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。
\[ \begin{equation}\begin{split}【1/1】求函数log_{x}^{e^{\frac{x}{sqrt(1 + xx)}}} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = log_{x}^{e^{\frac{x}{sqrt(x^{2} + 1)}}}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( log_{x}^{e^{\frac{x}{sqrt(x^{2} + 1)}}}\right)}{dx}\\=&(\frac{(\frac{(e^{\frac{x}{sqrt(x^{2} + 1)}}(\frac{1}{sqrt(x^{2} + 1)} + \frac{x*-(2x + 0)*\frac{1}{2}}{(x^{2} + 1)(x^{2} + 1)^{\frac{1}{2}}}))}{(e^{\frac{x}{sqrt(x^{2} + 1)}})} - \frac{(1)log_{x}^{e^{\frac{x}{sqrt(x^{2} + 1)}}}}{(x)})}{(ln(x))})\\=&\frac{1}{ln(x)sqrt(x^{2} + 1)} - \frac{x^{2}}{(x^{2} + 1)^{\frac{3}{2}}ln(x)} - \frac{log_{x}^{e^{\frac{x}{sqrt(x^{2} + 1)}}}}{xln(x)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{1}{ln(x)sqrt(x^{2} + 1)} - \frac{x^{2}}{(x^{2} + 1)^{\frac{3}{2}}ln(x)} - \frac{log_{x}^{e^{\frac{x}{sqrt(x^{2} + 1)}}}}{xln(x)}\right)}{dx}\\=&\frac{-1}{ln^{2}(x)(x)sqrt(x^{2} + 1)} + \frac{-(2x + 0)*\frac{1}{2}}{ln(x)(x^{2} + 1)(x^{2} + 1)^{\frac{1}{2}}} - \frac{(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} + 1)^{\frac{5}{2}}})x^{2}}{ln(x)} - \frac{2x}{(x^{2} + 1)^{\frac{3}{2}}ln(x)} - \frac{x^{2}*-1}{(x^{2} + 1)^{\frac{3}{2}}ln^{2}(x)(x)} - \frac{-log_{x}^{e^{\frac{x}{sqrt(x^{2} + 1)}}}}{x^{2}ln(x)} - \frac{(\frac{(\frac{(e^{\frac{x}{sqrt(x^{2} + 1)}}(\frac{1}{sqrt(x^{2} + 1)} + \frac{x*-(2x + 0)*\frac{1}{2}}{(x^{2} + 1)(x^{2} + 1)^{\frac{1}{2}}}))}{(e^{\frac{x}{sqrt(x^{2} + 1)}})} - \frac{(1)log_{x}^{e^{\frac{x}{sqrt(x^{2} + 1)}}}}{(x)})}{(ln(x))})}{xln(x)} - \frac{log_{x}^{e^{\frac{x}{sqrt(x^{2} + 1)}}}*-1}{xln^{2}(x)(x)}\\=&\frac{-2}{xln^{2}(x)sqrt(x^{2} + 1)} - \frac{3x}{(x^{2} + 1)^{\frac{3}{2}}ln(x)} + \frac{3x^{3}}{(x^{2} + 1)^{\frac{5}{2}}ln(x)} + \frac{2x}{(x^{2} + 1)^{\frac{3}{2}}ln^{2}(x)} + \frac{log_{x}^{e^{\frac{x}{sqrt(x^{2} + 1)}}}}{x^{2}ln(x)} + \frac{2log_{x}^{e^{\frac{x}{sqrt(x^{2} + 1)}}}}{x^{2}ln^{2}(x)}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-2}{xln^{2}(x)sqrt(x^{2} + 1)} - \frac{3x}{(x^{2} + 1)^{\frac{3}{2}}ln(x)} + \frac{3x^{3}}{(x^{2} + 1)^{\frac{5}{2}}ln(x)} + \frac{2x}{(x^{2} + 