本次共计算 1 个题目：每一题对 x 求 2 阶导数。
    注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数\frac{(1 + (l - 2)x + (1 - 2l){x}^{2} + l{x}^{3})}{(1 + (l - 2)x + (2 - l){x}^{2})} 关于 x 的 2 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{lx}{(lx - 2x + 2x^{2} - lx^{2} + 1)} - \frac{2x}{(lx - 2x + 2x^{2} - lx^{2} + 1)} + \frac{x^{2}}{(lx - 2x + 2x^{2} - lx^{2} + 1)} - \frac{2lx^{2}}{(lx - 2x + 2x^{2} - lx^{2} + 1)} + \frac{lx^{3}}{(lx - 2x + 2x^{2} - lx^{2} + 1)} + \frac{1}{(lx - 2x + 2x^{2} - lx^{2} + 1)}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( \frac{lx}{(lx - 2x + 2x^{2} - lx^{2} + 1)} - \frac{2x}{(lx - 2x + 2x^{2} - lx^{2} + 1)} + \frac{x^{2}}{(lx - 2x + 2x^{2} - lx^{2} + 1)} - \frac{2lx^{2}}{(lx - 2x + 2x^{2} - lx^{2} + 1)} + \frac{lx^{3}}{(lx - 2x + 2x^{2} - lx^{2} + 1)} + \frac{1}{(lx - 2x + 2x^{2} - lx^{2} + 1)}\right)}{dx}\\=&(\frac{-(l - 2 + 2*2x - l*2x + 0)}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}})lx + \frac{l}{(lx - 2x + 2x^{2} - lx^{2} + 1)} - 2(\frac{-(l - 2 + 2*2x - l*2x + 0)}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}})x - \frac{2}{(lx - 2x + 2x^{2} - lx^{2} + 1)} + (\frac{-(l - 2 + 2*2x - l*2x + 0)}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}})x^{2} + \frac{2x}{(lx - 2x + 2x^{2} - lx^{2} + 1)} - 2(\frac{-(l - 2 + 2*2x - l*2x + 0)}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}})lx^{2} - \frac{2l*2x}{(lx - 2x + 2x^{2} - lx^{2} + 1)} + (\frac{-(l - 2 + 2*2x - l*2x + 0)}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}})lx^{3} + \frac{l*3x^{2}}{(lx - 2x + 2x^{2} - lx^{2} + 1)} + (\frac{-(l - 2 + 2*2x - l*2x + 0)}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}})\\=&\frac{4l^{2}x^{2}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} - \frac{13lx^{2}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} - \frac{5l^{2}x^{3}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} - \frac{4lx}{(lx - 2x + 2x^{2} - lx^{2} + 1)} + \frac{12lx^{3}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} + \frac{10x^{2}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} - \frac{4x^{3}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} + \frac{2x}{(lx - 2x + 2x^{2} - lx^{2} + 1)} - \frac{4lx^{4}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} - \frac{8x}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} + \frac{6lx}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} + \frac{3lx^{2}}{(lx - 2x + 2x^{2} - lx^{2} + 1)} + \frac{2l^{2}x^{4}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} - \frac{l^{2}x}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} + \frac{l}{(lx - 2x + 2x^{2} - lx^{2} + 1)} - \frac{l}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} - \frac{2}{(lx - 2x + 2x^{2} - lx^{2} + 1)} + \frac{2}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{4l^{2}x^{2}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} - \frac{13lx^{2}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} - \frac{5l^{2}x^{3}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} - \frac{4lx}{(lx - 2x + 2x^{2} - lx^{2} + 1)} + \frac{12lx^{3}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} + \frac{10x^{2}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} - \frac{4x^{3}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} + \frac{2x}{(lx - 2x + 2x^{2} - lx^{2} + 1)} - \frac{4lx^{4}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} - \frac{8x}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} + \frac{6lx}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} + \frac{3lx^{2}}{(lx - 2x + 2x^{2} - lx^{2} + 1)} + \frac{2l^{2}x^{4}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} - \frac{l^{2}x}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} + \frac{l}{(lx - 2x + 2x^{2} - lx^{2} + 1)} - \frac{l}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} - \frac{2}{(lx - 2x + 2x^{2} - lx^{2} + 1)} + \frac{2}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}}\right)}{dx}\\=&4(\frac{-2(l - 2 + 2*2x - l*2x + 0)}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{3}})l^{2}x^{2} + \frac{4l^{2}*2x}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} - 13(\frac{-2(l - 2 + 2*2x - l*2x + 0)}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{3}})lx^{2} - \frac{13l*2x}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} - 5(\frac{-2(l - 2 + 2*2x - l*2x + 0)}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{3}})l^{2}x^{3} - \frac{5l^{2}*3x^{2}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} - 4(\frac{-(l - 2 + 2*2x - l*2x + 0)}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}})lx - \frac{4l}{(lx - 2x + 2x^{2} - lx^{2} + 1)} + 12(\frac{-2(l - 2 + 2*2x - l*2x + 0)}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{3}})lx^{3} + \frac{12l*3x^{2}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} + 10(\frac{-2(l - 2 + 2*2x - l*2x + 0)}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{3}})x^{2} + \frac{10*2x}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} - 4(\frac{-2(l - 2 + 2*2x - l*2x + 0)}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{3}})x^{3} - \frac{4*3x^{2}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} + 2(\frac{-(l - 2 + 2*2x - l*2x + 0)}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}})x + \frac{2}{(lx - 2x + 2x^{2} - lx^{2} + 1)} - 4(\frac{-2(l - 2 + 2*2x - l*2x + 0)}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{3}})lx^{4} - \frac{4l*4x^{3}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} - 8(\frac{-2(l - 2 + 2*2x - l*2x + 0)}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{3}})x - \frac{8}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} + 6(\frac{-2(l - 2 + 2*2x - l*2x + 0)}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{3}})lx + \frac{6l}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} + 3(\frac{-(l - 2 + 2*2x - l*2x + 0)}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}})lx^{2} + \frac{3l*2x}{(lx - 2x + 2x^{2} - lx^{2} + 1)} + 2(\frac{-2(l - 2 + 2*2x - l*2x + 0)}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{3}})l^{2}x^{4} + \frac{2l^{2}*4x^{3}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} - (\frac{-2(l - 2 + 2*2x - l*2x + 0)}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{3}})l^{2}x - \frac{l^{2}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} + (\frac{-(l - 2 + 2*2x - l*2x + 0)}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}})l + 0 - (\frac{-2(l - 2 + 2*2x - l*2x + 0)}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{3}})l + 0 - 2(\frac{-(l - 2 + 2*2x - l*2x + 0)}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}}) + 2(\frac{-2(l - 2 + 2*2x - l*2x + 0)}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{3}})\\=&\frac{26l^{3}x^{3}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{3}} - \frac{128l^{2}x^{3}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{3}} + \frac{200lx^{3}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{3}} + \frac{14l^{2}x}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} - \frac{24l^{3}x^{4}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{3}} + \frac{104l^{2}x^{4}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{3}} - \frac{128lx^{4}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{3}} - \frac{44lx}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} - \frac{32l^{2}x^{5}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{3}} + \frac{74l^{2}x^{2}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{3}} - \frac{26l^{2}x^{2}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} + \frac{62lx^{2}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} + \frac{14l^{2}x^{3}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} + \frac{6lx}{(lx - 2x + 2x^{2} - lx^{2} + 1)} + \frac{8l^{3}x^{5}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{3}} + \frac{32lx^{5}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{3}} - \frac{12l^{3}x^{2}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{3}} - \frac{152lx^{2}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{3}} - \frac{96x^{3}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{3}} + \frac{32x^{4}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{3}} + \frac{32x}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} + \frac{104x^{2}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{3}} - \frac{20x^{2}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} - \frac{28lx^{3}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} + \frac{56lx}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{3}} - \frac{48x}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{3}} - \frac{20l^{2}x}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{3}} + \frac{2l^{3}x}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{3}} - \frac{4l}{(lx - 2x + 2x^{2} - lx^{2} + 1)} + \frac{10l}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} - \frac{2l^{2}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} + \frac{2l^{2}}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{3}} - \frac{8l}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{3}} - \frac{12}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{2}} + \frac{2}{(lx - 2x + 2x^{2} - lx^{2} + 1)} + \frac{8}{(lx - 2x + 2x^{2} - lx^{2} + 1)^{3}}\\ \end{split}\end{equation} \]

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