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。
\[ \begin{equation}\begin{split}【1/1】求函数log_{x}^{2x}sin(2)xcos(x) + {e}^{x}cos(2)x 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = xlog_{x}^{2x}sin(2)cos(x) + x{e}^{x}cos(2)\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( xlog_{x}^{2x}sin(2)cos(x) + x{e}^{x}cos(2)\right)}{dx}\\=&log_{x}^{2x}sin(2)cos(x) + x(\frac{(\frac{(2)}{(2x)} - \frac{(1)log_{x}^{2x}}{(x)})}{(ln(x))})sin(2)cos(x) + xlog_{x}^{2x}cos(2)*0cos(x) + xlog_{x}^{2x}sin(2)*-sin(x) + {e}^{x}cos(2) + x({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))cos(2) + x{e}^{x}*-sin(2)*0\\=&log_{x}^{2x}sin(2)cos(x) - \frac{log_{x}^{2x}sin(2)cos(x)}{ln(x)} + \frac{sin(2)cos(x)}{ln(x)} - xlog_{x}^{2x}sin(x)sin(2) + {e}^{x}cos(2) + x{e}^{x}cos(2)\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( log_{x}^{2x}sin(2)cos(x) - \frac{log_{x}^{2x}sin(2)cos(x)}{ln(x)} + \frac{sin(2)cos(x)}{ln(x)} - xlog_{x}^{2x}sin(x)sin(2) + {e}^{x}cos(2) + x{e}^{x}cos(2)\right)}{dx}\\=&(\frac{(\frac{(2)}{(2x)} - \frac{(1)log_{x}^{2x}}{(x)})}{(ln(x))})sin(2)cos(x) + log_{x}^{2x}cos(2)*0cos(x) + log_{x}^{2x}sin(2)*-sin(x) - \frac{(\frac{(\frac{(2)}{(2x)} - \frac{(1)log_{x}^{2x}}{(x)})}{(ln(x))})sin(2)cos(x)}{ln(x)} - \frac{log_{x}^{2x}*-sin(2)cos(x)}{ln^{2}(x)(x)} - \frac{log_{x}^{2x}cos(2)*0cos(x)}{ln(x)} - \frac{log_{x}^{2x}sin(2)*-sin(x)}{ln(x)} + \frac{-sin(2)cos(x)}{ln^{2}(x)(x)} + \frac{cos(2)*0cos(x)}{ln(x)} + \frac{sin(2)*-sin(x)}{ln(x)} - log_{x}^{2x}sin(x)sin(2) - x(\frac{(\frac{(2)}{(2x)} - \frac{(1)log_{x}^{2x}}{(x)})}{(ln(x))})sin(x)sin(2) - xlog_{x}^{2x}cos(x)sin(2) - xlog_{x}^{2x}sin(x)cos(2)*0 + ({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))cos(2) + {e}^{x}*-sin(2)*0 + {e}^{x}cos(2) + x({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))cos(2) + x{e}^{x}*-sin(2)*0\\=&\frac{-log_{x}^{2x}sin(2)cos(x)}{xln(x)} + \frac{sin(2)cos(x)}{xln(x)} - 2log_{x}^{2x}sin(x)sin(2) + \frac{2log_{x}^{2x}sin(2)cos(x)}{xln^{2}(x)} - \frac{2sin(2)cos(x)}{xln^{2}(x)} + \frac{2log_{x}^{2x}sin(x)sin(2)}{ln(x)} - \frac{2sin(x)sin(2)}{ln(x)} - xlog_{x}^{2x}sin(2)cos(x) + 2{e}^{x}cos(2) + x{e}^{x}cos(2)\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-log_{x}^{2x}sin(2)cos(x)}{xln(x)} + \frac{sin(2)cos(x)}{xln(x)} - 