There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ {x}^{(\frac{a}{x})}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( {x}^{(\frac{a}{x})}\right)}{dx}\\=&({x}^{(\frac{a}{x})}((\frac{a*-1}{x^{2}})ln(x) + \frac{(\frac{a}{x})(1)}{(x)}))\\=&\frac{-a{x}^{(\frac{a}{x})}ln(x)}{x^{2}} + \frac{a{x}^{(\frac{a}{x})}}{x^{2}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-a{x}^{(\frac{a}{x})}ln(x)}{x^{2}} + \frac{a{x}^{(\frac{a}{x})}}{x^{2}}\right)}{dx}\\=&\frac{-a*-2{x}^{(\frac{a}{x})}ln(x)}{x^{3}} - \frac{a({x}^{(\frac{a}{x})}((\frac{a*-1}{x^{2}})ln(x) + \frac{(\frac{a}{x})(1)}{(x)}))ln(x)}{x^{2}} - \frac{a{x}^{(\frac{a}{x})}}{x^{2}(x)} + \frac{a*-2{x}^{(\frac{a}{x})}}{x^{3}} + \frac{a({x}^{(\frac{a}{x})}((\frac{a*-1}{x^{2}})ln(x) + \frac{(\frac{a}{x})(1)}{(x)}))}{x^{2}}\\=&\frac{2a{x}^{(\frac{a}{x})}ln(x)}{x^{3}} + \frac{a^{2}{x}^{(\frac{a}{x})}ln^{2}(x)}{x^{4}} - \frac{2a^{2}{x}^{(\frac{a}{x})}ln(x)}{x^{4}} - \frac{3a{x}^{(\frac{a}{x})}}{x^{3}} + \frac{a^{2}{x}^{(\frac{a}{x})}}{x^{4}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{2a{x}^{(\frac{a}{x})}ln(x)}{x^{3}} + \frac{a^{2}{x}^{(\frac{a}{x})}ln^{2}(x)}{x^{4}} - \frac{2a^{2}{x}^{(\frac{a}{x})}ln(x)}{x^{4}} - \frac{3a{x}^{(\frac{a}{x})}}{x^{3}} + \frac{a^{2}{x}^{(\frac{a}{x})}}{x^{4}}\right)}{dx}\\=&\frac{2a*-3{x}^{(\frac{a}{x})}ln(x)}{x^{4}} + \frac{2a({x}^{(\frac{a}{x})}((\frac{a*-1}{x^{2}})ln(x) + \frac{(\frac{a}{x})(1)}{(x)}))ln(x)}{x^{3}} + \frac{2a{x}^{(\frac{a}{x})}}{x^{3}(x)} + \frac{a^{2}*-4{x}^{(\frac{a}{x})}ln^{2}(x)}{x^{5}} + \frac{a^{2}({x}^{(\frac{a}{x})}((\frac{a*-1}{x^{2}})ln(x) + \frac{(\frac{a}{x})(1)}{(x)}))ln^{2}(x)}{x^{4}} + \frac{a^{2}{x}^{(\frac{a}{x})}*2ln(x)}{x^{4}(x)} - \frac{2a^{2}*-4{x}^{(\frac{a}{x})}ln(x)}{x^{5}} - \frac{2a^{2}({x}^{(\frac{a}{x})}((\frac{a*-1}{x^{2}})ln(x) + \frac{(\frac{a}{x})(1)}{(x)}))ln(x)}{x^{4}} - \frac{2a^{2}{x}^{(\frac{a}{x})}}{x^{4}(x)} - \frac{3a*-3{x}^{(\frac{a}{x})}}{x^{4}} - \frac{3a({x}^{(\frac{a}{x})}((\frac{a*-1}{x^{2}})ln(x) + \frac{(\frac{a}{x})(1)}{(x)}))}{x^{3}} + \frac{a^{2}*-4{x}^{(\frac{a}{x})}}{x^{5}} + \frac{a^{2}({x}^{(\frac{a}{x})}((\frac{a*-1}{x^{2}})ln(x) + \frac{(\frac{a}{x})(1)}{(x)}))}{x^{4}}\\=&\frac{-6a{x}^{(\frac{a}{x})}ln(x)}{x^{4}} - \frac{6a^{2}{x}^{(\frac{a}{x})}ln^{2}(x)}{x^{5}} - \frac{a^{3}{x}^{(\frac{a}{x})}ln^{3}(x)}{x^{6}} + \frac{15a^{2}{x}^{(\frac{a}{x})}ln(x)}{x^{5}} + \frac{3a^{3}{x}^{(\frac{a}{x})}ln^{2}(x)}{x^{6}} - \frac{3a^{3}{x}^{(\frac{a}{x})}ln(x)}{x^{6}} + \frac{11a{x}^{(\frac{a}{x})}}{x^{4}} - \frac{9a^{2}{x}^{(\frac{a}{x})}}{x^{5}} + \frac{a^{3}{x}^{(\frac{a}{x})}}{x^{6}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-6a{x}^{(\frac{a}{x})}ln(x)}{x^{4}} - \frac{6a^{2}{x}^{(\frac{a}{x})}ln^{2}(x)}{x^{5}} - \frac{a^{3}{x}^