}^{x}ln^{6}(x)ln(x)}{x^{4}} + \frac{180180{x}^{x}ln^{7}(x)ln(x)}{x} + \frac{180180{x}^{x}ln^{6}(x)ln(x)}{x} + \frac{1261260{x}^{x}ln^{6}(x)ln(x)}{x^{3}} + \frac{120120{x}^{x}ln^{9}(x)ln(x)}{x^{5}} - \frac{180180{x}^{x}ln^{8}(x)ln(x)}{x^{5}} - \frac{1261260{x}^{x}ln^{7}(x)ln(x)}{x^{5}} + 6435{x}^{x}ln^{7}(x)ln(x) + 5005{x}^{x}ln^{6}(x)ln(x) + \frac{990990{x}^{x}ln^{7}(x)ln(x)}{x^{2}} + \frac{1471470{x}^{x}ln^{6}(x)ln(x)}{x^{2}} + 3003{x}^{x}ln^{5}(x)ln(x) + 1365{x}^{x}ln^{4}(x)ln(x) + \frac{4099095{x}^{x}ln^{5}(x)ln(x)}{x^{5}} + \frac{4099095{x}^{x}ln^{4}(x)ln(x)}{x^{5}} + \frac{1531530{x}^{x}ln^{5}(x)ln(x)}{x^{2}} + \frac{1126125{x}^{x}ln^{4}(x)ln(x)}{x^{2}} - \frac{2963961{x}^{x}ln^{5}(x)ln(x)}{x^{4}} + 455{x}^{x}ln^{3}(x)ln(x) + 105{x}^{x}ln^{2}(x)ln(x) - \frac{2207205{x}^{x}ln^{4}(x)ln(x)}{x^{4}} - \frac{772200{x}^{x}ln^{8}(x)ln(x)}{x^{6}} + \frac{1209780{x}^{x}ln^{7}(x)ln(x)}{x^{6}} + \frac{4724720{x}^{x}ln^{6}(x)ln(x)}{x^{6}} - \frac{2462460{x}^{x}ln^{5}(x)ln(x)}{x^{6}} + \frac{135135{x}^{x}ln^{5}(x)ln(x)}{x} + \frac{75075{x}^{x}ln^{4}(x)ln(x)}{x} + \frac{3153150{x}^{x}ln^{5}(x)ln(x)}{x^{3}} + \frac{4054050{x}^{x}ln^{4}(x)ln(x)}{x^{3}} + \frac{30030{x}^{x}ln^{3}(x)ln(x)}{x} + \frac{8190{x}^{x}ln^{2}(x)ln(x)}{x} - \frac{10435425{x}^{x}ln^{4}(x)ln(x)}{x^{6}} + \frac{3708705{x}^{x}ln^{2}(x)ln(x)}{x^{6}} + \frac{3153150{x}^{x}ln^{3}(x)ln(x)}{x^{3}} + \frac{1501500{x}^{x}ln^{2}(x)ln(x)}{x^{3}} - \frac{630630{x}^{x}ln^{3}(x)ln(x)}{x^{5}} + \frac{1365{x}^{x}ln(x)ln(x)}{x} - \frac{2612610{x}^{x}ln^{2}(x)ln(x)}{x^{5}} + \frac{4633200{x}^{x}ln^{7}(x)ln(x)}{x^{7}} - \frac{6846840{x}^{x}ln^{6}(x)ln(x)}{x^{7}} - \frac{16036020{x}^{x}ln^{5}(x)ln(x)}{x^{7}} + \frac{12312300{x}^{x}ln^{4}(x)ln(x)}{x^{7}} + \frac{20870850{x}^{x}ln^{3}(x)ln(x)}{x^{7}} + \frac{575575{x}^{x}ln^{3}(x)ln(x)}{x^{2}} + \frac{195195{x}^{x}ln^{2}(x)ln(x)}{x^{2}} + \frac{465465{x}^{x}ln^{3}(x)ln(x)}{x^{4}} + \frac{1666665{x}^{x}ln^{2}(x)ln(x)}{x^{4}} + \frac{39585{x}^{x}ln(x)ln(x)}{x^{2}} - \frac{4669665{x}^{x}ln(x)ln(x)}{x^{7}} + \frac{945945{x}^{x}ln(x)ln(x)}{x^{4}} + \frac{2125695{x}^{x}ln(x)ln(x)}{