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。
\[ \begin{equation}\begin{split}【1/1】求函数{x}^{\frac{1}{x}} 关于 x 的 15 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ \\ &\color{blue}{函数的 15 阶导数：} \\=&\frac{-1307674368000{x}^{\frac{1}{x}}ln(x)}{x^{16}} - \frac{9153720576000{x}^{\frac{1}{x}}ln^{2}(x)}{x^{17}} - \frac{19833061248000{x}^{\frac{1}{x}}ln^{3}(x)}{x^{18}} + \frac{45472329504000{x}^{\frac{1}{x}}ln(x)}{x^{17}} - \frac{19833061248000{x}^{\frac{1}{x}}ln^{4}(x)}{x^{19}} + \frac{124826701104000{x}^{\frac{1}{x}}ln^{2}(x)}{x^{18}} - \frac{10908183686400{x}^{\frac{1}{x}}ln^{5}(x)}{x^{20}} + \frac{147250132848000{x}^{\frac{1}{x}}ln^{3}(x)}{x^{19}} - \frac{3636061228800{x}^{\frac{1}{x}}ln^{6}(x)}{x^{21}} + \frac{91844104716000{x}^{\frac{1}{x}}ln^{4}(x)}{x^{20}} - \frac{251292776630400{x}^{\frac{1}{x}}ln(x)}{x^{18}} - \frac{779155977600{x}^{\frac{1}{x}}ln^{7}(x)}{x^{22}} + \frac{33866495863200{x}^{\frac{1}{x}}ln^{5}(x)}{x^{21}} - \frac{397490778921600{x}^{\frac{1}{x}}ln^{2}(x)}{x^{19}} - \frac{111307996800{x}^{\frac{1}{x}}ln^{8}(x)}{x^{23}} + \frac{7892334450000{x}^{\frac{1}{x}}ln^{6}(x)}{x^{22}} - \frac{301958372570400{x}^{\frac{1}{x}}ln^{3}(x)}{x^{20}} - \frac{10821610800{x}^{\frac{1}{x}}ln^{9}(x)}{x^{24}} + \frac{1211095314000{x}^{\frac{1}{x}}ln^{7}(x)}{x^{23}} - \frac{128939706735840{x}^{\frac{1}{x}}ln^{4}(x)}{x^{21}} - \frac{721440720{x}^{\frac{1}{x}}ln^{10}(x)}{x^{25}} + \frac{125299102500{x}^{\frac{1}{x}}ln^{8}(x)}{x^{24}} - \frac{33741142042608{x}^{\frac{1}{x}}ln^{5}(x)}{x^{22}} - \frac{32792760{x}^{\frac{1}{x}}ln^{11}(x)}{x^{26}} + \frac{8824315500{x}^{\frac{1}{x}}ln^{9}(x)}{x^{25}} + \frac{463632300069120{x}^{\frac{1}{x}}ln(x)}{x^{19}} - \frac{5695033584240{x}^{\frac{1}{x}}ln^{6}(x)}{x^{23}} - \frac{993720{x}^{\frac{1}{x}}ln^{12}(x)}{x^{27}} + \frac{421242822{x}^{\frac{1}{x}}ln^{10}(x)}{x^{26}} + \frac{485515050962760{x}^{\frac{1}{x}}ln^{2}(x)}{x^{20}} - \frac{638587237860{x}^{\frac{1}{x}}ln^{7}(x)}{x^{24}} - \frac{19110{x}^{\frac{1}{x}}ln^{13}(x)}{x^{28}} + \frac{13341510{x}^{\frac{1}{x}}ln^{11}(x)}{x^{27}} + \frac{257236270931640{x}^{\frac{1}{x}}ln^{3}(x)}{x^{21}} - \frac{48207699540{x}^{\frac{1}{x}}ln^{8}(x)}{x^{25}} - \frac{210{x}^{\frac{1}{x}}ln^{14}(x)}{x^{29}} + \frac{267085{x}^{\frac{1}{x}}ln^{12}(x)}{x^{28}} + \frac{79010565974340{x}^{\frac{1}{x}}ln^{4}(x)}{x^{22}} - \frac{2445813370{x}^{\frac{1}{x}}ln^{9}(x)}{x^{26}} - \frac{{x}^{\frac{1}{x}}ln^{15}(x)}{x^{30}} + \frac{3045{x}^{\frac{1}{x}}