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。
\[ \begin{equation}\begin{split}【1/1】求函数sec(x)tan(sin(x)) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = tan(sin(x))sec(x)\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( tan(sin(x))sec(x)\right)}{dx}\\=&sec^{2}(sin(x))(cos(x))sec(x) + tan(sin(x))sec(x)tan(x)\\=&cos(x)sec^{2}(sin(x))sec(x) + tan(x)tan(sin(x))sec(x)\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( cos(x)sec^{2}(sin(x))sec(x) + tan(x)tan(sin(x))sec(x)\right)}{dx}\\=&-sin(x)sec^{2}(sin(x))sec(x) + cos(x)*2sec^{2}(sin(x))tan(sin(x))cos(x)sec(x) + cos(x)sec^{2}(sin(x))sec(x)tan(x) + sec^{2}(x)(1)tan(sin(x))sec(x) + tan(x)sec^{2}(sin(x))(cos(x))sec(x) + tan(x)tan(sin(x))sec(x)tan(x)\\=&-sin(x)sec^{2}(sin(x))sec(x) + 2cos^{2}(x)tan(sin(x))sec^{2}(sin(x))sec(x) + 2cos(x)tan(x)sec^{2}(sin(x))sec(x) + tan(sin(x))sec^{3}(x) + tan(sin(x))tan^{2}(x)sec(x)\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( -sin(x)sec^{2}(sin(x))sec(x) + 2cos^{2}(x)tan(sin(x))sec^{2}(sin(x))sec(x) + 2cos(x)tan(x)sec^{2}(sin(x))sec(x) + tan(sin(x))sec^{3}(x) + tan(sin(x))tan^{2}(x)sec(x)\right)}{dx}\\=&-cos(x)sec^{2}(sin(x))sec(x) - sin(x)*2sec^{2}(sin(x))tan(sin(x))cos(x)sec(x) - sin(x)sec^{2}(sin(x))sec(x)tan(x) + 2*-2cos(x)sin(x)tan(sin(x))sec^{2}(sin(x))sec(x) + 2cos^{2}(x)sec^{2}(sin(x))(cos(x))sec^{2}(sin(x))sec(x) + 2cos^{2}(x)tan(sin(x))*2sec^{2}(sin(x))tan(sin(x))cos(x)sec(x) + 2cos^{2}(x)tan(sin(x))sec^{2}(sin(x))sec(x)tan(x) + 2*-sin(x)tan(x)sec^{2}(sin(x))sec(x) + 2cos(x)sec^{2}(x)(1)sec^{2}(sin(x))sec(x) + 2cos(x)tan(x)*2sec^{2}(sin(x))tan(sin(x))cos(x)sec(x) + 2cos(x)tan(x)sec^{2}(sin(x))sec(x)tan(x) + sec^{2}(sin(x))(cos(x))sec^{3}(x) + tan(sin(x))*3sec^{3}(x)tan(x) + sec^{2}(sin(x))(cos(x))tan^{2}(x)sec(x) + tan(sin(x))*2tan(x)sec^{2}(x)(1)sec(x) + tan(sin(x))tan^{2}(x)sec(x)tan(x)\\=&2cos(x)sec^{3}(x)sec^{2}(sin(x)) - 6sin(x)cos(x)tan(sin(x))sec^{2}(sin(x))sec(x) - 3sin(x)tan(x)sec^{2}(sin(x))sec(x) + 2cos^{3}(x)sec^{4}(sin(x))sec(x) + 4cos^{3}(x)tan^{2}(sin(x))sec^{2}(sin(x))sec(x) + 2cos^{2}(x)tan(x)tan(sin(x))sec^{2}(sin(x))sec(x) - cos(x)sec^{2}(sin(x))sec(x) + 