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。
\[ \begin{equation}\begin{split}【1/1】求函数e^{\frac{x}{(lg(x) + a)}} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = e^{\frac{x}{(lg(x) + a)}}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( e^{\frac{x}{(lg(x) + a)}}\right)}{dx}\\=&e^{\frac{x}{(lg(x) + a)}}((\frac{-(\frac{1}{ln{10}(x)} + 0)}{(lg(x) + a)^{2}})x + \frac{1}{(lg(x) + a)})\\=&\frac{-e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{2}ln{10}} + \frac{e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{-e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{2}ln{10}} + \frac{e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)}\right)}{dx}\\=&\frac{-(\frac{-2(\frac{1}{ln{10}(x)} + 0)}{(lg(x) + a)^{3}})e^{\frac{x}{(lg(x) + a)}}}{ln{10}} - \frac{e^{\frac{x}{(lg(x) + a)}}((\frac{-(\frac{1}{ln{10}(x)} + 0)}{(lg(x) + a)^{2}})x + \frac{1}{(lg(x) + a)})}{(lg(x) + a)^{2}ln{10}} - \frac{e^{\frac{x}{(lg(x) + a)}}*-0}{(lg(x) + a)^{2}ln^{2}{10}} + (\frac{-(\frac{1}{ln{10}(x)} + 0)}{(lg(x) + a)^{2}})e^{\frac{x}{(lg(x) + a)}} + \frac{e^{\frac{x}{(lg(x) + a)}}((\frac{-(\frac{1}{ln{10}(x)} + 0)}{(lg(x) + a)^{2}})x + \frac{1}{(lg(x) + a)})}{(lg(x) + a)}\\=&\frac{2e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{3}xln^{2}{10}} + \frac{e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{4}ln^{2}{10}} - \frac{2e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{3}ln{10}} - \frac{e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{2}xln{10}} + \frac{e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{2e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{3}xln^{2}{10}} + \frac{e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{4}ln^{2}{10}} - \frac{2e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{3}ln{10}} - \frac{e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{2}xln{10}} + \frac{e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{2}}\right)}{dx}\\=&\frac{2(\frac{-3(\frac{1}{ln{10}(x)} + 0)}{(lg(x) + a)^{4}})e^{\frac{x}{(lg(x) + a)}}}{xln^{2}{10}} + \frac{2*-e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{3}x^{2}ln^{2}{10}} + \frac{2e^{\frac{x}{(lg(x) + a)}}((\frac{-(\frac{1}{ln{10}(x)} + 0)}{(lg(x) + a)^{2}})x + \frac{1}{(lg(x) + a)})}{(lg(x) + a)^{3}xln^{2}{10}} + \frac{2e^{\frac{x}{(lg(x) + a)}}*-2*0}{(lg(x) + a)^{3}xln^{3}{10}} + \frac{(\frac{-4(\frac{1}{ln{10}(x)} + 0)}{(lg(x) + a)^{5}})e^{\frac{x}{(lg(x) + a)}}}{ln^{2}{10}} + \frac{e^{\frac{x}{(lg(x) + a)}}((\frac{-(\frac{1}{ln{10}(x)} + 0)}{(lg(x) + a)^{2}})x + \frac{1}{(lg(x) + a)})}{(lg(x) + a)^{4}ln^{2}{10}} + \frac{e^{\frac{x}{(lg(x) + a)}}*-2*0}{(lg(x) + a)^{4}ln^{3}{10}} - \frac{2(\frac{-3(\frac{1}{ln{10}(x)} + 0)}{(lg(x) + a)^{4}})e^{\frac{x}{(lg(x) + a)}}}{ln{10}} - \frac{2e^{\frac{x}{(lg(x) + a)}}((\frac{-(\frac{1}{ln{10}(x)} + 0)}{(lg(x) + a)^{2}})x + \frac{1}{(lg(x) + a)})}{(lg(x) + a)^{3}ln{10}} - \frac{2e^{\frac{x}{(lg(x) + a)}}*-0}{(lg(x) + a)^{3}ln^{2}{10}} - \frac{(\frac{-2(\frac{1}{ln{10}(x)} + 0)}{(lg(x) + a)^{3}})e^{\frac{x}{(lg(x) + a)}}}{xln{10}} - \frac{-e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{2}x^{2}ln{10}} - \frac{e^{\frac{x}{(lg(x) + a)}}((\frac{-(\frac{1}{ln{10}(x)} + 