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。
\[ \begin{equation}\begin{split}【1/1】求函数{(\frac{({x}^{2})}{({x}^{3} - 4x)})}^{3} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{x^{6}}{(x^{3} - 4x)^{3}}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( \frac{x^{6}}{(x^{3} - 4x)^{3}}\right)}{dx}\\=&(\frac{-3(3x^{2} - 4)}{(x^{3} - 4x)^{4}})x^{6} + \frac{6x^{5}}{(x^{3} - 4x)^{3}}\\=&\frac{-9x^{8}}{(x^{3} - 4x)^{4}} + \frac{12x^{6}}{(x^{3} - 4x)^{4}} + \frac{6x^{5}}{(x^{3} - 4x)^{3}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{-9x^{8}}{(x^{3} - 4x)^{4}} + \frac{12x^{6}}{(x^{3} - 4x)^{4}} + \frac{6x^{5}}{(x^{3} - 4x)^{3}}\right)}{dx}\\=&-9(\frac{-4(3x^{2} - 4)}{(x^{3} - 4x)^{5}})x^{8} - \frac{9*8x^{7}}{(x^{3} - 4x)^{4}} + 12(\frac{-4(3x^{2} - 4)}{(x^{3} - 4x)^{5}})x^{6} + \frac{12*6x^{5}}{(x^{3} - 4x)^{4}} + 6(\frac{-3(3x^{2} - 4)}{(x^{3} - 4x)^{4}})x^{5} + \frac{6*5x^{4}}{(x^{3} - 4x)^{3}}\\=&\frac{108x^{10}}{(x^{3} - 4x)^{5}} - \frac{288x^{8}}{(x^{3} - 4x)^{5}} - \frac{126x^{7}}{(x^{3} - 4x)^{4}} + \frac{192x^{6}}{(x^{3} - 4x)^{5}} + \frac{144x^{5}}{(x^{3} - 4x)^{4}} + \frac{30x^{4}}{(x^{3} - 4x)^{3}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{108x^{10}}{(x^{3} - 4x)^{5}} - \frac{288x^{8}}{(x^{3} - 4x)^{5}} - \frac{126x^{7}}{(x^{3} - 4x)^{4}} + \frac{192x^{6}}{(x^{3} - 4x)^{5}} + \frac{144x^{5}}{(x^{3} - 4x)^{4}} + \frac{30x^{4}}{(x^{3} - 4x)^{3}}\right)}{dx}\\=&108(\frac{-5(3x^{2} - 4)}{(x^{3} - 4x)^{6}})x^{10} + \frac{108*10x^{9}}{(x^{3} - 4x)^{5}} - 288(\frac{-5(3x^{2} - 4)}{(x^{3} - 4x)^{6}})x^{8} - \frac{288*8x^{7}}{(x^{3} - 4x)^{5}} - 126(\frac{-4(3x^{2} - 4)}{(x^{3} - 4x)^{5}})x^{7} - \frac{126*7x^{6}}{(x^{3} - 4x)^{4}} + 192(\frac{-5(3x^{2} - 4)}{(x^{3} - 4x)^{6}})x^{6} + \frac{192*6x^{5}}{(x^{3} - 4x)^{5}} + 144(\frac{-4(3x^{2} - 4)}{(x^{3} - 4x)^{5}})x^{5} + \frac{144*5x^{4}}{(x^{3} - 4x)^{4}} + 30(\frac{-3(3x^{2} - 4)}{(x^{3} - 4x)^{4}})x^{4} + \frac{30*4x^{3}}{(x^{3} - 4x)^{3}}\\=&\frac{-1620x^{12}}{(x^{3} - 4x)^{6}} + \frac{6480x^{10}}{(x^{3} - 4x)^{6}} + \frac{2592x^{9}}{(x^{3} - 4x)^{5}} - \frac{8640x^{8}}{(x^{3} - 4x)^{6}} - \frac{6048x^{7}}{(x^{3} - 4x)^{5}} - \frac{1152x^{6}}{(x^{3} - 