4cos^{2}(x)tan(sin(x))tan(x)sec^{2}(sin(x))sec(x) + 3cos(x)tan^{2}(x)sec^{2}(sin(x))sec(x) + cos(x)sec^{2}(sin(x))sec^{3}(x) + 5tan(x)tan(sin(x))sec^{3}(x) + tan^{3}(x)tan(sin(x))sec(x)\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( 2cos(x)sec^{3}(x)sec^{2}(sin(x)) - 6sin(x)cos(x)tan(sin(x))sec^{2}(sin(x))sec(x) - 3sin(x)tan(x)sec^{2}(sin(x))sec(x) + 2cos^{3}(x)sec^{4}(sin(x))sec(x) + 4cos^{3}(x)tan^{2}(sin(x))sec^{2}(sin(x))sec(x) + 2cos^{2}(x)tan(x)tan(sin(x))sec^{2}(sin(x))sec(x) - cos(x)sec^{2}(sin(x))sec(x) + 4cos^{2}(x)tan(sin(x))tan(x)sec^{2}(sin(x))sec(x) + 3cos(x)tan^{2}(x)sec^{2}(sin(x))sec(x) + cos(x)sec^{2}(sin(x))sec^{3}(x) + 5tan(x)tan(sin(x))sec^{3}(x) + tan^{3}(x)tan(sin(x))sec(x)\right)}{dx}\\=&2*-sin(x)sec^{3}(x)sec^{2}(sin(x)) + 2cos(x)*3sec^{3}(x)tan(x)sec^{2}(sin(x)) + 2cos(x)sec^{3}(x)*2sec^{2}(sin(x))tan(sin(x))cos(x) - 6cos(x)cos(x)tan(sin(x))sec^{2}(sin(x))sec(x) - 6sin(x)*-sin(x)tan(sin(x))sec^{2}(sin(x))sec(x) - 6sin(x)cos(x)sec^{2}(sin(x))(cos(x))sec^{2}(sin(x))sec(x) - 6sin(x)cos(x)tan(sin(x))*2sec^{2}(sin(x))tan(sin(x))cos(x)sec(x) - 6sin(x)cos(x)tan(sin(x))sec^{2}(sin(x))sec(x)tan(x) - 3cos(x)tan(x)sec^{2}(sin(x))sec(x) - 3sin(x)sec^{2}(x)(1)sec^{2}(sin(x))sec(x) - 3sin(x)tan(x)*2sec^{2}(sin(x))tan(sin(x))cos(x)sec(x) - 3sin(x)tan(x)sec^{2}(sin(x))sec(x)tan(x) + 2*-3cos^{2}(x)sin(x)sec^{4}(sin(x))sec(x) + 2cos^{3}(x)*4sec^{4}(sin(x))tan(sin(x))cos(x)sec(x) + 2cos^{3}(x)sec^{4}(sin(x))sec(x)tan(x) + 4*-3cos^{2}(x)sin(x)tan^{2}(sin(x))sec^{2}(sin(x))sec(x) + 4cos^{3}(x)*2tan(sin(x))sec^{2}(sin(x))(cos(x))sec^{2}(sin(x))sec(x) + 4cos^{3}(x)tan^{2}(sin(x))*2sec^{2}(sin(x))tan(sin(x))cos(x)sec(x) + 4cos^{3}(x)tan^{2}(sin(x))sec^{2}(sin(x))sec(x)tan(x) + 2*-2cos(x)sin(x)tan(x)tan(sin(x))sec^{2}(sin(x))sec(x) + 2cos^{2}(x)sec^{2}(x)(1)tan(sin(x))sec^{2}(sin(x))sec(x) + 2cos^{2}(x)tan(x)sec^{2}(sin(x))(cos(x))sec^{2}(sin(x))sec(x) + 2cos^{2}(x)tan(x)tan(sin(x))*2sec^{2}(sin(x))tan(sin(x))cos(x)sec(x) + 2cos^{2}(x)tan(x)tan(sin(x))sec^{2}(sin(x))sec(x)tan(x) - -sin(x)sec^{2}(sin(x))sec(x) - cos(x)*2sec^{2}(sin(x))tan(sin(x))cos(x)sec(x) - cos(x)sec^{2}(sin(x))sec(x)tan(x) + 