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。
\[ \begin{equation}\begin{split}【1/1】求函数(({x}^{2}){\frac{1}{({x}^{3} - 4x)}}^{3}) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{x^{2}}{(x^{3} - 4x)^{3}}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( \frac{x^{2}}{(x^{3} - 4x)^{3}}\right)}{dx}\\=&(\frac{-3(3x^{2} - 4)}{(x^{3} - 4x)^{4}})x^{2} + \frac{2x}{(x^{3} - 4x)^{3}}\\=&\frac{-9x^{4}}{(x^{3} - 4x)^{4}} + \frac{12x^{2}}{(x^{3} - 4x)^{4}} + \frac{2x}{(x^{3} - 4x)^{3}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{-9x^{4}}{(x^{3} - 4x)^{4}} + \frac{12x^{2}}{(x^{3} - 4x)^{4}} + \frac{2x}{(x^{3} - 4x)^{3}}\right)}{dx}\\=&-9(\frac{-4(3x^{2} - 4)}{(x^{3} - 4x)^{5}})x^{4} - \frac{9*4x^{3}}{(x^{3} - 4x)^{4}} + 12(\frac{-4(3x^{2} - 4)}{(x^{3} - 4x)^{5}})x^{2} + \frac{12*2x}{(x^{3} - 4x)^{4}} + 2(\frac{-3(3x^{2} - 4)}{(x^{3} - 4x)^{4}})x + \frac{2}{(x^{3} - 4x)^{3}}\\=&\frac{108x^{6}}{(x^{3} - 4x)^{5}} - \frac{288x^{4}}{(x^{3} - 4x)^{5}} - \frac{54x^{3}}{(x^{3} - 4x)^{4}} + \frac{192x^{2}}{(x^{3} - 4x)^{5}} + \frac{48x}{(x^{3} - 4x)^{4}} + \frac{2}{(x^{3} - 4x)^{3}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{108x^{6}}{(x^{3} - 4x)^{5}} - \frac{288x^{4}}{(x^{3} - 4x)^{5}} - \frac{54x^{3}}{(x^{3} - 4x)^{4}} + \frac{192x^{2}}{(x^{3} - 4x)^{5}} + \frac{48x}{(x^{3} - 4x)^{4}} + \frac{2}{(x^{3} - 4x)^{3}}\right)}{dx}\\=&108(\frac{-5(3x^{2} - 4)}{(x^{3} - 4x)^{6}})x^{6} + \frac{108*6x^{5}}{(x^{3} - 4x)^{5}} - 288(\frac{-5(3x^{2} - 4)}{(x^{3} - 4x)^{6}})x^{4} - \frac{288*4x^{3}}{(x^{3} - 4x)^{5}} - 54(\frac{-4(3x^{2} - 4)}{(x^{3} - 4x)^{5}})x^{3} - \frac{54*3x^{2}}{(x^{3} - 4x)^{4}} + 192(\frac{-5(3x^{2} - 4)}{(x^{3} - 4x)^{6}})x^{2} + \frac{192*2x}{(x^{3} - 4x)^{5}} + 48(\frac{-4(3x^{2} - 4)}{(x^{3} - 4x)^{5}})x + \frac{48}{(x^{3} - 4x)^{4}} + 2(\frac{-3(3x^{2} - 4)}{(x^{3} - 4x)^{4}})\\=&\frac{-1620x^{8}}{(x^{3} - 4x)^{6}} + \frac{6480x^{6}}{(x^{3} - 4x)^{6}} + \frac{1296x^{5}}{(x^{3} - 4x)^{5}} - \frac{8640x^{4}}{(x^{3} - 4x)^{6}} - \frac{2592x^{3}}{(x^{3} - 4x)^{5}} - \frac{180x^{2}}{(x^{3} - 4x)^{4}} + \frac{3840x^{2}}{(x^{3} - 4x)^{6}} + \frac{1152x}{(x^{3} - 4x)^{5}} + \frac{72}{(x^{3} - 4x)^{4}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{-1620x^{8}}{(x^{3} - 4x)^{6}} + \frac{6480x^{6}}{(x^{3} - 4x)^{6}} + \frac{1296x^{5}}{(x^{3} - 4x)^{5}} - \frac{8640x^{4}}{(x^{3} - 4x)^{6}} - \frac{2592x^{3}}{(x^{3} - 4x)^{5}} - \frac{180x^{2}}{(x^{3} - 4x)^{4}} + \frac{3840x^{2}}{(x^{3} - 4x)^{6}} + \frac{1152x}{(x^{3} - 4x)^{5}} + \frac{72}{(x^{3} - 4x)^{4}}\right)}{dx}\\=&-1620(\frac{-6(3x^{2} - 4)}{(x^{3} - 4x)^{7}})x^{8} - \frac{1620*8x^{7}}{(x^{3} - 4x)^{6}} + 6480(\frac{-6(3x^{2} - 4)}{(x^{3} - 4x)^{7}})x^{6} + \frac{6480*6x^{5}}{(x^{3} - 4x)^{6}} + 1296(\frac{-5(3x^{2} - 4)}{(x^{3} - 4x)^{6}})x^{5} + \frac{1296*5x^{4}}{(x^{3} - 4x)^{5}} - 8640(\frac{-6(3x^{2} - 4)}{(x^{3} - 4x)^{7}})x^{4} - \frac{8640*4x^{3}}{(x^{3} - 4x)^{6}} - 2592(\frac{-5(3x^{2} - 4)}{(x^{3} - 4x)^{6}})x^{3} - \frac{2592*3x^{2}}{(x^{3} - 4x)^{5}} - 180(\frac{-4(3x^{2} - 4)}{(x^{3} - 4x)^{5}})x^{2} - \frac{180*2x}{(x^{3} - 4x)^{4}} + 3840(\frac{-6(3x^{2} - 4)}{(x^{3} - 4x)^{7}})x^{2} + \frac{3840*2x}{(x^{3} - 4x)^{6}} + 1152(\frac{-5(3x^{2} - 4)}{(x^{3} - 4x)^{6}})x + \frac{1152}{(x^{3} - 4x)^{5}} + 72(\frac{-4(3x^{2} - 4)}{(x^{3} - 4x)^{5}})\\=&\frac{29160x^{10}}{(x^{3} - 4x)^{7}} - \frac{155520x^{8}}{(x^{3} - 4x)^{7}} - \frac{32400x^{7}}{(x^{3} - 4x)^{6}} + \frac{311040x^{6}}{(x^{3} - 4x)^{7}} + \frac{103680x^{5}}{(x^{3} - 4x)^{6}} - \frac{276480x^{4}}{(x^{3} - 4x)^{7}} + \frac{8640x^{4}}{(x^{3} - 4x)^{5}} - \frac{103680x^{3}}{(x^{3} - 4x)^{6}} - \frac{11520x^{2}}{(x^{3} - 4x)^{5}} - \frac{360x}{(x^{3} - 4x)^{4}} + \frac{92160x^{2}}{(x^{3} - 4x)^{7}} + \frac{30720x}{(x^{3} - 4x)^{6}} + \frac{2304}{(x^{3} - 4x)^{5}}\\ \end{split}\end{equation} \]

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