1)^{\frac{3}{2}}ln^{2}(x)} + \frac{log_{x}^{e^{\frac{x}{sqrt(x^{2} + 1)}}}}{x^{2}ln(x)} + \frac{2log_{x}^{e^{\frac{x}{sqrt(x^{2} + 1)}}}}{x^{2}ln^{2}(x)}\right)}{dx}\\=&\frac{-2*-1}{x^{2}ln^{2}(x)sqrt(x^{2} + 1)} - \frac{2*-2}{xln^{3}(x)(x)sqrt(x^{2} + 1)} - \frac{2*-(2x + 0)*\frac{1}{2}}{xln^{2}(x)(x^{2} + 1)(x^{2} + 1)^{\frac{1}{2}}} - \frac{3(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} + 1)^{\frac{5}{2}}})x}{ln(x)} - \frac{3}{(x^{2} + 1)^{\frac{3}{2}}ln(x)} - \frac{3x*-1}{(x^{2} + 1)^{\frac{3}{2}}ln^{2}(x)(x)} + \frac{3(\frac{\frac{-5}{2}(2x + 0)}{(x^{2} + 1)^{\frac{7}{2}}})x^{3}}{ln(x)} + \frac{3*3x^{2}}{(x^{2} + 1)^{\frac{5}{2}}ln(x)} + \frac{3x^{3}*-1}{(x^{2} + 1)^{\frac{5}{2}}ln^{2}(x)(x)} + \frac{2(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} + 1)^{\frac{5}{2}}})x}{ln^{2}(x)} + \frac{2}{(x^{2} + 1)^{\frac{3}{2}}ln^{2}(x)} + \frac{2x*-2}{(x^{2} + 1)^{\frac{3}{2}}ln^{3}(x)(x)} + \frac{-2log_{x}^{e^{\frac{x}{sqrt(x^{2} + 1)}}}}{x^{3}ln(x)} + \frac{(\frac{(\frac{(e^{\frac{x}{sqrt(x^{2} + 1)}}(\frac{1}{sqrt(x^{2} + 1)} + \frac{x*-(2x + 0)*\frac{1}{2}}{(x^{2} + 1)(x^{2} + 1)^{\frac{1}{2}}}))}{(e^{\frac{x}{sqrt(x^{2} + 1)}})} - \frac{(1)log_{x}^{e^{\frac{x}{sqrt(x^{2} + 1)}}}}{(x)})}{(ln(x))})}{x^{2}ln(x)} + \frac{log_{x}^{e^{\frac{x}{sqrt(x^{2} + 1)}}}*-1}{x^{2}ln^{2}(x)(x)} + \frac{2*-2log_{x}^{e^{\frac{x}{sqrt(x^{2} + 1)}}}}{x^{3}ln^{2}(x)} + \frac{2(\frac{(\frac{(e^{\frac{x}{sqrt(x^{2} + 1)}}(\frac{1}{sqrt(x^{2} + 1)} + \frac{x*-(2x + 0)*\frac{1}{2}}{(x^{2} + 1)(x^{2} + 1)^{\frac{1}{2}}}))}{(e^{\frac{x}{sqrt(x^{2} + 1)}})} - \frac{(1)log_{x}^{e^{\frac{x}{sqrt(x^{2} + 1)}}}}{(x)})}{(ln(x))})}{x^{2}ln^{2}(x)} + \frac{2log_{x}^{e^{\frac{x}{sqrt(x^{2} + 1)}}}*-2}{x^{2}ln^{3}(x)(x)}\\=&\frac{3}{x^{2}ln^{2}(x)sqrt(x^{2} + 1)} + \frac{6}{x^{2}ln^{3}(x)sqrt(x^{2} + 1)} + \frac{6}{(x^{2} + 1)^{\frac{3}{2}}ln^{2}(x)} + \frac{18x^{2}}{(x^{2} + 1)^{\frac{5}{2}}ln(x)} - \frac{3}{(x^{2} + 1)^{\frac{3}{2}}ln(x)} - \frac{15x^{4}}{(x^{2} + 1)^{\frac{7}{2}}ln(x)} - \frac{9x^{2}}{(x^{2} + 1)^{\frac{5}{2}}ln^{2}(x)} - \frac{6}{(x^{2} + 1)^{\frac{3}{2}}ln^{3}(x)} - \frac{2log_{x}^{e^{\frac{x}{sqrt(x^{2} + 