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。
\[ \begin{equation}\begin{split}【1/1】求函数{2}^{(\frac{x}{(x - 1)})} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = {2}^{(\frac{x}{(x - 1)})}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( {2}^{(\frac{x}{(x - 1)})}\right)}{dx}\\=&({2}^{(\frac{x}{(x - 1)})}(((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)})ln(2) + \frac{(\frac{x}{(x - 1)})(0)}{(2)}))\\=&\frac{-x{2}^{(\frac{x}{(x - 1)})}ln(2)}{(x - 1)^{2}} + \frac{{2}^{(\frac{x}{(x - 1)})}ln(2)}{(x - 1)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{-x{2}^{(\frac{x}{(x - 1)})}ln(2)}{(x - 1)^{2}} + \frac{{2}^{(\frac{x}{(x - 1)})}ln(2)}{(x - 1)}\right)}{dx}\\=&-(\frac{-2(1 + 0)}{(x - 1)^{3}})x{2}^{(\frac{x}{(x - 1)})}ln(2) - \frac{{2}^{(\frac{x}{(x - 1)})}ln(2)}{(x - 1)^{2}} - \frac{x({2}^{(\frac{x}{(x - 1)})}(((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)})ln(2) + \frac{(\frac{x}{(x - 1)})(0)}{(2)}))ln(2)}{(x - 1)^{2}} - \frac{x{2}^{(\frac{x}{(x - 1)})}*0}{(x - 1)^{2}(2)} + (\frac{-(1 + 0)}{(x - 1)^{2}}){2}^{(\frac{x}{(x - 1)})}ln(2) + \frac{({2}^{(\frac{x}{(x - 1)})}(((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)})ln(2) + \frac{(\frac{x}{(x - 1)})(0)}{(2)}))ln(2)}{(x - 1)} + \frac{{2}^{(\frac{x}{(x - 1)})}*0}{(x - 1)(2)}\\=&\frac{2x{2}^{(\frac{x}{(x - 1)})}ln(2)}{(x - 1)^{3}} - \frac{2 * {2}^{(\frac{x}{(x - 1)})}ln(2)}{(x - 1)^{2}} + \frac{x^{2}{2}^{(\frac{x}{(x - 1)})}ln^{2}(2)}{(x - 1)^{4}} - \frac{2x{2}^{(\frac{x}{(x - 1)})}ln^{2}(2)}{(x - 1)^{3}} + \frac{{2}^{(\frac{x}{(x - 1)})}ln^{2}(2)}{(x - 1)^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{2x{2}^{(\frac{x}{(x - 1)})}ln(2)}{(x - 1)^{3}} - \frac{2 * {2}^{(\frac{x}{(x - 1)})}ln(2)}{(x - 1)^{2}} + \frac{x^{2}{2}^{(\frac{x}{(x - 1)})}ln^{2}(2)}{(x - 1)^{4}} - \frac{2x{2}^{(\frac{x}{(x - 1)})}ln^{2}(2)}{(x - 1)^{3}} + \frac{{2}^{(\frac{x}{(x - 1)})}ln^{2}(2)}{(x - 1)^{2}}\right)}{dx}\\=&2(\frac{-3(1 + 0)}{(x - 1)^{4}})x{2}^{(\frac{x}{(x - 1)})}ln(2) + \frac{2 * {2}^{(\frac{x}{(x - 1)})}ln(2)}{(x - 1)^{3}} + \frac{2x({2}^{(\frac{x}{(x - 1)})}(((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)})ln(2) + \frac{(\frac{x}{(x - 1)})(0)}{(2)}))ln(2)}{(x - 1)^{3}} + \frac{2x{2}^{(\frac{x}{(x - 1)})}*0}{(x - 1)^{3}(2)} - 2(\frac{-2(1 + 0)}{(x - 1)^{3}}){2}^{(\frac{x}{(x - 1)})}ln(2) - \frac{2({2}^{(\frac{x}{(x - 1)})}(((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)})ln(2) + \frac{(\frac{x}{(x - 1)})(0)}{(2)}))ln(2)}{(x - 1)^{2}} - \frac{2 * {2}^{(\frac{x}{(x - 1)})}*0}{(x - 1)^{2}(2)} + (\frac{-4(1 + 0)}{(x - 1)^{5}})x^{2}{2}^{(\frac{x}{(x - 1)})}ln^{2}(2) + \frac{2x{2}^{(\frac{x}{(x - 1)})}ln^{2}(2)}{(x - 1)^{4}} + \frac{x^{2}({2}^{(\frac{x}{(x - 1)})}(((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)})ln(2) + \frac{(\frac{x}{(x - 1)})(0)}{(2)}))ln^{2}(2)}{(x - 1)^{4}} + \frac{x^{2}{2}^{(\frac{x}{(x - 1)})}*2ln(2)*0}{(x - 1)^{4}(2)} - 2(\frac{-3(1 + 0)}{(x - 1)^{4}})x{2}^{(\frac{x}{(x - 1)})}ln^{2}(2) - \frac{2 * {2}^{(\frac{x}{(x - 1)})}ln^{2}(2)}{(x - 1)^{3}} - \frac{2x({2}^{(\frac{x}{(x - 1)})}(((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)})ln(2) + \frac{(\frac{x}{(x - 1)})(0)}{(2)}))ln^{2}(2)}{(x - 1)^{3}} - \frac{2x{2}^{(\frac{x}{(x - 1)})}*2ln(2)*0}{(x - 1)^{3}(2)} + (\frac{-2(1 + 0)}{(x - 1)^{3}}){2}^{(\frac{x}{(x - 1)})}ln^{2}(2) + \frac{({2}^{(\frac{x}{(x - 1)})}(((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)})ln(2) + \frac{(\frac{x}{(x - 1)})(0)}{(2)}))ln^{2}(2)}{(x - 1)^{2}} + \frac{{2}^{(\frac{x}{(x - 1)})}*2ln(2)*0}{(x - 1)^{2}(2)}\\=&\frac{-6x{2}^{(\frac{x}{(x - 1)})}ln(2)}{(x - 1)^{4}} + \frac{6 * {2}^{(\frac{x}{(x - 1)})}ln(2)}{(x - 1)^{3}} - \frac{6x^{2}{2}^{(\frac{x}{(x - 1)})}ln^{2}(2)}{(x - 1)^{5}} + \frac{12x{2}^{(\frac{x}{(x - 1)})}ln^{2}(2)}{(x - 1)^{4}} - \frac{6 * {2}^{(\frac{x}{(x - 1)})}ln^{2}(2)}{(x - 1)^{3}} - \frac{x^{3}{2}^{(\frac{x}{(x - 1)})}ln^{3}(2)}{(x - 1)^{6}} + \frac{3x^{2}{2}^{(\frac{x}{(x - 1)})}ln^{3}(2)}{(x - 1)^{5}} - \frac{3x{2}^{(\frac{x}{(x - 1)})}ln^{3}(2)}{(x - 1)^{4}} + \frac{{2}^{(\frac{x}{(x - 1)})}ln^{3}(2)}{(x - 1)^{3}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{-6x{2}^{(\frac{x}{(x - 1)})}ln(2)}{(x - 1)^{4}} + \frac{6 * {2}^{(\frac{x}{(x - 1)})}ln(2)}{(x - 1)^{3}} - \frac{6x^{2}{2}^{(\frac{x}{(x - 1)})}ln^{2}(2)}{(x - 1)^{5}} + \frac{12x{2}^{(\frac{x}{(x - 1)})}ln^{2}(2)}{(x - 1)^{4}} - \frac{6 * {2}^{(\frac{x}{(x - 1)})}ln^{2}(2)}{(x - 1)^{3}} - \frac{x^{3}{2}^{(\frac{x}{(x - 1)})}ln^{3}(2)}{(x - 1)^{6}} + \frac{3x^{2}{2}^{(\frac{x}{(x - 1)})}ln^{3}(2)}{(x - 1)^{5}} - \frac{3x{2}^{(\frac{x}{(x - 1)})}ln^{3}(2)}{(x - 1)^{4}} + \frac{{2}^{(\frac{x}{(x - 1)})}ln^{3}(2)}{(x - 1)^{3}}\right)}{dx}\\=&-6(\frac{-4(1 + 0)}{(x - 1)^{5}})x{2}^{(\frac{x}{(x - 1)})}ln(2) - \frac{6 * {2}^{(\frac{x}{(x - 1)})}ln(2)}{(x - 1)^{4}} - \frac{6x({2}^{(\frac{x}{(x - 1)})}(((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)})ln(2) + \frac{(\frac{x}{(x - 1)})(0)}{(2)}))ln(2)}{(x - 1)^{4}} - \frac{6x{2}^{(\frac{x}{(x - 1)})}*0}{(x - 1)^{4}(2)} + 6(\frac{-3(1 + 0)}{(x - 1)^{4}}){2}^{(\frac{x}{(x - 1)})}ln(2) + \frac{6({2}^{(\frac{x}{(x - 1)})}(((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)})ln(2) + \frac{(\frac{x}{(x - 1)})(0)}{(2)}))ln(2)}{(x - 1)^{3}} + \frac{6 * {2}^{(\frac{x}{(x - 1)})}*0}{(x - 1)^{3}(2)} - 6(\frac{-5(1 + 0)}{(x - 1)^{6}})x^{2}{2}^{(\frac{x}{(x - 1)})}ln^{2}(2) - \frac{6*2x{2}^{(\frac{x}{(x - 1)})}ln^{2}(2)}{(x - 1)^{5}} - \frac{6x^{2}({2}^{(\frac{x}{(x - 1)})}(((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)})ln(2) + \frac{(\frac{x}{(x - 1)})(0)}{(2)}))ln^{2}(2)}{(x - 1)^{5}} - \frac{6x^{2}{2}^{(\frac{x}{(x - 1)})}*2ln(2)*0}{(x - 1)^{5}(2)} + 12(\frac{-4(1 + 0)}{(x - 1)^{5}})x{2}^{(\frac{x}{(x - 1)})}ln^{2}(2) + \frac{12 * {2}^{(\frac{x}{(x - 1)})}ln^{2}(2)}{(x - 1)^{4}} + \frac{12x({2}^{(\frac{x}{(x - 1)})}(((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)})ln(2) + \frac{(\frac{x}{(x - 1)})(0)}{(2)}))ln^{2}(2)}{(x - 1)^{4}} + \frac{12x{2}^{(\frac{x}{(x - 1)})}*2ln(2)*0}{(x - 1)^{4}(2)} - 6(\frac{-3(1 + 0)}{(x - 1)^{4}}){2}^{(\frac{x}{(x - 1)})}ln^{2}(2) - \frac{6({2}^{(\frac{x}{(x - 1)})}(((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)})ln(2) + \frac{(\frac{x}{(x - 1)})(0)}{(2)}))ln^{2}(2)}{(x - 1)^{3}} - \frac{6 * {2}^{(\frac{x}{(x - 1)})}*2ln(2)*0}{(x - 1)^{3}(2)} - (\frac{-6(1 + 0)}{(x - 1)^{7}})x^{3}{2}^{(\frac{x}{(x - 1)})}ln^{3}(2) - \frac{3x^{2}{2}^{(\frac{x}{(x - 1)})}ln^{3}(2)}{(x - 1)^{6}} - \frac{x^{3}({2}^{(\frac{x}{(x - 