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。
\[ \begin{equation}\begin{split}【1/1】求函数{cos({cos(x)}^{2})}^{2} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = cos^{2}(cos^{2}(x))\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( cos^{2}(cos^{2}(x))\right)}{dx}\\=&-2cos(cos^{2}(x))sin(cos^{2}(x))*-2cos(x)sin(x)\\=&4sin(x)sin(cos^{2}(x))cos(cos^{2}(x))cos(x)\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( 4sin(x)sin(cos^{2}(x))cos(cos^{2}(x))cos(x)\right)}{dx}\\=&4cos(x)sin(cos^{2}(x))cos(cos^{2}(x))cos(x) + 4sin(x)cos(cos^{2}(x))*-2cos(x)sin(x)cos(cos^{2}(x))cos(x) + 4sin(x)sin(cos^{2}(x))*-sin(cos^{2}(x))*-2cos(x)sin(x)cos(x) + 4sin(x)sin(cos^{2}(x))cos(cos^{2}(x))*-sin(x)\\=&4sin(cos^{2}(x))cos^{2}(x)cos(cos^{2}(x)) - 8sin^{2}(x)cos^{2}(cos^{2}(x))cos^{2}(x) + 8sin^{2}(cos^{2}(x))sin^{2}(x)cos^{2}(x) - 4sin^{2}(x)sin(cos^{2}(x))cos(cos^{2}(x))\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( 4sin(cos^{2}(x))cos^{2}(x)cos(cos^{2}(x)) - 8sin^{2}(x)cos^{2}(cos^{2}(x))cos^{2}(x) + 8sin^{2}(cos^{2}(x))sin^{2}(x)cos^{2}(x) - 4sin^{2}(x)sin(cos^{2}(x))cos(cos^{2}(x))\right)}{dx}\\=&4cos(cos^{2}(x))*-2cos(x)sin(x)cos^{2}(x)cos(cos^{2}(x)) + 4sin(cos^{2}(x))*-2cos(x)sin(x)cos(cos^{2}(x)) + 4sin(cos^{2}(x))cos^{2}(x)*-sin(cos^{2}(x))*-2cos(x)sin(x) - 8*2sin(x)cos(x)cos^{2}(cos^{2}(x))cos^{2}(x) - 8sin^{2}(x)*-2cos(cos^{2}(x))sin(cos^{2}(x))*-2cos(x)sin(x)cos^{2}(x) - 8sin^{2}(x)cos^{2}(cos^{2}(x))*-2cos(x)sin(x) + 8*2sin(cos^{2}(x))cos(cos^{2}(x))*-2cos(x)sin(x)sin^{2}(x)cos^{2}(x) + 8sin^{2}(cos^{2}(x))*2sin(x)cos(x)cos^{2}(x) + 8sin^{2}(cos^{2}(x))sin^{2}(x)*-2cos(x)sin(x) - 4*2sin(x)cos(x)sin(cos^{2}(x))cos(cos^{2}(x)) - 4sin^{2}(x)cos(cos^{2}(x))*-2cos(x)sin(x)cos(cos^{2}(x)) - 4sin^{2}(x)sin(cos^{2}(x))*-sin(cos^{2}(x))*-2cos(x)sin(x)\\=&-8sin(x)cos^{2}(cos^{2}(x))cos^{3}(x) - 16sin(x)sin(cos^{2}(x))cos(x)cos(cos^{2}(x)) - 32sin(cos^{2}(x))sin^{3}(x)cos(cos^{2}(x))cos^{3}(x) - 16sin(x)cos^{3}(x)cos^{2}(cos^{2}(x)) - 32sin^{3}(x)sin(cos^{2}(x))cos^{3}(x)cos(cos^{2}(x)) + 24sin^{3}(x)cos^{2}(cos^{2}(x))cos(x) + 8sin^{2}(cos^{2}(x))sin(x)cos^{3}(x) + 16sin(x)sin^{2}(cos^{2}(x))cos^{3}(x) - 24sin^{2}(cos^{2}(x))sin^{3}(x)cos(x)\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( -8sin(x)cos^{2}(cos^{2}(x))cos^{3}(x) - 16sin(x)sin(cos^{2}(x))cos(x)cos(cos^{2}(x)) - 32sin(cos^{2}(x))sin^{3}(x)cos(cos^{2}(x))cos^{3}(x) - 16sin(x)cos^{3}(x)cos^{2}(cos^{2}(x)) - 32sin^{3}(x)sin(cos^{2}(x))cos^{3}(x)cos(cos^{2}(x)) + 24sin^{3}(x)cos^{2}(cos^{2}(x))cos(x) + 8sin^{2}(cos^{2}(x))sin(x)cos^{3}(x) + 16sin(x)sin^{2}(cos^{2}(x))cos^{3}(x) - 24sin^{2}(cos^{2}(x))sin^{3}(x)cos(x)\right)}{dx}\\=&-8cos(x)cos^{2}(cos^{2}(x))cos^{3}(x) - 