1)}}}}{x^{3}ln(x)} - \frac{6log_{x}^{e^{\frac{x}{sqrt(x^{2} + 1)}}}}{x^{3}ln^{2}(x)} - \frac{6log_{x}^{e^{\frac{x}{sqrt(x^{2} + 1)}}}}{x^{3}ln^{3}(x)}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{3}{x^{2}ln^{2}(x)sqrt(x^{2} + 1)} + \frac{6}{x^{2}ln^{3}(x)sqrt(x^{2} + 1)} + \frac{6}{(x^{2} + 1)^{\frac{3}{2}}ln^{2}(x)} + \frac{18x^{2}}{(x^{2} + 1)^{\frac{5}{2}}ln(x)} - \frac{3}{(x^{2} + 1)^{\frac{3}{2}}ln(x)} - \frac{15x^{4}}{(x^{2} + 1)^{\frac{7}{2}}ln(x)} - \frac{9x^{2}}{(x^{2} + 1)^{\frac{5}{2}}ln^{2}(x)} - \frac{6}{(x^{2} + 1)^{\frac{3}{2}}ln^{3}(x)} - \frac{2log_{x}^{e^{\frac{x}{sqrt(x^{2} + 1)}}}}{x^{3}ln(x)} - \frac{6log_{x}^{e^{\frac{x}{sqrt(x^{2} + 1)}}}}{x^{3}ln^{2}(x)} - \frac{6log_{x}^{e^{\frac{x}{sqrt(x^{2} + 1)}}}}{x^{3}ln^{3}(x)}\right)}{dx}\\=&\frac{3*-2}{x^{3}ln^{2}(x)sqrt(x^{2} + 1)} + \frac{3*-2}{x^{2}ln^{3}(x)(x)sqrt(x^{2} + 1)} + \frac{3*-(2x + 0)*\frac{1}{2}}{x^{2}ln^{2}(x)(x^{2} + 1)(x^{2} + 1)^{\frac{1}{2}}} + \frac{6*-2}{x^{3}ln^{3}(x)sqrt(x^{2} + 1)} + \frac{6*-3}{x^{2}ln^{4}(x)(x)sqrt(x^{2} + 1)} + \frac{6*-(2x + 0)*\frac{1}{2}}{x^{2}ln^{3}(x)(x^{2} + 1)(x^{2} + 1)^{\frac{1}{2}}} + \frac{6(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} + 1)^{\frac{5}{2}}})}{ln^{2}(x)} + \frac{6*-2}{(x^{2} + 1)^{\frac{3}{2}}ln^{3}(x)(x)} + \frac{18(\frac{\frac{-5}{2}(2x + 0)}{(x^{2} + 1)^{\frac{7}{2}}})x^{2}}{ln(x)} + \frac{18*2x}{(x^{2} + 1)^{\frac{5}{2}}ln(x)} + \frac{18x^{2}*-1}{(x^{2} + 1)^{\frac{5}{2}}ln^{2}(x)(x)} - \frac{3(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} + 1)^{\frac{5}{2}}})}{ln(x)} - \frac{3*-1}{(x^{2} + 1)^{\frac{3}{2}}ln^{2}(x)(x)} - \frac{15(\frac{\frac{-7}{2}(2x + 0)}{(x^{2} + 1)^{\frac{9}{2}}})x^{4}}{ln(x)} - \frac{15*4x^{3}}{(x^{2} + 1)^{\frac{7}{2}}ln(x)} - \frac{15x^{4}*-1}{(x^{2} + 1)^{\frac{7}{2}}ln^{2}(x)(x)} - \frac{9(\frac{\frac{-5}{2}(2x + 0)}{(x^{2} + 1)^{\frac{7}{2}}})x^{2}}{ln^{2}(x)} - \frac{9*2x}{(x^{2} + 1)^{\frac{5}{2}}ln^{2}(x)} - \frac{9x^{2}*-2}{(x^{2} + 1)^{\frac{5}{2}}ln^{3}(x)(x)} - \frac{6(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} + 1)^{\frac{5}{2}}})}{ln^{3}(x)} - \frac{6*-3}{(x^{2} + 