4*-2cos(x)sin(x)tan(sin(x))tan(x)sec^{2}(sin(x))sec(x) + 4cos^{2}(x)sec^{2}(sin(x))(cos(x))tan(x)sec^{2}(sin(x))sec(x) + 4cos^{2}(x)tan(sin(x))sec^{2}(x)(1)sec^{2}(sin(x))sec(x) + 4cos^{2}(x)tan(sin(x))tan(x)*2sec^{2}(sin(x))tan(sin(x))cos(x)sec(x) + 4cos^{2}(x)tan(sin(x))tan(x)sec^{2}(sin(x))sec(x)tan(x) + 3*-sin(x)tan^{2}(x)sec^{2}(sin(x))sec(x) + 3cos(x)*2tan(x)sec^{2}(x)(1)sec^{2}(sin(x))sec(x) + 3cos(x)tan^{2}(x)*2sec^{2}(sin(x))tan(sin(x))cos(x)sec(x) + 3cos(x)tan^{2}(x)sec^{2}(sin(x))sec(x)tan(x) + -sin(x)sec^{2}(sin(x))sec^{3}(x) + cos(x)*2sec^{2}(sin(x))tan(sin(x))cos(x)sec^{3}(x) + cos(x)sec^{2}(sin(x))*3sec^{3}(x)tan(x) + 5sec^{2}(x)(1)tan(sin(x))sec^{3}(x) + 5tan(x)sec^{2}(sin(x))(cos(x))sec^{3}(x) + 5tan(x)tan(sin(x))*3sec^{3}(x)tan(x) + 3tan^{2}(x)sec^{2}(x)(1)tan(sin(x))sec(x) + tan^{3}(x)sec^{2}(sin(x))(cos(x))sec(x) + tan^{3}(x)tan(sin(x))sec(x)tan(x)\\=&-5sin(x)sec^{3}(x)sec^{2}(sin(x)) + 12cos(x)tan(x)sec^{3}(x)sec^{2}(sin(x)) + 16cos^{4}(x)tan(sin(x))sec^{4}(sin(x))sec(x) - 8cos^{2}(x)tan(sin(x))sec^{2}(sin(x))sec(x) + 6sin^{2}(x)tan(sin(x))sec^{2}(sin(x))sec(x) - 12sin(x)cos^{2}(x)sec^{4}(sin(x))sec(x) - 24sin(x)cos^{2}(x)tan^{2}(sin(x))sec^{2}(sin(x))sec(x) - 10sin(x)cos(x)tan(x)tan(sin(x))sec^{2}(sin(x))sec(x) + 10cos^{2}(x)tan(sin(x))sec^{3}(x)sec^{2}(sin(x)) - 14sin(x)cos(x)tan(sin(x))tan(x)sec^{2}(sin(x))sec(x) - 6sin(x)tan^{2}(x)sec^{2}(sin(x))sec(x) + 8cos^{4}(x)tan^{3}(sin(x))sec^{2}(sin(x))sec(x) + 12cos^{3}(x)tan(x)tan^{2}(sin(x))sec^{2}(sin(x))sec(x) - 4cos(x)tan(x)sec^{2}(sin(x))sec(x) + 8cos^{3}(x)tan(x)sec^{4}(sin(x))sec(x) + 4cos^{3}(x)tan^{2}(sin(x))tan(x)sec^{2}(sin(x))sec(x) + 8cos^{2}(x)tan(sin(x))tan^{2}(x)sec^{2}(sin(x))sec(x) + sin(x)sec^{2}(sin(x))sec(x) + 4cos^{2}(x)tan^{2}(x)tan(sin(x))sec^{2}(sin(x))sec(x) + 4cos(x)tan^{3}(x)sec^{2}(sin(x))sec(x) - sin(x)sec^{2}(sin(x))sec^{3}(x) + 2cos^{2}(x)tan(sin(x))sec^{2}(sin(x))sec^{3}(x) + 8cos(x)tan(x)sec^{2}(sin(x))sec^{3}(x) + 5tan(sin(x))sec^{5}(x) + 18tan^{2}(x)tan(sin(x))sec^{3}(x) + tan(sin(x))tan^{4}(x)sec(x)\\ \end{split}\end{equation} \]

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