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。
\[ \begin{equation}\begin{split}【1/1】求函数\frac{(xsin({x}^{2}) - 2(1 - cos(x))sin(x))}{({x}^{4})} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{2sin(x)cos(x)}{x^{4}} - \frac{2sin(x)}{x^{4}} + \frac{sin(x^{2})}{x^{3}}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( \frac{2sin(x)cos(x)}{x^{4}} - \frac{2sin(x)}{x^{4}} + \frac{sin(x^{2})}{x^{3}}\right)}{dx}\\=&\frac{2*-4sin(x)cos(x)}{x^{5}} + \frac{2cos(x)cos(x)}{x^{4}} + \frac{2sin(x)*-sin(x)}{x^{4}} - \frac{2*-4sin(x)}{x^{5}} - \frac{2cos(x)}{x^{4}} + \frac{-3sin(x^{2})}{x^{4}} + \frac{cos(x^{2})*2x}{x^{3}}\\=& - \frac{8sin(x)cos(x)}{x^{5}} + \frac{2cos^{2}(x)}{x^{4}} - \frac{2sin^{2}(x)}{x^{4}} + \frac{8sin(x)}{x^{5}} - \frac{2cos(x)}{x^{4}} - \frac{3sin(x^{2})}{x^{4}} + \frac{2cos(x^{2})}{x^{2}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( - \frac{8sin(x)cos(x)}{x^{5}} + \frac{2cos^{2}(x)}{x^{4}} - \frac{2sin^{2}(x)}{x^{4}} + \frac{8sin(x)}{x^{5}} - \frac{2cos(x)}{x^{4}} - \frac{3sin(x^{2})}{x^{4}} + \frac{2cos(x^{2})}{x^{2}}\right)}{dx}\\=& - \frac{8*-5sin(x)cos(x)}{x^{6}} - \frac{8cos(x)cos(x)}{x^{5}} - \frac{8sin(x)*-sin(x)}{x^{5}} + \frac{2*-4cos^{2}(x)}{x^{5}} + \frac{2*-2cos(x)sin(x)}{x^{4}} - \frac{2*-4sin^{2}(x)}{x^{5}} - \frac{2*2sin(x)cos(x)}{x^{4}} + \frac{8*-5sin(x)}{x^{6}} + \frac{8cos(x)}{x^{5}} - \frac{2*-4cos(x)}{x^{5}} - \frac{2*-sin(x)}{x^{4}} - \frac{3*-4sin(x^{2})}{x^{5}} - \frac{3cos(x^{2})*2x}{x^{4}} + \frac{2*-2cos(x^{2})}{x^{3}} + \frac{2*-sin(x^{2})*2x}{x^{2}}\\=&\frac{40sin(x)cos(x)}{x^{6}} - \frac{16cos^{2}(x)}{x^{5}} - \frac{8sin(x)cos(x)}{x^{4}} + \frac{16sin^{2}(x)}{x^{5}} - \frac{40sin(x)}{x^{6}} + \frac{16cos(x)}{x^{5}} + \frac{2sin(x)}{x^{4}} + \frac{12sin(x^{2})}{x^{5}} - \frac{10cos(x^{2})}{x^{3}} - \frac{4sin(x^{2})}{x}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{40sin(x)cos(x)}{x^{6}} - \frac{16cos^{2}(x)}{x^{5}} - \frac{8sin(x)cos(x)}{x^{4}} + \frac{16sin^{2}(x)}{x^{5}} - \frac{40sin(x)}{x^{6}} + \frac{16cos(x)}{x^{5}} + \frac{2sin(x)}{x^{4}} + \frac{12sin(x^{2})}{x^{5}} - \frac{10cos(x^{2})}{x^{3}} - \frac{4sin(x^{2})}{x}\right)}{dx}\\=&\frac{40*-6sin(x)cos(x)}{x^{7}} + \frac{40cos(x)cos(x)}{x^{6}} + \frac{40sin(x)*-sin(x)}{x^{6}} - \frac{16*-5cos^{2}(x)}{x^{6}} - \frac{16*-2cos(x)sin(x)}{x^{5}} - \frac{8*-4sin(x)cos(x)}{x^{5}} - \frac{8cos(x)cos(x)}{x^{4}} - \frac{8sin(x)*-sin(x)}{x^{4}} + \frac{16*-5sin^{2}(x)}{x^{6}} + \frac{16*2sin(x)cos(x)}{x^{5}} - \frac{40*-6sin(x)}{x^{7}} - \frac{40cos(x)}{x^{6}} + \frac{16*-5cos(x)}{x^{6}} + \frac{16*-sin(x)}{x^{5}} + \frac{2*-4sin(x)}{x^{5}} + \frac{2cos(x)}{x^{4}} + \frac{12*-5sin(x^{2})}{x^{6}} + \frac{12cos(x^{2})*2x}{x^{5}} - \frac{10*-3cos(x^{2})}{x^{4}} - \frac{10*-sin(x^{2})*2x}{x^{3}} - \frac{4*-sin(x^{2})}{x^{2}} - \frac{4cos(x^{2})*2x}{x}\\=& - \frac{240sin(x)cos(x)}{x^{7}} + \frac{120cos^{2}(x)}{x^{6}} + \frac{96sin(x)cos(x)}{x^{5}} - \frac{8cos^{2}(x)}{x^{4}} - \frac{120sin^{2}(x)}{x^{6}} + \frac{8sin^{2}(x)}{x^{4}} + \frac{240sin(x)}{x^{7}} - \frac{120cos(x)}{x^{6}} - \frac{24sin(x)}{x^{5}} + \frac{2cos(x)}{x^{4}} - \frac{60sin(x^{2})}{x^{6}} + \frac{54cos(x^{2})}{x^{4}} + \frac{24sin(x^{2})}{x^{2}} - 8cos(x^{2})\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( - \frac{240sin(x)cos(x)}{x^{7}} + \frac{120cos^{2}(x)}{x^{6}} + \frac{96sin(x)cos(x)}{x^{5}} - \frac{8cos^{2}(x)}{x^{4}} - \frac{120sin^{2}(x)}{x^{6}} + \frac{8sin^{2}(x)}{x^{4}} + \frac{240sin(x)}{x^{7}} - \frac{120cos(x)}{x^{6}} - \frac{24sin(x)}{x^{5}} + \frac{2cos(x)}{x^{4}} - \frac{60sin(x^{2})}{x^{6}} + \frac{54cos(x^{2})}{x^{4}} + \frac{24sin(x^{2})}{x^{2}} - 8cos(x^{2})\right)}{dx}\\=& - \frac{240*-7sin(x)cos(x)}{x^{8}} - \frac{240cos(x)cos(x)}{x^{7}} - \frac{240sin(x)*-sin(x)}{x^{7}} + \frac{120*-6cos^{2}(x)}{x^{7}} + \frac{120*-2cos(x)sin(x)}{x^{6}} + \frac{96*-5sin(x)cos(x)}{x^{6}} + \frac{96cos(x)cos(x)}{x^{5}} + \frac{96sin(x)*-sin(x)}{x^{5}} - \frac{8*-4cos^{2}(x)}{x^{5}} - \frac{8*-2cos(x)sin(x)}{x^{4}} - \frac{120*-6sin^{2}(x)}{x^{7}} - \frac{120*2sin(x)cos(x)}{x^{6}} + \frac{8*-4sin^{2}(x)}{x^{5}} + \frac{8*2sin(x)cos(x)}{x^{4}} + \frac{240*-7sin(x)}{x^{8}} + \frac{240cos(x)}{x^{7}} - \frac{120*-6cos(x)}{x^{7}} - \frac{120*-sin(x)}{x^{6}} - \frac{24*-5sin(x)}{x^{6}} - \frac{24cos(x)}{x^{5}} + \frac{2*-4cos(x)}{x^{5}} + \frac{2*-sin(x)}{x^{4}} - \frac{60*-6sin(x^{2})}{x^{7}} - \frac{60cos(x^{2})*2x}{x^{6}} + \frac{54*-4cos(x^{2})}{x^{5}} + \frac{54*-sin(x^{2})*2x}{x^{4}} + \frac{24*-2sin(x^{2})}{x^{3}} + \frac{24cos(x^{2})*2x}{x^{2}} - 8*-sin(x^{2})*2x\\=&\frac{1680sin(x)cos(x)}{x^{8}} - \frac{960cos^{2}(x)}{x^{7}} - \frac{960sin(x)cos(x)}{x^{6}} + \frac{128cos^{2}(x)}{x^{5}} + \frac{32sin(x)cos(x)}{x^{4}} - \frac{128sin^{2}(x)}{x^{5}} + \frac{960sin^{2}(x)}{x^{7}} - \frac{1680sin(x)}{x^{8}} + \frac{960cos(x)}{x^{7}} + \frac{240sin(x)}{x^{6}} - \frac{32cos(x)}{x^{5}} - \frac{2sin(x)}{x^{4}} + \frac{360sin(x^{2})}{x^{7}} - \frac{336cos(x^{2})}{x^{5}} - \frac{156sin(x^{2})}{x^{3}} + \frac{48cos(x^{2})}{x} + 16xsin(x^{2})\\ \end{split}\end{equation} \]

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