本次共计算 6 个题目：每一题对 x 求 4 阶导数。
    注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/6】求函数xe^{x} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( xe^{x}\right)}{dx}\\=&e^{x} + xe^{x}\\=&e^{x} + xe^{x}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( e^{x} + xe^{x}\right)}{dx}\\=&e^{x} + e^{x} + xe^{x}\\=&2e^{x} + xe^{x}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( 2e^{x} + xe^{x}\right)}{dx}\\=&2e^{x} + e^{x} + xe^{x}\\=&3e^{x} + xe^{x}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( 3e^{x} + xe^{x}\right)}{dx}\\=&3e^{x} + e^{x} + xe^{x}\\=&4e^{x} + xe^{x}\\ \end{split}\end{equation} \]

\[ \begin{equation}\begin{split}【2/6】求函数\frac{x}{e^{x}} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( \frac{x}{e^{x}}\right)}{dx}\\=&\frac{1}{e^{x}} + \frac{x*-e^{x}}{e^{{x}*{2}}}\\=&\frac{1}{e^{x}} - \frac{x}{e^{x}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{1}{e^{x}} - \frac{x}{e^{x}}\right)}{dx}\\=&\frac{-e^{x}}{e^{{x}*{2}}} - \frac{1}{e^{x}} - \frac{x*-e^{x}}{e^{{x}*{2}}}\\=&\frac{-2}{e^{x}} + \frac{x}{e^{x}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-2}{e^{x}} + \frac{x}{e^{x}}\right)}{dx}\\=&\frac{-2*-e^{x}}{e^{{x}*{2}}} + \frac{1}{e^{x}} + \frac{x*-e^{x}}{e^{{x}*{2}}}\\=&\frac{3}{e^{x}} - \frac{x}{e^{x}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{3}{e^{x}} - \frac{x}{e^{x}}\right)}{dx}\\=&\frac{3*-e^{x}}{e^{{x}*{2}}} - \frac{1}{e^{x}} - \frac{x*-e^{x}}{e^{{x}*{2}}}\\=&\frac{-4}{e^{x}} + \frac{x}{e^{x}}\\ \end{split}\end{equation} \]

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

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

\[ \begin{equation}\begin{split}【5/6】求函数\frac{ln(x)}{x} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{ln(x)}{x}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( \frac{ln(x)}{x}\right)}{dx}\\=&\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)}\\=&\frac{-ln(x)}{x^{2}} + \frac{1}{x^{2}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{-ln(x)}{x^{2}} + \frac{1}{x^{2}}\right)}{dx}\\=&\frac{--2ln(x)}{x^{3}} - \frac{1}{x^{2}(x)} + \frac{-2}{x^{3}}\\=&\frac{2ln(x)}{x^{3}} - \frac{3}{x^{3}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{2ln(x)}{x^{3}} - \frac{3}{x^{3}}\right)}{dx}\\=&\frac{2*-3ln(x)}{x^{4}} + \frac{2}{x^{3}(x)} - \frac{3*-3}{x^{4}}\\=&\frac{-6ln(x)}{x^{4}} + \frac{11}{x^{4}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{-6ln(x)}{x^{4}} + \frac{11}{x^{4}}\right)}{dx}\\=&\frac{-6*-4ln(x)}{x^{5}} - \frac{6}{x^{4}(x)} + \frac{11*-4}{x^{5}}\\=&\frac{24ln(x)}{x^{5}} - \frac{50}{x^{5}}\\ \end{split}\end{equation} \]

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

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