ln^{13}(x)}{x^{29}} + \frac{15130271656500{x}^{\frac{1}{x}}ln^{5}(x)}{x^{23}} - \frac{81771690{x}^{\frac{1}{x}}ln^{10}(x)}{x^{27}} + \frac{15{x}^{\frac{1}{x}}ln^{14}(x)}{x^{30}} - \frac{105{x}^{\frac{1}{x}}ln^{13}(x)}{x^{30}} + \frac{1881496897280{x}^{\frac{1}{x}}ln^{6}(x)}{x^{24}} - \frac{1718535{x}^{\frac{1}{x}}ln^{11}(x)}{x^{28}} - \frac{20475{x}^{\frac{1}{x}}ln^{12}(x)}{x^{29}} + \frac{455{x}^{\frac{1}{x}}ln^{12}(x)}{x^{30}} + \frac{154979845020{x}^{\frac{1}{x}}ln^{7}(x)}{x^{25}} + \frac{84630{x}^{\frac{1}{x}}ln^{11}(x)}{x^{29}} - \frac{1365{x}^{\frac{1}{x}}ln^{11}(x)}{x^{30}} + \frac{6741735{x}^{\frac{1}{x}}ln^{10}(x)}{x^{28}} + \frac{8475982515{x}^{\frac{1}{x}}ln^{8}(x)}{x^{26}} - \frac{240240{x}^{\frac{1}{x}}ln^{10}(x)}{x^{29}} + \frac{3003{x}^{\frac{1}{x}}ln^{10}(x)}{x^{30}} + \frac{302627325{x}^{\frac{1}{x}}ln^{9}(x)}{x^{27}} - \frac{17992975{x}^{\frac{1}{x}}ln^{9}(x)}{x^{28}} - \frac{382455486307440{x}^{\frac{1}{x}}ln(x)}{x^{20}} + \frac{495495{x}^{\frac{1}{x}}ln^{9}(x)}{x^{29}} - \frac{5005{x}^{\frac{1}{x}}ln^{9}(x)}{x^{30}} - \frac{753377625{x}^{\frac{1}{x}}ln^{8}(x)}{x^{27}} + \frac{34504470{x}^{\frac{1}{x}}ln^{8}(x)}{x^{28}} - \frac{283992908078880{x}^{\frac{1}{x}}ln^{2}(x)}{x^{21}} - \frac{765765{x}^{\frac{1}{x}}ln^{8}(x)}{x^{29}} + \frac{6435{x}^{\frac{1}{x}}ln^{8}(x)}{x^{30}} - \frac{19486782315{x}^{\frac{1}{x}}ln^{7}(x)}{x^{26}} + \frac{1329368040{x}^{\frac{1}{x}}ln^{7}(x)}{x^{27}} - \frac{109559566696580{x}^{\frac{1}{x}}ln^{3}(x)}{x^{22}} - \frac{48918870{x}^{\frac{1}{x}}ln^{7}(x)}{x^{28}} + \frac{900900{x}^{\frac{1}{x}}ln^{7}(x)}{x^{29}} - \frac{24859539404700{x}^{\frac{1}{x}}ln^{4}(x)}{x^{23}} - \frac{6435{x}^{\frac{1}{x}}ln^{7}(x)}{x^{30}} - \frac{324843188670{x}^{\frac{1}{x}}ln^{6}(x)}{x^{25}} + \frac{31217551365{x}^{\frac{1}{x}}ln^{6}(x)}{x^{26}} - \frac{1705223520{x}^{\frac{1}{x}}ln^{6}(x)}{x^{27}} - \frac{3533966706270{x}^{\frac{1}{x}}ln^{5}(x)}{x^{24}} + \frac{51921870{x}^{\frac{1}{x}}ln^{6}(x)}{x^{28}} - \frac{810810{x}^{\frac{1}{x}}ln^{6}(x)}{x^{29}} + \frac{5005{x}^{\frac{1}{x}}ln^{6}(x)}{x^{30}} + \frac{464055707115{x}^{\frac{1}{x}}ln^{5}(x)}{x^{25}} - \frac{35568351819{x}^{\frac{1}{x}}ln^{5}(x)}{x^{26}} + \frac{1602430830{x}^{\frac{1}{x}}ln^{5}(x)}{x^{27}} - \frac{41261220{x}^{\frac{1}{x}}ln^{5}(x)}{x^{28}} + \frac{555555{x}^{\frac{1}{x}}ln^{5}(x)}{x^{29}} - \frac{3003{x}^{\frac{1}{x}}ln^{5}(x)}{x^{30}} + \frac{4390694182875{x}^{\frac{1}{x}}ln^{4}(x)}{x^{24}} - \frac{457747515225{x}^{\