本次共计算 3 个题目：每一题对 x 求 4 阶导数。
    注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/3】求函数sin(x)sin(x) - sin(sin(x)) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = sin^{2}(x) - sin(sin(x))\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( sin^{2}(x) - sin(sin(x))\right)}{dx}\\=&2sin(x)cos(x) - cos(sin(x))cos(x)\\=&2sin(x)cos(x) - cos(x)cos(sin(x))\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( 2sin(x)cos(x) - cos(x)cos(sin(x))\right)}{dx}\\=&2cos(x)cos(x) + 2sin(x)*-sin(x) - -sin(x)cos(sin(x)) - cos(x)*-sin(sin(x))cos(x)\\=&2cos^{2}(x) + sin(x)cos(sin(x)) + sin(sin(x))cos^{2}(x) - 2sin^{2}(x)\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( 2cos^{2}(x) + sin(x)cos(sin(x)) + sin(sin(x))cos^{2}(x) - 2sin^{2}(x)\right)}{dx}\\=&2*-2cos(x)sin(x) + cos(x)cos(sin(x)) + sin(x)*-sin(sin(x))cos(x) + cos(sin(x))cos(x)cos^{2}(x) + sin(sin(x))*-2cos(x)sin(x) - 2*2sin(x)cos(x)\\=&-8sin(x)cos(x) + cos(x)cos(sin(x)) - 3sin(x)sin(sin(x))cos(x) + cos^{3}(x)cos(sin(x))\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( -8sin(x)cos(x) + cos(x)cos(sin(x)) - 3sin(x)sin(sin(x))cos(x) + cos^{3}(x)cos(sin(x))\right)}{dx}\\=&-8cos(x)cos(x) - 8sin(x)*-sin(x) + -sin(x)cos(sin(x)) + cos(x)*-sin(sin(x))cos(x) - 3cos(x)sin(sin(x))cos(x) - 3sin(x)cos(sin(x))cos(x)cos(x) - 3sin(x)sin(sin(x))*-sin(x) + -3cos^{2}(x)sin(x)cos(sin(x)) + cos^{3}(x)*-sin(sin(x))cos(x)\\=&-8cos^{2}(x) + 3sin^{2}(x)sin(sin(x)) - 6sin(x)cos^{2}(x)cos(sin(x)) - 4sin(sin(x))cos^{2}(x) - sin(x)cos(sin(x)) - sin(sin(x))cos^{4}(x) + 8sin^{2}(x)\\ \end{split}\end{equation} \]

\[ \begin{equation}\begin{split}【2/3】求函数cos(x)cos(x) - cos(cos(x)) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = cos^{2}(x) - cos(cos(x))\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( cos^{2}(x) - cos(cos(x))\right)}{dx}\\=&-2cos(x)sin(x) - -sin(cos(x))*-sin(x)\\=&-2sin(x)cos(x) - sin(x)sin(cos(x))\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( -2sin(x)cos(x) - sin(x)sin(cos(x))\right)}{dx}\\=&-2cos(x)cos(x) - 2sin(x)*-sin(x) - cos(x)sin(cos(x)) - sin(x)cos(cos(x))*-sin(x)\\=&-2cos^{2}(x) - sin(cos(x))cos(x) + sin^{2}(x)cos(cos(x)) + 2sin^{2}(x)\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( -2cos^{2}(x) - sin(cos(x))cos(x) + sin^{2}(x)cos(cos(x)) + 