2log_{x}^{2x}sin(x)sin(2) + \frac{2log_{x}^{2x}sin(2)cos(x)}{xln^{2}(x)} - \frac{2sin(2)cos(x)}{xln^{2}(x)} + \frac{2log_{x}^{2x}sin(x)sin(2)}{ln(x)} - \frac{2sin(x)sin(2)}{ln(x)} - xlog_{x}^{2x}sin(2)cos(x) + 2{e}^{x}cos(2) + x{e}^{x}cos(2)\right)}{dx}\\=&\frac{--log_{x}^{2x}sin(2)cos(x)}{x^{2}ln(x)} - \frac{(\frac{(\frac{(2)}{(2x)} - \frac{(1)log_{x}^{2x}}{(x)})}{(ln(x))})sin(2)cos(x)}{xln(x)} - \frac{log_{x}^{2x}*-sin(2)cos(x)}{xln^{2}(x)(x)} - \frac{log_{x}^{2x}cos(2)*0cos(x)}{xln(x)} - \frac{log_{x}^{2x}sin(2)*-sin(x)}{xln(x)} + \frac{-sin(2)cos(x)}{x^{2}ln(x)} + \frac{-sin(2)cos(x)}{xln^{2}(x)(x)} + \frac{cos(2)*0cos(x)}{xln(x)} + \frac{sin(2)*-sin(x)}{xln(x)} - 2(\frac{(\frac{(2)}{(2x)} - \frac{(1)log_{x}^{2x}}{(x)})}{(ln(x))})sin(x)sin(2) - 2log_{x}^{2x}cos(x)sin(2) - 2log_{x}^{2x}sin(x)cos(2)*0 + \frac{2*-log_{x}^{2x}sin(2)cos(x)}{x^{2}ln^{2}(x)} + \frac{2(\frac{(\frac{(2)}{(2x)} - \frac{(1)log_{x}^{2x}}{(x)})}{(ln(x))})sin(2)cos(x)}{xln^{2}(x)} + \frac{2log_{x}^{2x}*-2sin(2)cos(x)}{xln^{3}(x)(x)} + \frac{2log_{x}^{2x}cos(2)*0cos(x)}{xln^{2}(x)} + \frac{2log_{x}^{2x}sin(2)*-sin(x)}{xln^{2}(x)} - \frac{2*-sin(2)cos(x)}{x^{2}ln^{2}(x)} - \frac{2*-2sin(2)cos(x)}{xln^{3}(x)(x)} - \frac{2cos(2)*0cos(x)}{xln^{2}(x)} - \frac{2sin(2)*-sin(x)}{xln^{2}(x)} + \frac{2(\frac{(\frac{(2)}{(2x)} - \frac{(1)log_{x}^{2x}}{(x)})}{(ln(x))})sin(x)sin(2)}{ln(x)} + \frac{2log_{x}^{2x}*-sin(x)sin(2)}{ln^{2}(x)(x)} + \frac{2log_{x}^{2x}cos(x)sin(2)}{ln(x)} + \frac{2log_{x}^{2x}sin(x)cos(2)*0}{ln(x)} - \frac{2*-sin(x)sin(2)}{ln^{2}(x)(x)} - \frac{2cos(x)sin(2)}{ln(x)} - \frac{2sin(x)cos(2)*0}{ln(x)} - log_{x}^{2x}sin(2)cos(x) - x(\frac{(\frac{(2)}{(2x)} - \frac{(1)log_{x}^{2x}}{(x)})}{(ln(x))})sin(2)cos(x) - xlog_{x}^{2x}cos(2)*0cos(x) - xlog_{x}^{2x}sin(2)*-sin(x) + 2({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))cos(2) + 2{e}^{x}*-sin(2)*0 + {e}^{x}cos(2) + x({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))cos(2) + x{e}^{x}*-sin(2)*0\\=&\frac{log_{x}^{2x}sin(2)cos(x)}{x^{2}ln(x)} + \frac{3log_{x}^{2x}sin(x)sin(2)}{xln(x)} - \frac{sin(2)cos(x)}{x^{2}ln(x)} - \frac{3sin(x)sin(2)}{xln(x)} - 