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。
\[ \begin{equation}\begin{split}【1/1】求函数cos(x)sin(x)cos(x)xxsin(sin(x)) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = x^{2}sin(x)sin(sin(x))cos^{2}(x)\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( x^{2}sin(x)sin(sin(x))cos^{2}(x)\right)}{dx}\\=&2xsin(x)sin(sin(x))cos^{2}(x) + x^{2}cos(x)sin(sin(x))cos^{2}(x) + x^{2}sin(x)cos(sin(x))cos(x)cos^{2}(x) + x^{2}sin(x)sin(sin(x))*-2cos(x)sin(x)\\=&2xsin(x)sin(sin(x))cos^{2}(x) + x^{2}sin(x)cos^{3}(x)cos(sin(x)) + x^{2}sin(sin(x))cos^{3}(x) - 2x^{2}sin^{2}(x)sin(sin(x))cos(x)\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( 2xsin(x)sin(sin(x))cos^{2}(x) + x^{2}sin(x)cos^{3}(x)cos(sin(x)) + x^{2}sin(sin(x))cos^{3}(x) - 2x^{2}sin^{2}(x)sin(sin(x))cos(x)\right)}{dx}\\=&2sin(x)sin(sin(x))cos^{2}(x) + 2xcos(x)sin(sin(x))cos^{2}(x) + 2xsin(x)cos(sin(x))cos(x)cos^{2}(x) + 2xsin(x)sin(sin(x))*-2cos(x)sin(x) + 2xsin(x)cos^{3}(x)cos(sin(x)) + x^{2}cos(x)cos^{3}(x)cos(sin(x)) + x^{2}sin(x)*-3cos^{2}(x)sin(x)cos(sin(x)) + x^{2}sin(x)cos^{3}(x)*-sin(sin(x))cos(x) + 2xsin(sin(x))cos^{3}(x) + x^{2}cos(sin(x))cos(x)cos^{3}(x) + x^{2}sin(sin(x))*-3cos^{2}(x)sin(x) - 2*2xsin^{2}(x)sin(sin(x))cos(x) - 2x^{2}*2sin(x)cos(x)sin(sin(x))cos(x) - 2x^{2}sin^{2}(x)cos(sin(x))cos(x)cos(x) - 2x^{2}sin^{2}(x)sin(sin(x))*-sin(x)\\=&2sin(x)sin(sin(x))cos^{2}(x) + 4xsin(x)cos^{3}(x)cos(sin(x)) - 5x^{2}sin^{2}(x)cos^{2}(x)cos(sin(x)) - 8xsin^{2}(x)sin(sin(x))cos(x) + 2x^{2}cos^{4}(x)cos(sin(x)) + 4xsin(sin(x))cos^{3}(x) - x^{2}sin(x)sin(sin(x))cos^{4}(x) - 7x^{2}sin(x)sin(sin(x))cos^{2}(x) + 2x^{2}sin^{3}(x)sin(sin(x))\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( 2sin(x)sin(sin(x))cos^{2}(x) + 4xsin(x)cos^{3}(x)cos(sin(x)) - 5x^{2}sin^{2}(x)cos^{2}(x)cos(sin(x)) - 8xsin^{2}(x)sin(sin(x))cos(x) + 2x^{2}cos^{4}(x)cos(sin(x)) + 4xsin(sin(x))cos^{3}(x) - x^{2}sin(x)sin(sin(x))cos^{4}(x) - 7x^{2}sin(x)sin(sin(x))cos^{2}(x) + 2x^{2}sin^{3}(x)sin(sin(x))\right)}{dx}\\=&2cos(x)sin(sin(x))cos^{2}(x) + 2sin(x)cos(sin(x))cos(x)cos^{2}(x) + 2sin(x)sin(sin(x))*-2cos(x)sin(x) + 4sin(x)cos^{3}(x)cos(sin(x)) + 4xcos(x)cos^{3}(x)cos(sin(x)) + 4xsin(x)*-3cos^{2}(x)sin(x)cos(sin(x)) + 4xsin(x)cos^{3}(x)*-sin(sin(x))cos(x) - 5*2xsin^{2}(x)cos^{2}(x)cos(sin(x)) - 5x^{2}*2sin(x)cos(x)cos^{2}(x)cos(sin(x)) - 