2sin^{2}(x)\right)}{dx}\\=&-2*-2cos(x)sin(x) - cos(cos(x))*-sin(x)cos(x) - sin(cos(x))*-sin(x) + 2sin(x)cos(x)cos(cos(x)) + sin^{2}(x)*-sin(cos(x))*-sin(x) + 2*2sin(x)cos(x)\\=&2sin(x)cos(x)cos(cos(x)) + sin(x)cos(cos(x))cos(x) + sin(x)sin(cos(x)) + 8sin(x)cos(x) + sin(cos(x))sin^{3}(x)\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( 2sin(x)cos(x)cos(cos(x)) + sin(x)cos(cos(x))cos(x) + sin(x)sin(cos(x)) + 8sin(x)cos(x) + sin(cos(x))sin^{3}(x)\right)}{dx}\\=&2cos(x)cos(x)cos(cos(x)) + 2sin(x)*-sin(x)cos(cos(x)) + 2sin(x)cos(x)*-sin(cos(x))*-sin(x) + cos(x)cos(cos(x))cos(x) + sin(x)*-sin(cos(x))*-sin(x)cos(x) + sin(x)cos(cos(x))*-sin(x) + cos(x)sin(cos(x)) + sin(x)cos(cos(x))*-sin(x) + 8cos(x)cos(x) + 8sin(x)*-sin(x) + cos(cos(x))*-sin(x)sin^{3}(x) + sin(cos(x))*3sin^{2}(x)cos(x)\\=&3cos^{2}(x)cos(cos(x)) - 4sin^{2}(x)cos(cos(x)) + 3sin(cos(x))sin^{2}(x)cos(x) + sin(cos(x))cos(x) + 8cos^{2}(x) + 3sin^{2}(x)sin(cos(x))cos(x) - sin^{4}(x)cos(cos(x)) - 8sin^{2}(x)\\ \end{split}\end{equation} \]

\[ \begin{equation}\begin{split}【3/3】求函数tan(x)tan(x) - tan(tan(x)) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = tan^{2}(x) - tan(tan(x))\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( tan^{2}(x) - tan(tan(x))\right)}{dx}\\=&2tan(x)sec^{2}(x)(1) - sec^{2}(tan(x))(sec^{2}(x)(1))\\=&2tan(x)sec^{2}(x) - sec^{2}(x)sec^{2}(tan(x))\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( 2tan(x)sec^{2}(x) - sec^{2}(x)sec^{2}(tan(x))\right)}{dx}\\=&2sec^{2}(x)(1)sec^{2}(x) + 2tan(x)*2sec^{2}(x)tan(x) - 2sec^{2}(x)tan(x)sec^{2}(tan(x)) - sec^{2}(x)*2sec^{2}(tan(x))tan(tan(x))sec^{2}(x)(1)\\=&2sec^{4}(x) - 2tan(x)sec^{2}(x)sec^{2}(tan(x)) - 2tan(tan(x))sec^{2}(tan(x))sec^{4}(x) + 4tan^{2}(x)sec^{2}(x)\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( 2sec^{4}(x) - 2tan(x)sec^{2}(x)sec^{2}(tan(x)) - 2tan(tan(x))sec^{2}(tan(x))sec^{4}(x) + 4tan^{2}(x)sec^{2}(x)\right)}{dx}\\=&2*4sec^{4}(x)tan(x) - 2sec^{2}(x)(1)sec^{2}(x)sec^{2}(tan(x)) - 2tan(x)*2sec^{2}(x)tan(x)sec^{2}(tan(x)) - 2tan(x)sec^{2}(x)*2sec^{2}(tan(x))tan(tan(x))sec^{2}(x)(1) - 2sec^{2}(tan(x))(sec^{2}(x)(1))sec^{2}(tan(x))sec^{4}(x) - 2tan(tan(x))*2sec^{2}(tan(x))tan(tan(x))sec^{2}(x)(1)sec^{4}(x) - 