frac{1}{x}}ln^{4}(x)}{x^{25}} + \frac{28830196395{x}^{\frac{1}{x}}ln^{4}(x)}{x^{26}} - \frac{1095043950{x}^{\frac{1}{x}}ln^{4}(x)}{x^{27}} + \frac{24249225{x}^{\frac{1}{x}}ln^{4}(x)}{x^{28}} - \frac{285285{x}^{\frac{1}{x}}ln^{4}(x)}{x^{29}} + \frac{1365{x}^{\frac{1}{x}}ln^{4}(x)}{x^{30}} + \frac{25884975466350{x}^{\frac{1}{x}}ln^{3}(x)}{x^{23}} - \frac{3610233451825{x}^{\frac{1}{x}}ln^{3}(x)}{x^{24}} + \frac{307963746090{x}^{\frac{1}{x}}ln^{3}(x)}{x^{25}} - \frac{16295944665{x}^{\frac{1}{x}}ln^{3}(x)}{x^{26}} + \frac{530780250{x}^{\frac{1}{x}}ln^{3}(x)}{x^{27}} - \frac{10245235{x}^{\frac{1}{x}}ln^{3}(x)}{x^{28}} + \frac{106470{x}^{\frac{1}{x}}ln^{3}(x)}{x^{29}} - \frac{455{x}^{\frac{1}{x}}ln^{3}(x)}{x^{30}} + \frac{90044435210730{x}^{\frac{1}{x}}ln^{2}(x)}{x^{22}} - \frac{16691450725950{x}^{\frac{1}{x}}ln^{2}(x)}{x^{23}} + \frac{1895283975585{x}^{\frac{1}{x}}ln^{2}(x)}{x^{24}} - \frac{135287071920{x}^{\frac{1}{x}}ln^{2}(x)}{x^{25}} + \frac{6118777665{x}^{\frac{1}{x}}ln^{2}(x)}{x^{26}} - \frac{173243070{x}^{\frac{1}{x}}ln^{2}(x)}{x^{27}} + \frac{2947035{x}^{\frac{1}{x}}ln^{2}(x)}{x^{28}} - \frac{27300{x}^{\frac{1}{x}}ln^{2}(x)}{x^{29}} + \frac{105{x}^{\frac{1}{x}}ln^{2}(x)}{x^{30}} + \frac{164705459919000{x}^{\frac{1}{x}}ln(x)}{x^{21}} - \frac{40648502384490{x}^{\frac{1}{x}}ln(x)}{x^{22}} + \frac{6097774698015{x}^{\frac{1}{x}}ln(x)}{x^{23}} - \frac{576679837305{x}^{\frac{1}{x}}ln(x)}{x^{24}} + \frac{35051811795{x}^{\frac{1}{x}}ln(x)}{x^{25}} - \frac{1373827455{x}^{\frac{1}{x}}ln(x)}{x^{26}} + \frac{34191885{x}^{\frac{1}{x}}ln(x)}{x^{27}} - \frac{517335{x}^{\frac{1}{x}}ln(x)}{x^{28}} + \frac{4305{x}^{\frac{1}{x}}ln(x)}{x^{29}} - \frac{15{x}^{\frac{1}{x}}ln(x)}{x^{30}} - \frac{53180752331520{x}^{\frac{1}{x}}}{x^{17}} + \frac{118264573470680{x}^{\frac{1}{x}}}{x^{20}} - \frac{197632289814960{x}^{\frac{1}{x}}}{x^{19}} + \frac{4339163001600{x}^{\frac{1}{x}}}{x^{16}} - \frac{39245689643160{x}^{\frac{1}{x}}}{x^{21}} + \frac{7781055114833{x}^{\frac{1}{x}}}{x^{22}} - \frac{966785423175{x}^{\frac{1}{x}}}{x^{23}} + \frac{77514685820{x}^{\frac{1}{x}}}{x^{24}} - \frac{4068509445{x}^{\frac{1}{x}}}{x^{25}} + \frac{139761622{x}^{\frac{1}{x}}}{x^{26}} - \frac{3086265{x}^{\frac{1}{x}}}{x^{27}} + \frac{41860{x}^{\frac{1}{x}}}{x^{28}} + \frac{162478082312064{x}^{\frac{1}{x}}}{x^{18}} - \frac{315{x}^{\frac{1}{x}}}{x^{29}} + \frac{{x}^{\frac{1}{x}}}{x^{30}}\\ \end{split}\end{equation} \]

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