0)}{(lg(x) + a)^{2}})x + \frac{1}{(lg(x) + a)})}{(lg(x) + a)^{2}xln{10}} - \frac{e^{\frac{x}{(lg(x) + a)}}*-0}{(lg(x) + a)^{2}xln^{2}{10}} + (\frac{-2(\frac{1}{ln{10}(x)} + 0)}{(lg(x) + a)^{3}})e^{\frac{x}{(lg(x) + a)}} + \frac{e^{\frac{x}{(lg(x) + a)}}((\frac{-(\frac{1}{ln{10}(x)} + 0)}{(lg(x) + a)^{2}})x + \frac{1}{(lg(x) + a)})}{(lg(x) + a)^{2}}\\=&\frac{-6e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{4}x^{2}ln^{3}{10}} - \frac{6e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{5}xln^{3}{10}} + \frac{9e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{4}xln^{2}{10}} - \frac{e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{6}ln^{3}{10}} + \frac{3e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{5}ln^{2}{10}} - \frac{3e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{4}ln{10}} + \frac{e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{2}x^{2}ln{10}} - \frac{3e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{3}xln{10}} + \frac{e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{3}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{-6e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{4}x^{2}ln^{3}{10}} - \frac{6e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{5}xln^{3}{10}} + \frac{9e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{4}xln^{2}{10}} - \frac{e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{6}ln^{3}{10}} + \frac{3e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{5}ln^{2}{10}} - \frac{3e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{4}ln{10}} + \frac{e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{2}x^{2}ln{10}} - \frac{3e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{3}xln{10}} + \frac{e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{3}}\right)}{dx}\\=&\frac{-6(\frac{-4(\frac{1}{ln{10}(x)} + 0)}{(lg(x) + a)^{5}})e^{\frac{x}{(lg(x) + a)}}}{x^{2}ln^{3}{10}} - \frac{6*-2e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{4}x^{3}ln^{3}{10}} - \frac{6e^{\frac{x}{(lg(x) + a)}}((\frac{-(\frac{1}{ln{10}(x)} + 0)}{(lg(x) + a)^{2}})x + \frac{1}{(lg(x) + a)})}{(lg(x) + a)^{4}x^{2}ln^{3}{10}} - \frac{6e^{\frac{x}{(lg(x) + a)}}*-3*0}{(lg(x) + a)^{4}x^{2}ln^{4}{10}} - \frac{6(\frac{-5(\frac{1}{ln{10}(x)} + 0)}{(lg(x) + a)^{6}})e^{\frac{x}{(lg(x) + a)}}}{xln^{3}{10}} - \frac{6*-e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{5}x^{2}ln^{3}{10}} - \frac{6e^{\frac{x}{(lg(x) + a)}}((\frac{-(\frac{1}{ln{10}(x)} + 0)}{(lg(x) + a)^{2}})x + \frac{1}{(lg(x) + a)})}{(lg(x) + a)^{5}xln^{3}{10}} - \frac{6e^{\frac{x}{(lg(x) + a)}}*-3*0}{(lg(x) + a)^{5}xln^{4}{10}} + \frac{9(\frac{-4(\frac{1}{ln{10}(x)} + 0)}{(lg(x) + a)^{5}})e^{\frac{x}{(lg(x) + a)}}}{xln^{2}{10}} + \frac{9*-e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{4}x^{2}ln^{2}{10}} + \frac{9e^{\frac{x}{(lg(x) + a)}}((\frac{-(\frac{1}{ln{10}(x)} + 0)}{(lg(x) + a)^{2}})x + \frac{1}{(lg(x) + a)})}{(lg(x) + a)^{4}xln^{2}{10}} + \frac{9e^{\frac{x}{(lg(x) + a)}}*-2*0}{(lg(x) + a)^{4}xln^{3}{10}} - \frac{(\frac{-6(\frac{1}{ln{10}(x)} + 0)}{(lg(x) + a)^{7}})e^{\frac{x}{(lg(x) + a)}}}{ln^{3}{10}} - \frac{e^{\frac{x}{(lg(x) + a)}}((\frac{-(\frac{1}{ln{10}(x)} + 