x^{6}} - \frac{25225200{x}^{x}ln^{6}(x)ln(x)}{x^{8}} + \frac{33081048{x}^{x}ln^{5}(x)ln(x)}{x^{8}} + \frac{47387340{x}^{x}ln^{4}(x)ln(x)}{x^{8}} - \frac{39659620{x}^{x}ln^{3}(x)ln(x)}{x^{8}} - \frac{31501470{x}^{x}ln^{2}(x)ln(x)}{x^{8}} + \frac{4444440{x}^{x}ln(x)ln(x)}{x^{8}} + \frac{405405{x}^{x}ln(x)ln(x)}{x^{3}} - \frac{1126125{x}^{x}ln(x)ln(x)}{x^{5}} + \frac{121080960{x}^{x}ln^{5}(x)ln(x)}{x^{9}} - \frac{134053920{x}^{x}ln^{4}(x)ln(x)}{x^{9}} - \frac{115915800{x}^{x}ln^{3}(x)ln(x)}{x^{9}} + \frac{90210120{x}^{x}ln^{2}(x)ln(x)}{x^{9}} + \frac{31891860{x}^{x}ln(x)ln(x)}{x^{9}} - \frac{495331200{x}^{x}ln^{4}(x)ln(x)}{x^{10}} + \frac{438852960{x}^{x}ln^{3}(x)ln(x)}{x^{10}} + \frac{218279880{x}^{x}ln^{2}(x)ln(x)}{x^{10}} - \frac{134479800{x}^{x}ln(x)ln(x)}{x^{10}} + \frac{1651104000{x}^{x}ln^{3}(x)ln(x)}{x^{11}} - \frac{1088942400{x}^{x}ln^{2}(x)ln(x)}{x^{11}} - \frac{279311760{x}^{x}ln(x)ln(x)}{x^{11}} - \frac{4191264000{x}^{x}ln^{2}(x)ln(x)}{x^{12}} + \frac{1820059200{x}^{x}ln(x)ln(x)}{x^{12}} + \frac{7185024000{x}^{x}ln(x)ln(x)}{x^{13}} - \frac{1365{x}^{x}ln^{11}(x)ln(x)}{x^{2}} + \frac{315315{x}^{x}ln^{7}(x)ln(x)}{x^{4}} - \frac{3478475{x}^{x}ln^{3}(x)ln(x)}{x^{6}} + \frac{900900{x}^{x}ln^{7}(x)}{x^{2}} + \frac{1365{x}^{x}ln^{2}(x)}{x} + \frac{15{x}^{x}ln^{14}(x)}{x} - \frac{10690680{x}^{x}ln^{3}(x)}{x^{5}} + \frac{1261260{x}^{x}ln^{7}(x)}{x^{5}} + \frac{30030{x}^{x}ln^{9}(x)}{x} - \frac{13436280{x}^{x}ln^{7}(x)}{x^{7}} + \frac{1365{x}^{x}ln^{12}(x)}{x} - \frac{19879860{x}^{x}ln^{6}(x)}{x^{7}} + \frac{4918914{x}^{x}ln^{5}(x)}{x^{5}} + \frac{43123080{x}^{x}ln^{5}(x)}{x^{7}} + \frac{50300250{x}^{x}ln^{4}(x)}{x^{7}} + \frac{3353350{x}^{x}ln^{3}(x)}{x^{3}} + \frac{1081080{x}^{x}ln^{6}(x)}{x^{2}} + \frac{600600{x}^{x}ln^{4}(x)}{x^{2}} + \frac{15015{x}^{x}ln^{10}(x)}{x} - \frac{5990985{x}^{x}ln^{4}(x)}{x^{5}} - \frac{100100{x}^{x}ln^{9}(x)}{x^{3}} + \frac{270270{x}^{x}ln^{3}(x)}{x^{2}} + \frac{945945{x}^{x}ln^{5}(x)}{x^{2}} + \frac{3603600{x}^{x}ln^{4}(x)}{x^{4}} - \frac{1801800{x}^{x}ln^{7}(x)}{x^{4}} + \frac{6756750{x}^{x}ln^{3}(x)}{x^{4}} + \frac{81900{x}^{x}ln