本次共计算 1 个题目：每一题对 x 求 3 阶导数。
    注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数ln(\frac{(1 - 2x)}{(1 + 3x)}) 关于 x 的 3 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = ln(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( ln(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})\right)}{dx}\\=&\frac{(-2(\frac{-(3 + 0)}{(3x + 1)^{2}})x - \frac{2}{(3x + 1)} + (\frac{-(3 + 0)}{(3x + 1)^{2}}))}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})}\\=&\frac{6x}{(3x + 1)^{2}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})} - \frac{3}{(3x + 1)^{2}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})} - \frac{2}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})(3x + 1)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{6x}{(3x + 1)^{2}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})} - \frac{3}{(3x + 1)^{2}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})} - \frac{2}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})(3x + 1)}\right)}{dx}\\=&\frac{6(\frac{-2(3 + 0)}{(3x + 1)^{3}})x}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})} + \frac{6(\frac{-(-2(\frac{-(3 + 0)}{(3x + 1)^{2}})x - \frac{2}{(3x + 1)} + (\frac{-(3 + 0)}{(3x + 1)^{2}}))}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{2}})x}{(3x + 1)^{2}} + \frac{6}{(3x + 1)^{2}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})} - \frac{3(\frac{-2(3 + 0)}{(3x + 1)^{3}})}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})} - \frac{3(\frac{-(-2(\frac{-(3 + 0)}{(3x + 1)^{2}})x - \frac{2}{(3x + 1)} + (\frac{-(3 + 0)}{(3x + 1)^{2}}))}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{2}})}{(3x + 1)^{2}} - \frac{2(\frac{-(-2(\frac{-(3 + 0)}{(3x + 1)^{2}})x - \frac{2}{(3x + 1)} + (\frac{-(3 + 0)}{(3x + 1)^{2}}))}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{2}})}{(3x + 1)} - \frac{2(\frac{-(3 + 0)}{(3x + 1)^{2}})}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})}\\=&\frac{-36x}{(3x + 1)^{3}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})} - \frac{36x^{2}}{(3x + 1)^{4}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{2}} + \frac{12x}{(3x + 1)^{3}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{2}} + \frac{36x}{(3x + 1)^{4}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{2}} + \frac{12x}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{2}(3x + 1)^{3}} + \frac{12}{(3x + 1)^{2}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})} - \frac{9}{(3x + 1)^{4}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{2}} + \frac{18}{(3x + 1)^{3}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})} - \frac{6}{(3x + 1)^{3}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{2}} - \frac{6}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{2}(3x + 1)^{3}} - \frac{4}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{2}(3x + 1)^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-36x}{(3x + 1)^{3}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})} - \frac{36x^{2}}{(3x + 1)^{4}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{2}} + \frac{12x}{(3x + 1)^{3}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{2}} + \frac{36x}{(3x + 1)^{4}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{2}} + \frac{12x}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{2}(3x + 1)^{3}} + \frac{12}{(3x + 1)^{2}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})} - \frac{9}{(3x + 1)^{4}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{2}} + \frac{18}{(3x + 1)^{3}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})} - \frac{6}{(3x + 1)^{3}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{2}} - \frac{6}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{2}(3x + 1)^{3}} - \frac{4}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{2}(3x + 1)^{2}}\right)}{dx}\\=&\frac{-36(\frac{-3(3 + 0)}{(3x + 1)^{4}})x}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})} - \frac{36(\frac{-(-2(\frac{-(3 + 0)}{(3x + 1)^{2}})x - \frac{2}{(3x + 1)} + (\frac{-(3 + 0)}{(3x + 1)^{2}}))}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{2}})x}{(3x + 1)^{3}} - \frac{36}{(3x + 1)^{3}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})} - \frac{36(\frac{-4(3 + 0)}{(3x + 1)^{5}})x^{2}}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{2}} - \frac{36(\frac{-2(-2(\frac{-(3 + 0)}{(3x + 1)^{2}})x - \frac{2}{(3x + 1)} + (\frac{-(3 + 0)}{(3x + 1)^{2}}))}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{3}})x^{2}}{(3x + 1)^{4}} - \frac{36*2x}{(3x + 1)^{4}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{2}} + \frac{12(\frac{-3(3 + 0)}{(3x + 1)^{4}})x}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{2}} + \frac{12(\frac{-2(-2(\frac{-(3 + 0)}{(3x + 1)^{2}})x - \frac{2}{(3x + 1)} + (\frac{-(3 + 0)}{(3x + 1)^{2}}))}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{3}})x}{(3x + 1)^{3}} + \frac{12}{(3x + 1)^{3}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{2}} + \frac{36(\frac{-4(3 + 0)}{(3x + 1)^{5}})x}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{2}} + \frac{36(\frac{-2(-2(\frac{-(3 + 0)}{(3x + 1)^{2}})x - \frac{2}{(3x + 1)} + (\frac{-(3 + 0)}{(3x + 1)^{2}}))}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{3}})x}{(3x + 1)^{4}} + \frac{36}{(3x + 1)^{4}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{2}} + \frac{12(\frac{-2(-2(\frac{-(3 + 0)}{(3x + 1)^{2}})x - \frac{2}{(3x + 1)} + (\frac{-(3 + 0)}{(3x + 1)^{2}}))}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{3}})x}{(3x + 1)^{3}} + \frac{12(\frac{-3(3 + 0)}{(3x + 1)^{4}})x}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{2}} + \frac{12}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{2}(3x + 1)^{3}} + \frac{12(\frac{-2(3 + 0)}{(3x + 1)^{3}})}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})} + \frac{12(\frac{-(-2(\frac{-(3 + 0)}{(3x + 1)^{2}})x - \frac{2}{(3x + 1)} + (\frac{-(3 + 0)}{(3x + 1)^{2}}))}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{2}})}{(3x + 1)^{2}} - \frac{9(\frac{-4(3 + 0)}{(3x + 1)^{5}})}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{2}} - \frac{9(\frac{-2(-2(\frac{-(3 + 0)}{(3x + 1)^{2}})x - \frac{2}{(3x + 1)} + (\frac{-(3 + 0)}{(3x + 1)^{2}}))}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{3}})}{(3x + 1)^{4}} + \frac{18(\frac{-3(3 + 0)}{(3x + 1)^{4}})}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})} + \frac{18(\frac{-(-2(\frac{-(3 + 0)}{(3x + 1)^{2}})x - \frac{2}{(3x + 1)} + (\frac{-(3 + 0)}{(3x + 1)^{2}}))}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{2}})}{(3x + 1)^{3}} - \frac{6(\frac{-3(3 + 0)}{(3x + 1)^{4}})}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{2}} - \frac{6(\frac{-2(-2(\frac{-(3 + 0)}{(3x + 1)^{2}})x - \frac{2}{(3x + 1)} + (\frac{-(3 + 0)}{(3x + 1)^{2}}))}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{3}})}{(3x + 1)^{3}} - \frac{6(\frac{-2(-2(\frac{-(3 + 0)}{(3x + 1)^{2}})x - \frac{2}{(3x + 1)} + (\frac{-(3 + 0)}{(3x + 1)^{2}}))}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{3}})}{(3x + 1)^{3}} - \frac{6(\frac{-3(3 + 0)}{(3x + 1)^{4}})}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{2}} - \frac{4(\frac{-2(-2(\frac{-(3 + 0)}{(3x + 1)^{2}})x - \frac{2}{(3x + 1)} + (\frac{-(3 + 0)}{(3x + 1)^{2}}))}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{3}})}{(3x + 1)^{2}} - \frac{4(\frac{-2(3 + 0)}{(3x + 1)^{3}})}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{2}}\\=&\frac{324x}{(3x + 1)^{4}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})} + \frac{648x^{2}}{(3x + 1)^{5}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{2}} - \frac{360x}{(3x + 1)^{4}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{2}} - \frac{648x}{(3x + 1)^{5}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{2}} - \frac{72x}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{2}(3x + 1)^{4}} + \frac{432x^{3}}{(3x + 1)^{6}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{3}} - \frac{288x^{2}}{(3x + 1)^{5}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{3}} - \frac{648x^{2}}{(3x + 1)^{6}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{3}} - \frac{144x^{2}}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{3}(3x + 1)^{5}} + \frac{48x}{(3x + 1)^{4}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{3}} + \frac{288x}{(3x + 1)^{5}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{3}} + \frac{324x}{(3x + 1)^{6}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{3}} + \frac{96x}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{3}(3x + 1)^{4}} + \frac{144x}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{3}(3x + 1)^{5}} + \frac{12}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{2}(3x + 1)^{3}} + \frac{216}{(3x + 1)^{4}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{2}} + \frac{162}{(3x + 1)^{5}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{2}} + \frac{60}{(3x + 1)^{3}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{2}} - \frac{54}{(3x + 1)^{6}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{3}} - \frac{162}{(3x + 1)^{4}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})} - \frac{108}{(3x + 1)^{3}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})} - \frac{72}{(3x + 1)^{5}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{3}} - \frac{36}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{3}(3x + 1)^{5}} - \frac{48}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{3}(3x + 1)^{4}} - \frac{24}{(3x + 1)^{4}(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{3}} - \frac{16}{(\frac{-2x}{(3x + 1)} + \frac{1}{(3x + 1)})^{3}(3x + 1)^{3}}\\ \end{split}\end{equation} \]

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