0)}{(lg(x) + a)^{2}})x + \frac{1}{(lg(x) + a)})}{(lg(x) + a)^{6}ln^{3}{10}} - \frac{e^{\frac{x}{(lg(x) + a)}}*-3*0}{(lg(x) + a)^{6}ln^{4}{10}} + \frac{3(\frac{-5(\frac{1}{ln{10}(x)} + 0)}{(lg(x) + a)^{6}})e^{\frac{x}{(lg(x) + a)}}}{ln^{2}{10}} + \frac{3e^{\frac{x}{(lg(x) + a)}}((\frac{-(\frac{1}{ln{10}(x)} + 0)}{(lg(x) + a)^{2}})x + \frac{1}{(lg(x) + a)})}{(lg(x) + a)^{5}ln^{2}{10}} + \frac{3e^{\frac{x}{(lg(x) + a)}}*-2*0}{(lg(x) + a)^{5}ln^{3}{10}} - \frac{3(\frac{-4(\frac{1}{ln{10}(x)} + 0)}{(lg(x) + a)^{5}})e^{\frac{x}{(lg(x) + a)}}}{ln{10}} - \frac{3e^{\frac{x}{(lg(x) + a)}}((\frac{-(\frac{1}{ln{10}(x)} + 0)}{(lg(x) + a)^{2}})x + \frac{1}{(lg(x) + a)})}{(lg(x) + a)^{4}ln{10}} - \frac{3e^{\frac{x}{(lg(x) + a)}}*-0}{(lg(x) + a)^{4}ln^{2}{10}} + \frac{(\frac{-2(\frac{1}{ln{10}(x)} + 0)}{(lg(x) + a)^{3}})e^{\frac{x}{(lg(x) + a)}}}{x^{2}ln{10}} + \frac{-2e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{2}x^{3}ln{10}} + \frac{e^{\frac{x}{(lg(x) + a)}}((\frac{-(\frac{1}{ln{10}(x)} + 0)}{(lg(x) + a)^{2}})x + \frac{1}{(lg(x) + a)})}{(lg(x) + a)^{2}x^{2}ln{10}} + \frac{e^{\frac{x}{(lg(x) + a)}}*-0}{(lg(x) + a)^{2}x^{2}ln^{2}{10}} - \frac{3(\frac{-3(\frac{1}{ln{10}(x)} + 0)}{(lg(x) + a)^{4}})e^{\frac{x}{(lg(x) + a)}}}{xln{10}} - \frac{3*-e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{3}x^{2}ln{10}} - \frac{3e^{\frac{x}{(lg(x) + a)}}((\frac{-(\frac{1}{ln{10}(x)} + 0)}{(lg(x) + a)^{2}})x + \frac{1}{(lg(x) + a)})}{(lg(x) + a)^{3}xln{10}} - \frac{3e^{\frac{x}{(lg(x) + a)}}*-0}{(lg(x) + a)^{3}xln^{2}{10}} + (\frac{-3(\frac{1}{ln{10}(x)} + 0)}{(lg(x) + a)^{4}})e^{\frac{x}{(lg(x) + a)}} + \frac{e^{\frac{x}{(lg(x) + a)}}((\frac{-(\frac{1}{ln{10}(x)} + 0)}{(lg(x) + a)^{2}})x + \frac{1}{(lg(x) + a)})}{(lg(x) + a)^{3}}\\=&\frac{24e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{5}x^{3}ln^{4}{10}} + \frac{12e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{4}x^{3}ln^{3}{10}} + \frac{36e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{6}x^{2}ln^{4}{10}} + \frac{12e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{7}xln^{4}{10}} - \frac{30e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{6}xln^{3}{10}} - \frac{36e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{5}x^{2}ln^{3}{10}} - \frac{e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{4}x^{2}ln^{2}{10}} + \frac{24e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{5}xln^{2}{10}} + \frac{e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{8}ln^{4}{10}} - \frac{4e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{7}ln^{3}{10}} + \frac{6e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{6}ln^{2}{10}} - \frac{4e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{5}ln{10}} - \frac{2e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{3}x^{3}ln^{2}{10}} - \frac{2e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{2}x^{3}ln{10}} + \frac{4e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{3}x^{2}ln{10}} - \frac{6e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{4}xln{10}} + \frac{e^{\frac{x}{(lg(x) + a)}}}{(lg(x) + a)^{4}}\\ \end{split}\end{equation} \]

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