2tan(tan(x))sec^{2}(tan(x))*4sec^{4}(x)tan(x) + 4*2tan(x)sec^{2}(x)(1)sec^{2}(x) + 4tan^{2}(x)*2sec^{2}(x)tan(x)\\=& - 4tan^{2}(x)sec^{2}(x)sec^{2}(tan(x)) - 2sec^{4}(x)sec^{2}(tan(x)) - 4tan^{2}(tan(x))sec^{6}(x)sec^{2}(tan(x)) - 12tan(x)tan(tan(x))sec^{2}(tan(x))sec^{4}(x) - 2sec^{6}(x)sec^{4}(tan(x)) + 16tan(x)sec^{4}(x) + 8tan^{3}(x)sec^{2}(x)\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( - 4tan^{2}(x)sec^{2}(x)sec^{2}(tan(x)) - 2sec^{4}(x)sec^{2}(tan(x)) - 4tan^{2}(tan(x))sec^{6}(x)sec^{2}(tan(x)) - 12tan(x)tan(tan(x))sec^{2}(tan(x))sec^{4}(x) - 2sec^{6}(x)sec^{4}(tan(x)) + 16tan(x)sec^{4}(x) + 8tan^{3}(x)sec^{2}(x)\right)}{dx}\\=& - 4*2tan(x)sec^{2}(x)(1)sec^{2}(x)sec^{2}(tan(x)) - 4tan^{2}(x)*2sec^{2}(x)tan(x)sec^{2}(tan(x)) - 4tan^{2}(x)sec^{2}(x)*2sec^{2}(tan(x))tan(tan(x))sec^{2}(x)(1) - 2*4sec^{4}(x)tan(x)sec^{2}(tan(x)) - 2sec^{4}(x)*2sec^{2}(tan(x))tan(tan(x))sec^{2}(x)(1) - 4*2tan(tan(x))sec^{2}(tan(x))(sec^{2}(x)(1))sec^{6}(x)sec^{2}(tan(x)) - 4tan^{2}(tan(x))*6sec^{6}(x)tan(x)sec^{2}(tan(x)) - 4tan^{2}(tan(x))sec^{6}(x)*2sec^{2}(tan(x))tan(tan(x))sec^{2}(x)(1) - 12sec^{2}(x)(1)tan(tan(x))sec^{2}(tan(x))sec^{4}(x) - 12tan(x)sec^{2}(tan(x))(sec^{2}(x)(1))sec^{2}(tan(x))sec^{4}(x) - 12tan(x)tan(tan(x))*2sec^{2}(tan(x))tan(tan(x))sec^{2}(x)(1)sec^{4}(x) - 12tan(x)tan(tan(x))sec^{2}(tan(x))*4sec^{4}(x)tan(x) - 2*6sec^{6}(x)tan(x)sec^{4}(tan(x)) - 2sec^{6}(x)*4sec^{4}(tan(x))tan(tan(x))sec^{2}(x)(1) + 16sec^{2}(x)(1)sec^{4}(x) + 16tan(x)*4sec^{4}(x)tan(x) + 8*3tan^{2}(x)sec^{2}(x)(1)sec^{2}(x) + 8tan^{3}(x)*2sec^{2}(x)tan(x)\\=& - 24tan(x)sec^{6}(x)sec^{4}(tan(x)) - 8tan^{3}(x)sec^{2}(x)sec^{2}(tan(x)) - 56tan^{2}(x)tan(tan(x))sec^{2}(tan(x))sec^{4}(x) - 16tan(x)sec^{4}(x)sec^{2}(tan(x)) - 8tan(tan(x))sec^{8}(x)sec^{4}(tan(x)) - 4tan(tan(x))sec^{2}(tan(x))sec^{6}(x) - 24tan(x)tan^{2}(tan(x))sec^{6}(x)sec^{2}(tan(x)) - 8tan^{3}(tan(x))sec^{2}(tan(x))sec^{8}(x) - 12tan(tan(x))sec^{6}(x)sec^{2}(tan(x)) - 24tan^{2}(tan(x))tan(x)sec^{6}(x)sec^{2}(tan(x)) - 8tan(tan(x))sec^{4}(tan(x))sec^{8}(x) + 16sec^{6}(x) + 88tan^{2}(x)sec^{4}(x) + 16tan^{4}(x)sec^{2}(x)\\ \end{split}\end{equation} \]

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