^{2}(x)}{x^{2}} + \frac{3640{x}^{x}ln(x)}{x^{2}} + \frac{15015{x}^{x}ln(x)}{x^{2}} - \frac{1891890{x}^{x}ln^{5}(x)}{x^{4}} - \frac{20870850{x}^{x}ln^{3}(x)}{x^{7}} + \frac{45045{x}^{x}ln^{8}(x)}{x} - \frac{667095{x}^{x}ln(x)}{x^{7}} - \frac{5336760{x}^{x}ln(x)}{x^{7}} + \frac{60060{x}^{x}ln^{10}(x)}{x^{2}} + \frac{4954950{x}^{x}ln^{2}(x)}{x^{4}} + \frac{182182{x}^{x}ln(x)}{x^{4}} + \frac{1786785{x}^{x}ln(x)}{x^{4}} + \frac{1981980{x}^{x}ln^{8}(x)}{x^{6}} + \frac{11171160{x}^{x}ln^{2}(x)}{x^{6}} + \frac{168740{x}^{x}ln(x)}{x^{6}} + \frac{405405{x}^{x}ln(x)}{x^{6}} + \frac{51480{x}^{x}ln^{7}(x)}{x} + \frac{4864860{x}^{x}ln^{7}(x)}{x^{6}} + \frac{80360280{x}^{x}ln^{6}(x)}{x^{8}} + \frac{5460{x}^{x}ln^{11}(x)}{x} + \frac{71171100{x}^{x}ln^{5}(x)}{x^{8}} - \frac{8828820{x}^{x}ln^{6}(x)}{x^{6}} - \frac{166366200{x}^{x}ln^{4}(x)}{x^{8}} + \frac{225225{x}^{x}ln^{8}(x)}{x^{3}} - \frac{96846750{x}^{x}ln^{3}(x)}{x^{8}} + \frac{45045{x}^{x}ln^{6}(x)}{x} + \frac{62612550{x}^{x}ln^{2}(x)}{x^{8}} + \frac{225225{x}^{x}ln^{9}(x)}{x^{2}} + \frac{1336335{x}^{x}ln^{2}(x)}{x^{3}} + \frac{47775{x}^{x}ln(x)}{x^{3}} + \frac{311220{x}^{x}ln(x)}{x^{3}} + \frac{30030{x}^{x}ln^{5}(x)}{x} - \frac{5270265{x}^{x}ln^{2}(x)}{x^{5}} - \frac{105105{x}^{x}ln(x)}{x^{5}} - \frac{450450{x}^{x}ln(x)}{x^{5}} + \frac{1621620{x}^{x}ln^{7}(x)}{x^{3}} + \frac{3993990{x}^{x}ln^{6}(x)}{x^{3}} - \frac{415999584{x}^{x}ln^{5}(x)}{x^{9}} + \frac{210{x}^{x}ln^{13}(x)}{x} - \frac{213693480{x}^{x}ln^{4}(x)}{x^{9}} + \frac{5765760{x}^{x}ln^{5}(x)}{x^{3}} + \frac{506826320{x}^{x}ln^{3}(x)}{x^{9}} + \frac{8190{x}^{x}ln^{11}(x)}{x^{2}} + \frac{133032900{x}^{x}ln^{2}(x)}{x^{9}} - \frac{4444440{x}^{x}ln(x)}{x^{9}} - \frac{98558460{x}^{x}ln(x)}{x^{9}} + \frac{15015{x}^{x}ln^{4}(x)}{x} - \frac{260260{x}^{x}ln^{9}(x)}{x^{5}} + \frac{1811890080{x}^{x}ln^{4}(x)}{x^{10}} + \frac{5460{x}^{x}ln^{3}(x)}{x} + \frac{507026520{x}^{x}ln^{3}(x)}{x^{10}} + \frac{30030{x}^{x}ln^{10}(x)}{x^{4}} - \frac{1159638480{x}^{x}ln^{2}(x)}{x^{10}} - \frac{16072420{x}^{x}ln(x)}{x^{10}} - \frac{106606500{x}^{x}ln(x)}{x^{10}} - \frac{1081080{x}^{x}ln^