1)})}(((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)})ln(2) + \frac{(\frac{x}{(x - 1)})(0)}{(2)}))ln^{3}(2)}{(x - 1)^{6}} - \frac{x^{3}{2}^{(\frac{x}{(x - 1)})}*3ln^{2}(2)*0}{(x - 1)^{6}(2)} + 3(\frac{-5(1 + 0)}{(x - 1)^{6}})x^{2}{2}^{(\frac{x}{(x - 1)})}ln^{3}(2) + \frac{3*2x{2}^{(\frac{x}{(x - 1)})}ln^{3}(2)}{(x - 1)^{5}} + \frac{3x^{2}({2}^{(\frac{x}{(x - 1)})}(((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)})ln(2) + \frac{(\frac{x}{(x - 1)})(0)}{(2)}))ln^{3}(2)}{(x - 1)^{5}} + \frac{3x^{2}{2}^{(\frac{x}{(x - 1)})}*3ln^{2}(2)*0}{(x - 1)^{5}(2)} - 3(\frac{-4(1 + 0)}{(x - 1)^{5}})x{2}^{(\frac{x}{(x - 1)})}ln^{3}(2) - \frac{3 * {2}^{(\frac{x}{(x - 1)})}ln^{3}(2)}{(x - 1)^{4}} - \frac{3x({2}^{(\frac{x}{(x - 1)})}(((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)})ln(2) + \frac{(\frac{x}{(x - 1)})(0)}{(2)}))ln^{3}(2)}{(x - 1)^{4}} - \frac{3x{2}^{(\frac{x}{(x - 1)})}*3ln^{2}(2)*0}{(x - 1)^{4}(2)} + (\frac{-3(1 + 0)}{(x - 1)^{4}}){2}^{(\frac{x}{(x - 1)})}ln^{3}(2) + \frac{({2}^{(\frac{x}{(x - 1)})}(((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)})ln(2) + \frac{(\frac{x}{(x - 1)})(0)}{(2)}))ln^{3}(2)}{(x - 1)^{3}} + \frac{{2}^{(\frac{x}{(x - 1)})}*3ln^{2}(2)*0}{(x - 1)^{3}(2)}\\=&\frac{24x{2}^{(\frac{x}{(x - 1)})}ln(2)}{(x - 1)^{5}} - \frac{24 * {2}^{(\frac{x}{(x - 1)})}ln(2)}{(x - 1)^{4}} + \frac{36x^{2}{2}^{(\frac{x}{(x - 1)})}ln^{2}(2)}{(x - 1)^{6}} - \frac{72x{2}^{(\frac{x}{(x - 1)})}ln^{2}(2)}{(x - 1)^{5}} + \frac{36 * {2}^{(\frac{x}{(x - 1)})}ln^{2}(2)}{(x - 1)^{4}} + \frac{12x^{3}{2}^{(\frac{x}{(x - 1)})}ln^{3}(2)}{(x - 1)^{7}} - \frac{36x^{2}{2}^{(\frac{x}{(x - 1)})}ln^{3}(2)}{(x - 1)^{6}} + \frac{36x{2}^{(\frac{x}{(x - 1)})}ln^{3}(2)}{(x - 1)^{5}} - \frac{12 * {2}^{(\frac{x}{(x - 1)})}ln^{3}(2)}{(x - 1)^{4}} + \frac{x^{4}{2}^{(\frac{x}{(x - 1)})}ln^{4}(2)}{(x - 1)^{8}} - \frac{4x^{3}{2}^{(\frac{x}{(x - 1)})}ln^{4}(2)}{(x - 1)^{7}} + \frac{6x^{2}{2}^{(\frac{x}{(x - 1)})}ln^{4}(2)}{(x - 1)^{6}} - \frac{4x{2}^{(\frac{x}{(x - 1)})}ln^{4}(2)}{(x - 1)^{5}} + \frac{{2}^{(\frac{x}{(x - 1)})}ln^{4}(2)}{(x - 1)^{4}}\\ \end{split}\end{equation} \]

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