4x)^{4}} + \frac{3840x^{6}}{(x^{3} - 4x)^{6}} + \frac{3456x^{5}}{(x^{3} - 4x)^{5}} + \frac{1080x^{4}}{(x^{3} - 4x)^{4}} + \frac{120x^{3}}{(x^{3} - 4x)^{3}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{-1620x^{12}}{(x^{3} - 4x)^{6}} + \frac{6480x^{10}}{(x^{3} - 4x)^{6}} + \frac{2592x^{9}}{(x^{3} - 4x)^{5}} - \frac{8640x^{8}}{(x^{3} - 4x)^{6}} - \frac{6048x^{7}}{(x^{3} - 4x)^{5}} - \frac{1152x^{6}}{(x^{3} - 4x)^{4}} + \frac{3840x^{6}}{(x^{3} - 4x)^{6}} + \frac{3456x^{5}}{(x^{3} - 4x)^{5}} + \frac{1080x^{4}}{(x^{3} - 4x)^{4}} + \frac{120x^{3}}{(x^{3} - 4x)^{3}}\right)}{dx}\\=&-1620(\frac{-6(3x^{2} - 4)}{(x^{3} - 4x)^{7}})x^{12} - \frac{1620*12x^{11}}{(x^{3} - 4x)^{6}} + 6480(\frac{-6(3x^{2} - 4)}{(x^{3} - 4x)^{7}})x^{10} + \frac{6480*10x^{9}}{(x^{3} - 4x)^{6}} + 2592(\frac{-5(3x^{2} - 4)}{(x^{3} - 4x)^{6}})x^{9} + \frac{2592*9x^{8}}{(x^{3} - 4x)^{5}} - 8640(\frac{-6(3x^{2} - 4)}{(x^{3} - 4x)^{7}})x^{8} - \frac{8640*8x^{7}}{(x^{3} - 4x)^{6}} - 6048(\frac{-5(3x^{2} - 4)}{(x^{3} - 4x)^{6}})x^{7} - \frac{6048*7x^{6}}{(x^{3} - 4x)^{5}} - 1152(\frac{-4(3x^{2} - 4)}{(x^{3} - 4x)^{5}})x^{6} - \frac{1152*6x^{5}}{(x^{3} - 4x)^{4}} + 3840(\frac{-6(3x^{2} - 4)}{(x^{3} - 4x)^{7}})x^{6} + \frac{3840*6x^{5}}{(x^{3} - 4x)^{6}} + 3456(\frac{-5(3x^{2} - 4)}{(x^{3} - 4x)^{6}})x^{5} + \frac{3456*5x^{4}}{(x^{3} - 4x)^{5}} + 1080(\frac{-4(3x^{2} - 4)}{(x^{3} - 4x)^{5}})x^{4} + \frac{1080*4x^{3}}{(x^{3} - 4x)^{4}} + 120(\frac{-3(3x^{2} - 4)}{(x^{3} - 4x)^{4}})x^{3} + \frac{120*3x^{2}}{(x^{3} - 4x)^{3}}\\=&\frac{29160x^{14}}{(x^{3} - 4x)^{7}} - \frac{155520x^{12}}{(x^{3} - 4x)^{7}} - \frac{58320x^{11}}{(x^{3} - 4x)^{6}} + \frac{311040x^{10}}{(x^{3} - 4x)^{7}} + \frac{207360x^{9}}{(x^{3} - 4x)^{6}} - \frac{276480x^{8}}{(x^{3} - 4x)^{7}} + \frac{37152x^{8}}{(x^{3} - 4x)^{5}} - \frac{241920x^{7}}{(x^{3} - 4x)^{6}} - \frac{73728x^{6}}{(x^{3} - 4x)^{5}} - \frac{7992x^{5}}{(x^{3} - 4x)^{4}} + \frac{92160x^{6}}{(x^{3} - 4x)^{7}} + \frac{92160x^{5}}{(x^{3} - 4x)^{6}} + \frac{34560x^{4}}{(x^{3} - 4x)^{5}} + \frac{5760x^{3}}{(x^{3} - 4x)^{4}} + \frac{360x^{2}}{(x^{3} - 4x)^{3}}\\ \end{split}\end{equation} \]

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