1)^{\frac{3}{2}}ln^{4}(x)(x)} - \frac{2*-3log_{x}^{e^{\frac{x}{sqrt(x^{2} + 1)}}}}{x^{4}ln(x)} - \frac{2(\frac{(\frac{(e^{\frac{x}{sqrt(x^{2} + 1)}}(\frac{1}{sqrt(x^{2} + 1)} + \frac{x*-(2x + 0)*\frac{1}{2}}{(x^{2} + 1)(x^{2} + 1)^{\frac{1}{2}}}))}{(e^{\frac{x}{sqrt(x^{2} + 1)}})} - \frac{(1)log_{x}^{e^{\frac{x}{sqrt(x^{2} + 1)}}}}{(x)})}{(ln(x))})}{x^{3}ln(x)} - \frac{2log_{x}^{e^{\frac{x}{sqrt(x^{2} + 1)}}}*-1}{x^{3}ln^{2}(x)(x)} - \frac{6*-3log_{x}^{e^{\frac{x}{sqrt(x^{2} + 1)}}}}{x^{4}ln^{2}(x)} - \frac{6(\frac{(\frac{(e^{\frac{x}{sqrt(x^{2} + 1)}}(\frac{1}{sqrt(x^{2} + 1)} + \frac{x*-(2x + 0)*\frac{1}{2}}{(x^{2} + 1)(x^{2} + 1)^{\frac{1}{2}}}))}{(e^{\frac{x}{sqrt(x^{2} + 1)}})} - \frac{(1)log_{x}^{e^{\frac{x}{sqrt(x^{2} + 1)}}}}{(x)})}{(ln(x))})}{x^{3}ln^{2}(x)} - \frac{6log_{x}^{e^{\frac{x}{sqrt(x^{2} + 1)}}}*-2}{x^{3}ln^{3}(x)(x)} - \frac{6*-3log_{x}^{e^{\frac{x}{sqrt(x^{2} + 1)}}}}{x^{4}ln^{3}(x)} - \frac{6(\frac{(\frac{(e^{\frac{x}{sqrt(x^{2} + 1)}}(\frac{1}{sqrt(x^{2} + 1)} + \frac{x*-(2x + 0)*\frac{1}{2}}{(x^{2} + 1)(x^{2} + 1)^{\frac{1}{2}}}))}{(e^{\frac{x}{sqrt(x^{2} + 1)}})} - \frac{(1)log_{x}^{e^{\frac{x}{sqrt(x^{2} + 1)}}}}{(x)})}{(ln(x))})}{x^{3}ln^{3}(x)} - \frac{6log_{x}^{e^{\frac{x}{sqrt(x^{2} + 1)}}}*-3}{x^{3}ln^{4}(x)(x)}\\=&\frac{-8}{x^{3}ln^{2}(x)sqrt(x^{2} + 1)} - \frac{24}{x^{3}ln^{3}(x)sqrt(x^{2} + 1)} - \frac{24}{x^{3}ln^{4}(x)sqrt(x^{2} + 1)} - \frac{12}{(x^{2} + 1)^{\frac{3}{2}}xln^{3}(x)} - \frac{54x}{(x^{2} + 1)^{\frac{5}{2}}ln^{2}(x)} - \frac{150x^{3}}{(x^{2} + 1)^{\frac{7}{2}}ln(x)} + \frac{45x}{(x^{2} + 1)^{\frac{5}{2}}ln(x)} + \frac{105x^{5}}{(x^{2} + 1)^{\frac{9}{2}}ln(x)} + \frac{60x^{3}}{(x^{2} + 1)^{\frac{7}{2}}ln^{2}(x)} + \frac{36x}{(x^{2} + 1)^{\frac{5}{2}}ln^{3}(x)} + \frac{24}{(x^{2} + 1)^{\frac{3}{2}}xln^{4}(x)} + \frac{6log_{x}^{e^{\frac{x}{sqrt(x^{2} + 1)}}}}{x^{4}ln(x)} + \frac{2}{(x^{2} + 1)^{\frac{3}{2}}xln^{2}(x)} + \frac{22log_{x}^{e^{\frac{x}{sqrt(x^{2} + 1)}}}}{x^{4}ln^{2}(x)} + \frac{36log_{x}^{e^{\frac{x}{sqrt(x^{2} + 1)}}}}{x^{4}ln^{3}(x)} + \frac{24log_{x}^{e^{\frac{x}{sqrt(x^{2} + 1)}}}}{x^{4}ln^{4}(x)}\\ \end{split}\end{equation} \]

你的问题在这里没有得到解决？请到 热门难题 里面看看吧！