3log_{x}^{2x}sin(2)cos(x) - \frac{6log_{x}^{2x}sin(2)cos(x)}{x^{2}ln^{3}(x)} + \frac{6sin(2)cos(x)}{x^{2}ln^{3}(x)} - \frac{6log_{x}^{2x}sin(x)sin(2)}{xln^{2}(x)} + \frac{6sin(x)sin(2)}{xln^{2}(x)} + \frac{3log_{x}^{2x}sin(2)cos(x)}{ln(x)} - \frac{3sin(2)cos(x)}{ln(x)} + xlog_{x}^{2x}sin(x)sin(2) + 3{e}^{x}cos(2) + x{e}^{x}cos(2)\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{log_{x}^{2x}sin(2)cos(x)}{x^{2}ln(x)} + \frac{3log_{x}^{2x}sin(x)sin(2)}{xln(x)} - \frac{sin(2)cos(x)}{x^{2}ln(x)} - \frac{3sin(x)sin(2)}{xln(x)} - 3log_{x}^{2x}sin(2)cos(x) - \frac{6log_{x}^{2x}sin(2)cos(x)}{x^{2}ln^{3}(x)} + \frac{6sin(2)cos(x)}{x^{2}ln^{3}(x)} - \frac{6log_{x}^{2x}sin(x)sin(2)}{xln^{2}(x)} + \frac{6sin(x)sin(2)}{xln^{2}(x)} + \frac{3log_{x}^{2x}sin(2)cos(x)}{ln(x)} - \frac{3sin(2)cos(x)}{ln(x)} + xlog_{x}^{2x}sin(x)sin(2) + 3{e}^{x}cos(2) + x{e}^{x}cos(2)\right)}{dx}\\=&\frac{-2log_{x}^{2x}sin(2)cos(x)}{x^{3}ln(x)} + \frac{(\frac{(\frac{(2)}{(2x)} - \frac{(1)log_{x}^{2x}}{(x)})}{(ln(x))})sin(2)cos(x)}{x^{2}ln(x)} + \frac{log_{x}^{2x}*-sin(2)cos(x)}{x^{2}ln^{2}(x)(x)} + \frac{log_{x}^{2x}cos(2)*0cos(x)}{x^{2}ln(x)} + \frac{log_{x}^{2x}sin(2)*-sin(x)}{x^{2}ln(x)} + \frac{3*-log_{x}^{2x}sin(x)sin(2)}{x^{2}ln(x)} + \frac{3(\frac{(\frac{(2)}{(2x)} - \frac{(1)log_{x}^{2x}}{(x)})}{(ln(x))})sin(x)sin(2)}{xln(x)} + \frac{3log_{x}^{2x}*-sin(x)sin(2)}{xln^{2}(x)(x)} + \frac{3log_{x}^{2x}cos(x)sin(2)}{xln(x)} + \frac{3log_{x}^{2x}sin(x)cos(2)*0}{xln(x)} - \frac{-2sin(2)cos(x)}{x^{3}ln(x)} - \frac{-sin(2)cos(x)}{x^{2}ln^{2}(x)(x)} - \frac{cos(2)*0cos(x)}{x^{2}ln(x)} - \frac{sin(2)*-sin(x)}{x^{2}ln(x)} - \frac{3*-sin(x)sin(2)}{x^{2}ln(x)} - \frac{3*-sin(x)sin(2)}{xln^{2}(x)(x)} - \frac{3cos(x)sin(2)}{xln(x)} - \frac{3sin(x)cos(2)*0}{xln(x)} - 3(\frac{(\frac{(2)}{(2x)} - \frac{(1)log_{x}^{2x}}{(x)})}{(ln(x))})sin(2)cos(x) - 3log_{x}^{2x}cos(2)*0cos(x) - 3log_{x}^{2x}sin(2)*-sin(x) - \frac{6*-2log_{x}^{2x}sin(2)cos(x)}{x^{3}ln^{3}(x)} - \frac{6(\frac{(\frac{(2)}{(2x)} - \frac{(1)log_{x}^{2x}}{(x)})}{(ln(x))})sin(2)cos(x)}{x^{2}ln^{3}(x)} - \frac{6log_{x}^{2x}*-3sin(2)cos(x)}{x^{2}ln^{4}(x)(x)} - \frac{6log_{x}^{2x}cos(2)*0cos(x)}{x^{2}ln^{3}(x)} - \frac{6log_{x}^{2x}sin(2)*-sin(x)}{x^{2}ln^{3}(x)} + \frac{6*-2sin(2)cos(x)}{x^{3}ln^{3}(x)} + \frac{6*-3sin(2)cos(x)}{x^{2}ln^{4}(x)(x)} + \frac{6cos(2)*0cos(x)}{x^{2}ln^{3}(x)} + \frac{6sin(2)*-sin(x)}{x^{2}ln^{3}(x)} - \frac{6*-log_{x}^{2x}sin(x)sin(2)}{x^{2}ln^{2}(x)} - \frac{6(\frac{(\frac{(2)}{(2x)} - \frac{(1)log_{x}^{2x}}{(x)})}{(ln(x))})sin(x)sin(2)}{xln^{2}(x)} - \frac{6log_{x}^{2x}*-2sin(x)sin(2)}{xln^{3}(x)(x)} - \frac{6log_{x}^{2x}cos(x)sin(2)}{xln^{2}(x)} - \frac{6log_{x}^{2x}sin(x)cos(2)*0}{xln^{2}(x)} + \frac{6*-sin(x)sin(2)}{x^{2}ln^{2}(x)} + \frac{6*-2sin(x)sin(2)}{xln^{3}(x)(x)} + \frac{6cos(x)sin(2)}{xln^{2}(x)} + \frac{6sin(x)cos(2)*0}{xln^{2}(x)} + \frac{3(\frac{(\frac{(2)}{(2x)} - \frac{(1)log_{x}^{2x}}{(x)})}{(ln(x))})sin(2)cos(x)}{ln(x)} + \frac{3log_{x}^{2x}*-sin(2)cos(x)}{ln^{2}(x)(x)} + \frac{3log_{x}^{2x}cos(2)*0cos(x)}{ln(x)} + \frac{3log_{x}^{2x}sin(2)*-sin(x)}{ln(x)} - \frac{3*-sin(2)cos(x)}{ln^{2}(x)(x)} - \frac{3cos(2)*0cos(x)}{ln(x)} - \frac{3sin(2)*-sin(x)}{ln(x)} + log_{x}^{2x}sin(x)sin(2) + x(\frac{(\frac{(2)}{(2x)} - \frac{(1)log_{x}^{2x}}{(x)})}{(ln(x))})sin(x)sin(2) + xlog_{x}^{2x}cos(x)sin(2) + xlog_{x}^{2x}sin(x)cos(2)*0 + 3({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))cos(2) + 3{e}^{x}*-sin(2)*0 + {e}^{x}cos(2) + x({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))cos(2) + x{e}^{x}*-sin(2)*0\\=&\frac{-2log_{x}^{2x}sin(2)cos(x)}{x^{3}ln(x)} - \frac{2log_{x}^{2x}sin(2)cos(x)}{x^{3}ln^{2}(x)} + \frac{2sin(2)cos(x)}{x^{3}ln^{2}(x)} - \frac{4log_{x}^{2x}sin(x)sin(2)}{x^{2}ln(x)} + \frac{6log_{x}^{2x}sin(2)cos(x)}{xln(x)} + \frac{2sin(2)cos(x)}{x^{3}ln(x)} + \frac{4sin(x)sin(2)}{x^{2}ln(x)} - \frac{6sin(2)cos(x)}{xln(x)} + 4log_{x}^{2x}sin(x)sin(2) + \frac{12log_{x}^{2x}sin(2)cos(x)}{x^{3}ln^{3}(x)} + \frac{24log_{x}^{2x}sin(2)cos(x)}{x^{3}ln^{4}(x)} - \frac{24sin(2)cos(x)}{x^{3}ln^{4}(x)} + \frac{24log_{x}^{2x}sin(x)sin(2)}{x^{2}ln^{3}(x)} - \frac{12sin(2)cos(x)}{x^{3}ln^{3}(x)} - \frac{24sin(x)sin(2)}{x^{2}ln^{3}(x)} - \frac{12log_{x}^{2x}sin(2)cos(x)}{xln^{2}(x)} + \frac{12sin(2)cos(x)}{xln^{2}(x)} - \frac{4log_{x}^{2x}sin(x)sin(2)}{ln(x)} + \frac{4sin(x)sin(2)}{ln(x)} + xlog_{x}^{2x}sin(2)cos(x) + 4{e}^{x}cos(2) + x{e}^{x}cos(2)\\ \end{split}\end{equation} \]

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