求导函数：
输入一个原函数（即需要求导的函数），然后设置需要求导的变量和求导的阶数，点击“下一步”按钮，即可获得该函数相应阶数的导函数。
注意，输入的函数支持数学函数和其它常量。
输入一个原函数（即需要求导的函数），然后设置需要求导的变量和求导的阶数，点击“下一步”按钮，即可获得该函数相应阶数的导函数。
注意，输入的函数支持数学函数和其它常量。
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。
\[ \begin{equation}\begin{split}【1/1】求函数ln(x) - \frac{(xx)}{(x - ln(x))} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = ln(x) - \frac{x^{2}}{(x - ln(x))}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( ln(x) - \frac{x^{2}}{(x - ln(x))}\right)}{dx}\\=&\frac{1}{(x)} - (\frac{-(1 - \frac{1}{(x)})}{(x - ln(x))^{2}})x^{2} - \frac{2x}{(x - ln(x))}\\=&\frac{1}{x} - \frac{x}{(x - ln(x))^{2}} + \frac{x^{2}}{(x - ln(x))^{2}} - \frac{2x}{(x - ln(x))}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{1}{x} - \frac{x}{(x - ln(x))^{2}} + \frac{x^{2}}{(x - ln(x))^{2}} - \frac{2x}{(x - ln(x))}\right)}{dx}\\=&\frac{-1}{x^{2}} - (\frac{-2(1 - \frac{1}{(x)})}{(x - ln(x))^{3}})x - \frac{1}{(x - ln(x))^{2}} + (\frac{-2(1 - \frac{1}{(x)})}{(x - ln(x))^{3}})x^{2} + \frac{2x}{(x - ln(x))^{2}} - 2(\frac{-(1 - \frac{1}{(x)})}{(x - ln(x))^{2}})x - \frac{2}{(x - ln(x))}\\=&\frac{-1}{x^{2}} + \frac{4x}{(x - ln(x))^{3}} + \frac{4x}{(x - ln(x))^{2}} - \frac{2x^{2}}{(x - ln(x))^{3}} - \frac{2}{(x - ln(x))^{3}} - \frac{3}{(x - ln(x))^{2}} - \frac{2}{(x - ln(x))}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-1}{x^{2}} + \frac{4x}{(x - ln(x))^{3}} + \frac{4x}{(x - ln(x))^{2}} - \frac{2x^{2}}{(x - ln(x))^{3}} - \frac{2}{(x - ln(x))^{3}} - \frac{3}{(x - ln(x))^{2}} - \frac{2}{(x - ln(x))}\right)}{dx}\\=&\frac{--2}{x^{3}} + 4(\frac{-3(1 - \frac{1}{(x)})}{(x - ln(x))^{4}})x + \frac{4}{(x - ln(x))^{3}} + 4(\frac{-2(1 - \frac{1}{(x)})}{(x - ln(x))^{3}})x + \frac{4}{(x - ln(x))^{2}} - 2(\frac{-3(1 - \frac{1}{(x)})}{(x - ln(x))^{4}})x^{2} - \frac{2*2x}{(x - ln(x))^{3}} - 2(\frac{-3(1 - \frac{1}{(x)})}{(x - ln(x))^{4}}) - 3(\frac{-2(1 - \frac{1}{(x)})}{(x - ln(x))^{3}}) - 2(\frac{-(1 - \frac{1}{(x)})}{(x - ln(x))^{2}})\\=&\frac{2}{x^{3}} - \frac{18x}{(x - ln(x))^{4}} - \frac{12x}{(x - ln(x))^{3}} + \frac{6x^{2}}{(x - ln(x))^{4}} - \frac{6}{(x - ln(x))^{4}x} - \frac{6}{(x - ln(x))^{3}x} - \frac{2}{(x - ln(x))^{2}x} + \frac{18}{(x - ln(x))^{3}} + \frac{18}{(x - ln(x))^{4}} + \frac{6}{(x - ln(x))^{2}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{2}{x^{3}} - \frac{18x}{(x - ln(x))^{4}} - \frac{12x}{(x - ln(x))^{3}} + \frac{6x^{2}}{(x - ln(x))^{4}} - \frac{6}{(x - ln(x))^{4}x} - \frac{6}{(x - ln(x))^{3}x} - \frac{2}{(x - ln(x))^{2}x} + \frac{18}{(x - ln(x))^{3}} + \frac{18}{(x - ln(x))^{4}} + \frac{6}{(x - ln(x))^{2}}\right)}{dx}\\=&\frac{2*-3}{x^{4}} - 18(\frac{-4(1 - \frac{1}{(x)})}{(x - ln(x))^{5}})x - \frac{18}{(x - ln(x))^{4}} - 12(\frac{-3(1 - \frac{1}{(x)})}{(x - ln(x))^{4}})x - \frac{12}{(x - ln(x))^{3}} + 6(\frac{-4(1 - \frac{1}{(x)})}{(x - ln(x))^{5}})x^{2} + \frac{6*2x}{(x - ln(x))^{4}} - \frac{6(\frac{-4(1 - \frac{1}{(x)})}{(x - ln(x))^{5}})}{x} - \frac{6*-1}{(x - ln(x))^{4}x^{2}} - \frac{6(\frac{-3(1 - \frac{1}{(x)})}{(x - ln(x))^{4}})}{x} - \frac{6*-1}{(x - ln(x))^{3}x^{2}} - \frac{2(\frac{-2(1 - \frac{1}{(x)})}{(x - ln(x))^{3}})}{x} - \frac{2*-1}{(x - ln(x))^{2}x^{2}} + 18(\frac{-3(1 - \frac{1}{(x)})}{(x - ln(x))^{4}}) + 18(\frac{-4(1 - \frac{1}{(x)})}{(x - ln(x))^{5}}) + 6(\frac{-2(1 - \frac{1}{(x)})}{(x - ln(x))^{3}})\\=&\frac{-6}{x^{4}} + \frac{96x}{(x - ln(x))^{5}} + \frac{48x}{(x - ln(x))^{4}} - \frac{12}{(x - ln(x))^{4}x^{2}} - \frac{24}{(x - ln(x))^{5}x^{2}} - \frac{24x^{2}}{(x - ln(x))^{5}} + \frac{96}{(x - ln(x))^{5}x} + \frac{2}{(x - ln(x))^{3}x^{2}} + \frac{72}{(x - ln(x))^{4}x} + \frac{16}{(x - ln(x))^{3}x} + \frac{2}{(x - ln(x))^{2}x^{2}} - \frac{24}{(x - ln(x))^{3}} - \frac{144}{(x - ln(x))^{5}} - \frac{108}{(x - ln(x))^{4}}\\ \end{split}\end{equation} \]
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数ln(x) - \frac{(xx)}{(x - ln(x))} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = ln(x) - \frac{x^{2}}{(x - ln(x))}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( ln(x) - \frac{x^{2}}{(x - ln(x))}\right)}{dx}\\=&\frac{1}{(x)} - (\frac{-(1 - \frac{1}{(x)})}{(x - ln(x))^{2}})x^{2} - \frac{2x}{(x - ln(x))}\\=&\frac{1}{x} - \frac{x}{(x - ln(x))^{2}} + \frac{x^{2}}{(x - ln(x))^{2}} - \frac{2x}{(x - ln(x))}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{1}{x} - \frac{x}{(x - ln(x))^{2}} + \frac{x^{2}}{(x - ln(x))^{2}} - \frac{2x}{(x - ln(x))}\right)}{dx}\\=&\frac{-1}{x^{2}} - (\frac{-2(1 - \frac{1}{(x)})}{(x - ln(x))^{3}})x - \frac{1}{(x - ln(x))^{2}} + (\frac{-2(1 - \frac{1}{(x)})}{(x - ln(x))^{3}})x^{2} + \frac{2x}{(x - ln(x))^{2}} - 2(\frac{-(1 - \frac{1}{(x)})}{(x - ln(x))^{2}})x - \frac{2}{(x - ln(x))}\\=&\frac{-1}{x^{2}} + \frac{4x}{(x - ln(x))^{3}} + \frac{4x}{(x - ln(x))^{2}} - \frac{2x^{2}}{(x - ln(x))^{3}} - \frac{2}{(x - ln(x))^{3}} - \frac{3}{(x - ln(x))^{2}} - \frac{2}{(x - ln(x))}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-1}{x^{2}} + \frac{4x}{(x - ln(x))^{3}} + \frac{4x}{(x - ln(x))^{2}} - \frac{2x^{2}}{(x - ln(x))^{3}} - \frac{2}{(x - ln(x))^{3}} - \frac{3}{(x - ln(x))^{2}} - \frac{2}{(x - ln(x))}\right)}{dx}\\=&\frac{--2}{x^{3}} + 4(\frac{-3(1 - \frac{1}{(x)})}{(x - ln(x))^{4}})x + \frac{4}{(x - ln(x))^{3}} + 4(\frac{-2(1 - \frac{1}{(x)})}{(x - ln(x))^{3}})x + \frac{4}{(x - ln(x))^{2}} - 2(\frac{-3(1 - \frac{1}{(x)})}{(x - ln(x))^{4}})x^{2} - \frac{2*2x}{(x - ln(x))^{3}} - 2(\frac{-3(1 - \frac{1}{(x)})}{(x - ln(x))^{4}}) - 3(\frac{-2(1 - \frac{1}{(x)})}{(x - ln(x))^{3}}) - 2(\frac{-(1 - \frac{1}{(x)})}{(x - ln(x))^{2}})\\=&\frac{2}{x^{3}} - \frac{18x}{(x - ln(x))^{4}} - \frac{12x}{(x - ln(x))^{3}} + \frac{6x^{2}}{(x - ln(x))^{4}} - \frac{6}{(x - ln(x))^{4}x} - \frac{6}{(x - ln(x))^{3}x} - \frac{2}{(x - ln(x))^{2}x} + \frac{18}{(x - ln(x))^{3}} + \frac{18}{(x - ln(x))^{4}} + \frac{6}{(x - ln(x))^{2}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{2}{x^{3}} - \frac{18x}{(x - ln(x))^{4}} - \frac{12x}{(x - ln(x))^{3}} + \frac{6x^{2}}{(x - ln(x))^{4}} - \frac{6}{(x - ln(x))^{4}x} - \frac{6}{(x - ln(x))^{3}x} - \frac{2}{(x - ln(x))^{2}x} + \frac{18}{(x - ln(x))^{3}} + \frac{18}{(x - ln(x))^{4}} + \frac{6}{(x - ln(x))^{2}}\right)}{dx}\\=&\frac{2*-3}{x^{4}} - 18(\frac{-4(1 - \frac{1}{(x)})}{(x - ln(x))^{5}})x - \frac{18}{(x - ln(x))^{4}} - 12(\frac{-3(1 - \frac{1}{(x)})}{(x - ln(x))^{4}})x - \frac{12}{(x - ln(x))^{3}} + 6(\frac{-4(1 - \frac{1}{(x)})}{(x - ln(x))^{5}})x^{2} + \frac{6*2x}{(x - ln(x))^{4}} - \frac{6(\frac{-4(1 - \frac{1}{(x)})}{(x - ln(x))^{5}})}{x} - \frac{6*-1}{(x - ln(x))^{4}x^{2}} - \frac{6(\frac{-3(1 - \frac{1}{(x)})}{(x - ln(x))^{4}})}{x} - \frac{6*-1}{(x - ln(x))^{3}x^{2}} - \frac{2(\frac{-2(1 - \frac{1}{(x)})}{(x - ln(x))^{3}})}{x} - \frac{2*-1}{(x - ln(x))^{2}x^{2}} + 18(\frac{-3(1 - \frac{1}{(x)})}{(x - ln(x))^{4}}) + 18(\frac{-4(1 - \frac{1}{(x)})}{(x - ln(x))^{5}}) + 6(\frac{-2(1 - \frac{1}{(x)})}{(x - ln(x))^{3}})\\=&\frac{-6}{x^{4}} + \frac{96x}{(x - ln(x))^{5}} + \frac{48x}{(x - ln(x))^{4}} - \frac{12}{(x - ln(x))^{4}x^{2}} - \frac{24}{(x - ln(x))^{5}x^{2}} - \frac{24x^{2}}{(x - ln(x))^{5}} + \frac{96}{(x - ln(x))^{5}x} + \frac{2}{(x - ln(x))^{3}x^{2}} + \frac{72}{(x - ln(x))^{4}x} + \frac{16}{(x - ln(x))^{3}x} + \frac{2}{(x - ln(x))^{2}x^{2}} - \frac{24}{(x - ln(x))^{3}} - \frac{144}{(x - ln(x))^{5}} - \frac{108}{(x - ln(x))^{4}}\\ \end{split}\end{equation} \]
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