5x^{2}sin^{2}(x)*-2cos(x)sin(x)cos(sin(x)) - 5x^{2}sin^{2}(x)cos^{2}(x)*-sin(sin(x))cos(x) - 8sin^{2}(x)sin(sin(x))cos(x) - 8x*2sin(x)cos(x)sin(sin(x))cos(x) - 8xsin^{2}(x)cos(sin(x))cos(x)cos(x) - 8xsin^{2}(x)sin(sin(x))*-sin(x) + 2*2xcos^{4}(x)cos(sin(x)) + 2x^{2}*-4cos^{3}(x)sin(x)cos(sin(x)) + 2x^{2}cos^{4}(x)*-sin(sin(x))cos(x) + 4sin(sin(x))cos^{3}(x) + 4xcos(sin(x))cos(x)cos^{3}(x) + 4xsin(sin(x))*-3cos^{2}(x)sin(x) - 2xsin(x)sin(sin(x))cos^{4}(x) - x^{2}cos(x)sin(sin(x))cos^{4}(x) - x^{2}sin(x)cos(sin(x))cos(x)cos^{4}(x) - x^{2}sin(x)sin(sin(x))*-4cos^{3}(x)sin(x) - 7*2xsin(x)sin(sin(x))cos^{2}(x) - 7x^{2}cos(x)sin(sin(x))cos^{2}(x) - 7x^{2}sin(x)cos(sin(x))cos(x)cos^{2}(x) - 7x^{2}sin(x)sin(sin(x))*-2cos(x)sin(x) + 2*2xsin^{3}(x)sin(sin(x)) + 2x^{2}*3sin^{2}(x)cos(x)sin(sin(x)) + 2x^{2}sin^{3}(x)cos(sin(x))cos(x)\\=&6sin(x)cos^{3}(x)cos(sin(x)) + 6sin(sin(x))cos^{3}(x) - 12sin^{2}(x)sin(sin(x))cos(x) + 12xcos^{4}(x)cos(sin(x)) - 30xsin^{2}(x)cos^{2}(x)cos(sin(x)) - 6xsin(x)sin(sin(x))cos^{4}(x) - 25x^{2}sin(x)cos^{3}(x)cos(sin(x)) + 12x^{2}sin^{3}(x)cos(x)cos(sin(x)) + 9x^{2}sin^{2}(x)sin(sin(x))cos^{3}(x) - 42xsin(x)sin(sin(x))cos^{2}(x) - x^{2}sin(x)cos^{5}(x)cos(sin(x)) - 3x^{2}sin(sin(x))cos^{5}(x) + 20x^{2}sin^{2}(x)sin(sin(x))cos(x) - 7x^{2}sin(sin(x))cos^{3}(x) + 12xsin^{3}(x)sin(sin(x))\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( 6sin(x)cos^{3}(x)cos(sin(x)) + 6sin(sin(x))cos^{3}(x) - 12sin^{2}(x)sin(sin(x))cos(x) + 12xcos^{4}(x)cos(sin(x)) - 30xsin^{2}(x)cos^{2}(x)cos(sin(x)) - 6xsin(x)sin(sin(x))cos^{4}(x) - 25x^{2}sin(x)cos^{3}(x)cos(sin(x)) + 12x^{2}sin^{3}(x)cos(x)cos(sin(x)) + 9x^{2}sin^{2}(x)sin(sin(x))cos^{3}(x) - 42xsin(x)sin(sin(x))cos^{2}(x) - x^{2}sin(x)cos^{5}(x)cos(sin(x)) - 3x^{2}sin(sin(x))cos^{5}(x) + 20x^{2}sin^{2}(x)sin(sin(x))cos(x) - 7x^{2}sin(sin(x))cos^{3}(x) + 12xsin^{3}(x)sin(sin(x))\right)}{dx}\\=&6cos(x)cos^{3}(x)cos(sin(x)) + 6sin(x)*-3cos^{2}(x)sin(x)cos(sin(x)) + 6sin(x)cos^{3}(x)*-sin(sin(x))cos(x) + 6cos(sin(x))cos(x)cos^{3}(x) + 6sin(sin(x))*-3cos^{2}(x)sin(x) - 12*2sin(x)cos(x)sin(sin(x))cos(x) - 12sin^{2}(x)cos(sin(x))cos(x)cos(x) - 12sin^{2}(x)sin(sin(x))*-sin(x) + 12cos^{4}(x)cos(sin(x)) + 12x*-4cos^{3}(x)sin(x)cos(sin(x)) + 12xcos^{4}(x)*-sin(sin(x))cos(x) - 30sin^{2}(x)cos^{2}(x)cos(sin(x)) - 30x*2sin(x)cos(x)cos^{2}(x)cos(sin(x)) - 