{(\frac{a}{x})}ln^{3}(x)}{x^{6}} + \frac{15a^{2}{x}^{(\frac{a}{x})}ln(x)}{x^{5}} + \frac{3a^{3}{x}^{(\frac{a}{x})}ln^{2}(x)}{x^{6}} - \frac{3a^{3}{x}^{(\frac{a}{x})}ln(x)}{x^{6}} + \frac{11a{x}^{(\frac{a}{x})}}{x^{4}} - \frac{9a^{2}{x}^{(\frac{a}{x})}}{x^{5}} + \frac{a^{3}{x}^{(\frac{a}{x})}}{x^{6}}\right)}{dx}\\=&\frac{-6a*-4{x}^{(\frac{a}{x})}ln(x)}{x^{5}} - \frac{6a({x}^{(\frac{a}{x})}((\frac{a*-1}{x^{2}})ln(x) + \frac{(\frac{a}{x})(1)}{(x)}))ln(x)}{x^{4}} - \frac{6a{x}^{(\frac{a}{x})}}{x^{4}(x)} - \frac{6a^{2}*-5{x}^{(\frac{a}{x})}ln^{2}(x)}{x^{6}} - \frac{6a^{2}({x}^{(\frac{a}{x})}((\frac{a*-1}{x^{2}})ln(x) + \frac{(\frac{a}{x})(1)}{(x)}))ln^{2}(x)}{x^{5}} - \frac{6a^{2}{x}^{(\frac{a}{x})}*2ln(x)}{x^{5}(x)} - \frac{a^{3}*-6{x}^{(\frac{a}{x})}ln^{3}(x)}{x^{7}} - \frac{a^{3}({x}^{(\frac{a}{x})}((\frac{a*-1}{x^{2}})ln(x) + \frac{(\frac{a}{x})(1)}{(x)}))ln^{3}(x)}{x^{6}} - \frac{a^{3}{x}^{(\frac{a}{x})}*3ln^{2}(x)}{x^{6}(x)} + \frac{15a^{2}*-5{x}^{(\frac{a}{x})}ln(x)}{x^{6}} + \frac{15a^{2}({x}^{(\frac{a}{x})}((\frac{a*-1}{x^{2}})ln(x) + \frac{(\frac{a}{x})(1)}{(x)}))ln(x)}{x^{5}} + \frac{15a^{2}{x}^{(\frac{a}{x})}}{x^{5}(x)} + \frac{3a^{3}*-6{x}^{(\frac{a}{x})}ln^{2}(x)}{x^{7}} + \frac{3a^{3}({x}^{(\frac{a}{x})}((\frac{a*-1}{x^{2}})ln(x) + \frac{(\frac{a}{x})(1)}{(x)}))ln^{2}(x)}{x^{6}} + \frac{3a^{3}{x}^{(\frac{a}{x})}*2ln(x)}{x^{6}(x)} - \frac{3a^{3}*-6{x}^{(\frac{a}{x})}ln(x)}{x^{7}} - \frac{3a^{3}({x}^{(\frac{a}{x})}((\frac{a*-1}{x^{2}})ln(x) + \frac{(\frac{a}{x})(1)}{(x)}))ln(x)}{x^{6}} - \frac{3a^{3}{x}^{(\frac{a}{x})}}{x^{6}(x)} + \frac{11a*-4{x}^{(\frac{a}{x})}}{x^{5}} + \frac{11a({x}^{(\frac{a}{x})}((\frac{a*-1}{x^{2}})ln(x) + \frac{(\frac{a}{x})(1)}{(x)}))}{x^{4}} - \frac{9a^{2}*-5{x}^{(\frac{a}{x})}}{x^{6}} - \frac{9a^{2}({x}^{(\frac{a}{x})}((\frac{a*-1}{x^{2}})ln(x) + \frac{(\frac{a}{x})(1)}{(x)}))}{x^{5}} + \frac{a^{3}*-6{x}^{(\frac{a}{x})}}{x^{7}} + \frac{a^{3}({x}^{(\frac{a}{x})}((\frac{a*-1}{x^{2}})ln(x) + \frac{(\frac{a}{x})(1)}{(x)}))}{x^{6}}\\=&\frac{24a{x}^{(\frac{a}{x})}ln(x)}{x^{5}} + \frac{36a^{2}{x}^{(\frac{a}{x})}ln^{2}(x)}{x^{6}} + \frac{12a^{3}{x}^{(\frac{a}{x})}ln^{3}(x)}{x^{7}} - \frac{104a^{2}{x}^{(\frac{a}{x})}ln(x)}{x^{6}} + \frac{a^{4}{x}^{(\frac{a}{x})}ln^{4}(x)}{x^{8}} - \frac{42a^{3}{x}^{(\frac{a}{x})}ln^{2}(x)}{x^{7}} - \frac{4a^{4}{x}^{(\frac{a}{x})}ln^{3}(x)}{x^{8}} + \frac{6a^{4}{x}^{(\frac{a}{x})}ln^{2}(x)}{x^{8}} + \frac{48a^{3}{x}^{(\frac{a}{x})}ln(x)}{x^{7}} - \frac{4a^{4}{x}^{(\frac{a}{x})}ln(x)}{x^{8}} - \frac{50a{x}^{(\frac{a}{x})}}{x^{5}} + \frac{71a^{2}{x}^{(\frac{a}{x})}}{x^{6}} - \frac{18a^{3}{x}^{(\frac{a}{x})}}{x^{7}} + \frac{a^{4}{x}^{(\frac{a}{x})}}{x^{8}}\\ \end{split}\end{equation} \]

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