{8}(x)}{x^{5}} - \frac{6369854400{x}^{x}ln^{3}(x)}{x^{11}} + \frac{19549530{x}^{x}ln^{3}(x)}{x^{6}} - \frac{866239920{x}^{x}ln^{2}(x)}{x^{11}} + \frac{100136400{x}^{x}ln(x)}{x^{11}} + \frac{1781357760{x}^{x}ln(x)}{x^{11}} - \frac{2730{x}^{x}ln^{11}(x)}{x^{3}} + \frac{5405400{x}^{x}ln^{4}(x)}{x^{3}} + \frac{16931678400{x}^{x}ln^{2}(x)}{x^{12}} + \frac{180973584{x}^{x}ln(x)}{x^{12}} + \frac{894544560{x}^{x}ln(x)}{x^{12}} + \frac{540540{x}^{x}ln^{8}(x)}{x^{2}} - \frac{1535880960{x}^{x}ln(x)}{x^{13}} - \frac{30223238400{x}^{x}ln(x)}{x^{13}} - \frac{45045{x}^{x}ln^{10}(x)}{x^{3}} + \frac{6831825{x}^{x}ln^{6}(x)}{x^{5}} - \frac{32687655{x}^{x}ln^{2}(x)}{x^{7}} + \frac{225225{x}^{x}ln^{9}(x)}{x^{4}} - \frac{105{x}^{x}ln^{13}(x)}{x^{2}} - \frac{20495475{x}^{x}ln^{5}(x)}{x^{6}} + \frac{32687655{x}^{x}ln(x)}{x^{8}} - \frac{3783780{x}^{x}ln^{6}(x)}{x^{4}} + \frac{2475473{x}^{x}ln(x)}{x^{8}} + \frac{910{x}^{x}ln^{12}(x)}{x^{3}} - \frac{8190{x}^{x}ln^{11}(x)}{x^{4}} + \frac{72072{x}^{x}ln^{10}(x)}{x^{5}} - \frac{600600{x}^{x}ln^{9}(x)}{x^{6}} + \frac{4633200{x}^{x}ln^{8}(x)}{x^{7}} - \frac{32432400{x}^{x}ln^{7}(x)}{x^{8}} + \frac{201801600{x}^{x}ln^{6}(x)}{x^{9}} - \frac{1089728640{x}^{x}ln^{5}(x)}{x^{10}} + \frac{4953312000{x}^{x}ln^{4}(x)}{x^{11}} - \frac{18162144000{x}^{x}ln^{3}(x)}{x^{12}} + \frac{50295168000{x}^{x}ln^{2}(x)}{x^{13}} - \frac{6227020800{x}^{x}ln(x)}{x^{14}} - \frac{93405312000{x}^{x}ln(x)}{x^{14}} + \frac{105{x}^{x}ln(x)}{x} + \frac{210{x}^{x}ln(x)}{x} + \frac{71891820{x}^{x}}{x^{10}} + \frac{32305{x}^{x}}{x^{3}} + \frac{26244400{x}^{x}}{x^{11}} - \frac{1380812160{x}^{x}}{x^{12}} - \frac{351623808{x}^{x}}{x^{13}} + \frac{217217{x}^{x}}{x^{5}} + \frac{1113255{x}^{x}}{x^{7}} + \frac{27151476480{x}^{x}}{x^{14}} - \frac{15359344{x}^{x}}{x^{9}} + \frac{1260{x}^{x}}{x^{2}} + \frac{262080{x}^{x}}{x^{4}} + 15{x}^{x}ln(x)ln(x) - \frac{600600{x}^{x}}{x^{6}} + {x}^{x}ln(x) + \frac{15{x}^{x}}{x} + \frac{87178291200{x}^{x}}{x^{15}} + \frac{87178291200}{x^{15}ln{10}} + {e}^{x} + cos(x)\\ \end{split}\end{equation} \]

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