30xsin^{2}(x)*-2cos(x)sin(x)cos(sin(x)) - 30xsin^{2}(x)cos^{2}(x)*-sin(sin(x))cos(x) - 6sin(x)sin(sin(x))cos^{4}(x) - 6xcos(x)sin(sin(x))cos^{4}(x) - 6xsin(x)cos(sin(x))cos(x)cos^{4}(x) - 6xsin(x)sin(sin(x))*-4cos^{3}(x)sin(x) - 25*2xsin(x)cos^{3}(x)cos(sin(x)) - 25x^{2}cos(x)cos^{3}(x)cos(sin(x)) - 25x^{2}sin(x)*-3cos^{2}(x)sin(x)cos(sin(x)) - 25x^{2}sin(x)cos^{3}(x)*-sin(sin(x))cos(x) + 12*2xsin^{3}(x)cos(x)cos(sin(x)) + 12x^{2}*3sin^{2}(x)cos(x)cos(x)cos(sin(x)) + 12x^{2}sin^{3}(x)*-sin(x)cos(sin(x)) + 12x^{2}sin^{3}(x)cos(x)*-sin(sin(x))cos(x) + 9*2xsin^{2}(x)sin(sin(x))cos^{3}(x) + 9x^{2}*2sin(x)cos(x)sin(sin(x))cos^{3}(x) + 9x^{2}sin^{2}(x)cos(sin(x))cos(x)cos^{3}(x) + 9x^{2}sin^{2}(x)sin(sin(x))*-3cos^{2}(x)sin(x) - 42sin(x)sin(sin(x))cos^{2}(x) - 42xcos(x)sin(sin(x))cos^{2}(x) - 42xsin(x)cos(sin(x))cos(x)cos^{2}(x) - 42xsin(x)sin(sin(x))*-2cos(x)sin(x) - 2xsin(x)cos^{5}(x)cos(sin(x)) - x^{2}cos(x)cos^{5}(x)cos(sin(x)) - x^{2}sin(x)*-5cos^{4}(x)sin(x)cos(sin(x)) - x^{2}sin(x)cos^{5}(x)*-sin(sin(x))cos(x) - 3*2xsin(sin(x))cos^{5}(x) - 3x^{2}cos(sin(x))cos(x)cos^{5}(x) - 3x^{2}sin(sin(x))*-5cos^{4}(x)sin(x) + 20*2xsin^{2}(x)sin(sin(x))cos(x) + 20x^{2}*2sin(x)cos(x)sin(sin(x))cos(x) + 20x^{2}sin^{2}(x)cos(sin(x))cos(x)cos(x) + 20x^{2}sin^{2}(x)sin(sin(x))*-sin(x) - 7*2xsin(sin(x))cos^{3}(x) - 7x^{2}cos(sin(x))cos(x)cos^{3}(x) - 7x^{2}sin(sin(x))*-3cos^{2}(x)sin(x) + 12sin^{3}(x)sin(sin(x)) + 12x*3sin^{2}(x)cos(x)sin(sin(x)) + 12xsin^{3}(x)cos(sin(x))cos(x)\\=&24cos^{4}(x)cos(sin(x)) - 60sin^{2}(x)cos^{2}(x)cos(sin(x)) - 12sin(x)sin(sin(x))cos^{4}(x) - 84sin(x)sin(sin(x))cos^{2}(x) + 24sin^{3}(x)sin(sin(x)) - 200xsin(x)cos^{3}(x)cos(sin(x)) - 8xsin(x)cos^{5}(x)cos(sin(x)) + 96xsin^{3}(x)cos(x)cos(sin(x)) + 131x^{2}sin^{2}(x)cos^{2}(x)cos(sin(x)) + 72xsin^{2}(x)sin(sin(x))cos^{3}(x) + 14x^{2}sin^{2}(x)cos^{4}(x)cos(sin(x)) - 32x^{2}cos^{4}(x)cos(sin(x)) + 58x^{2}sin(x)sin(sin(x))cos^{4}(x) - 24xsin(sin(x))cos^{5}(x) - 39x^{2}sin^{3}(x)sin(sin(x))cos^{2}(x) - 56xsin(sin(x))cos^{3}(x) + 160xsin^{2}(x)sin(sin(x))cos(x) - 4x^{2}cos^{6}(x)cos(sin(x)) + 61x^{2}sin(x)sin(sin(x))cos^{2}(x) + x^{2}sin(x)sin(sin(x))cos^{6}(x) - 12x^{2}sin^{4}(x)cos(sin(x)) - 20x^{2}sin^{3}(x)sin(sin(x))\\ \end{split}\end{equation} \]

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