8sin(x)*-2cos(cos^{2}(x))sin(cos^{2}(x))*-2cos(x)sin(x)cos^{3}(x) - 8sin(x)cos^{2}(cos^{2}(x))*-3cos^{2}(x)sin(x) - 16cos(x)sin(cos^{2}(x))cos(x)cos(cos^{2}(x)) - 16sin(x)cos(cos^{2}(x))*-2cos(x)sin(x)cos(x)cos(cos^{2}(x)) - 16sin(x)sin(cos^{2}(x))*-sin(x)cos(cos^{2}(x)) - 16sin(x)sin(cos^{2}(x))cos(x)*-sin(cos^{2}(x))*-2cos(x)sin(x) - 32cos(cos^{2}(x))*-2cos(x)sin(x)sin^{3}(x)cos(cos^{2}(x))cos^{3}(x) - 32sin(cos^{2}(x))*3sin^{2}(x)cos(x)cos(cos^{2}(x))cos^{3}(x) - 32sin(cos^{2}(x))sin^{3}(x)*-sin(cos^{2}(x))*-2cos(x)sin(x)cos^{3}(x) - 32sin(cos^{2}(x))sin^{3}(x)cos(cos^{2}(x))*-3cos^{2}(x)sin(x) - 16cos(x)cos^{3}(x)cos^{2}(cos^{2}(x)) - 16sin(x)*-3cos^{2}(x)sin(x)cos^{2}(cos^{2}(x)) - 16sin(x)cos^{3}(x)*-2cos(cos^{2}(x))sin(cos^{2}(x))*-2cos(x)sin(x) - 32*3sin^{2}(x)cos(x)sin(cos^{2}(x))cos^{3}(x)cos(cos^{2}(x)) - 32sin^{3}(x)cos(cos^{2}(x))*-2cos(x)sin(x)cos^{3}(x)cos(cos^{2}(x)) - 32sin^{3}(x)sin(cos^{2}(x))*-3cos^{2}(x)sin(x)cos(cos^{2}(x)) - 32sin^{3}(x)sin(cos^{2}(x))cos^{3}(x)*-sin(cos^{2}(x))*-2cos(x)sin(x) + 24*3sin^{2}(x)cos(x)cos^{2}(cos^{2}(x))cos(x) + 24sin^{3}(x)*-2cos(cos^{2}(x))sin(cos^{2}(x))*-2cos(x)sin(x)cos(x) + 24sin^{3}(x)cos^{2}(cos^{2}(x))*-sin(x) + 8*2sin(cos^{2}(x))cos(cos^{2}(x))*-2cos(x)sin(x)sin(x)cos^{3}(x) + 8sin^{2}(cos^{2}(x))cos(x)cos^{3}(x) + 8sin^{2}(cos^{2}(x))sin(x)*-3cos^{2}(x)sin(x) + 16cos(x)sin^{2}(cos^{2}(x))cos^{3}(x) + 16sin(x)*2sin(cos^{2}(x))cos(cos^{2}(x))*-2cos(x)sin(x)cos^{3}(x) + 16sin(x)sin^{2}(cos^{2}(x))*-3cos^{2}(x)sin(x) - 24*2sin(cos^{2}(x))cos(cos^{2}(x))*-2cos(x)sin(x)sin^{3}(x)cos(x) - 24sin^{2}(cos^{2}(x))*3sin^{2}(x)cos(x)cos(x) - 24sin^{2}(cos^{2}(x))sin^{3}(x)*-sin(x)\\=&-24cos^{4}(x)cos^{2}(cos^{2}(x)) - 96sin(cos^{2}(x))sin^{2}(x)cos(cos^{2}(x))cos^{4}(x) - 224sin^{2}(x)sin(cos^{2}(x))cos^{4}(x)cos(cos^{2}(x)) - 16sin(cos^{2}(x))cos^{2}(x)cos(cos^{2}(x)) + 56sin^{2}(x)cos^{2}(cos^{2}(x))cos^{2}(x) + 192sin(cos^{2}(x))sin^{4}(x)cos(cos^{2}(x))cos^{2}(x) + 192sin^{4}(x)sin(cos^{2}(x))cos^{2}(x)cos(cos^{2}(x)) + 64sin^{4}(x)cos^{4}(x)cos^{2}(cos^{2}(x)) - 64sin(cos^{2}(x))sin^{2}(x)cos^{4}(x)cos(cos^{2}(x)) - 56sin^{2}(cos^{2}(x))sin^{2}(x)cos^{2}(x) + 120sin^{2}(x)cos^{2}(x)cos^{2}(cos^{2}(x)) - 64sin^{4}(x)sin^{2}(cos^{2}(x))cos^{4}(x) + 64sin^{4}(x)cos^{2}(cos^{2}(x))cos^{4}(x) + 16sin^{2}(x)sin(cos^{2}(x))cos(cos^{2}(x)) - 64sin^{2}(cos^{2}(x))sin^{4}(x)cos^{4}(x) - 120sin^{2}(x)sin^{2}(cos^{2}(x))cos^{2}(x) + 24sin^{2}(cos^{2}(x))cos^{4}(x) - 24sin^{4}(x)cos^{2}(cos^{2}(x)) + 24sin^{2}(cos^{2}(x))sin^{4}(x)\\ \end{split}\end{equation} \]

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