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。
\[ \begin{equation}\begin{split}【1/1】求函数ln(\frac{(1 - {x}^{2})}{(nx - {x}^{2})}) 关于 x 的 2 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = ln(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( ln(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})\right)}{dx}\\=&\frac{(-(\frac{-(n - 2x)}{(nx - x^{2})^{2}})x^{2} - \frac{2x}{(nx - x^{2})} + (\frac{-(n - 2x)}{(nx - x^{2})^{2}}))}{(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})}\\=&\frac{nx^{2}}{(nx - x^{2})^{2}(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})} - \frac{2x^{3}}{(nx - x^{2})^{2}(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})} - \frac{2x}{(nx - x^{2})(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})} - \frac{n}{(nx - x^{2})^{2}(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})} + \frac{2x}{(nx - x^{2})^{2}(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{nx^{2}}{(nx - x^{2})^{2}(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})} - \frac{2x^{3}}{(nx - x^{2})^{2}(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})} - \frac{2x}{(nx - x^{2})(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})} - \frac{n}{(nx - x^{2})^{2}(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})} + \frac{2x}{(nx - x^{2})^{2}(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})}\right)}{dx}\\=&\frac{(\frac{-2(n - 2x)}{(nx - x^{2})^{3}})nx^{2}}{(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})} + \frac{(\frac{-(-(\frac{-(n - 2x)}{(nx - x^{2})^{2}})x^{2} - \frac{2x}{(nx - x^{2})} + (\frac{-(n - 2x)}{(nx - x^{2})^{2}}))}{(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})^{2}})nx^{2}}{(nx - x^{2})^{2}} + \frac{n*2x}{(nx - x^{2})^{2}(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})} - \frac{2(\frac{-2(n - 2x)}{(nx - x^{2})^{3}})x^{3}}{(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})} - \frac{2(\frac{-(-(\frac{-(n - 2x)}{(nx - x^{2})^{2}})x^{2} - \frac{2x}{(nx - x^{2})} + (\frac{-(n - 2x)}{(nx - x^{2})^{2}}))}{(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})^{2}})x^{3}}{(nx - x^{2})^{2}} - \frac{2*3x^{2}}{(nx - x^{2})^{2}(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})} - \frac{2(\frac{-(n - 2x)}{(nx - x^{2})^{2}})x}{(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})} - \frac{2(\frac{-(-(\frac{-(n - 2x)}{(nx - x^{2})^{2}})x^{2} - \frac{2x}{(nx - x^{2})} + (\frac{-(n - 2x)}{(nx - x^{2})^{2}}))}{(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})^{2}})x}{(nx - x^{2})} - \frac{2}{(nx - x^{2})(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})} - \frac{(\frac{-2(n - 2x)}{(nx - x^{2})^{3}})n}{(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})} - \frac{(\frac{-(-(\frac{-(n - 2x)}{(nx - x^{2})^{2}})x^{2} - \frac{2x}{(nx - x^{2})} + (\frac{-(n - 2x)}{(nx - x^{2})^{2}}))}{(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})^{2}})n}{(nx - x^{2})^{2}} + 0 + \frac{2(\frac{-2(n - 2x)}{(nx - x^{2})^{3}})x}{(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})} + \frac{2(\frac{-(-(\frac{-(n - 2x)}{(nx - x^{2})^{2}})x^{2} - \frac{2x}{(nx - x^{2})} + (\frac{-(n - 2x)}{(nx - x^{2})^{2}}))}{(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})^{2}})x}{(nx - x^{2})^{2}} + \frac{2}{(nx - x^{2})^{2}(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})}\\=&\frac{-2n^{2}x^{2}}{(nx - x^{2})^{3}(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})} + \frac{8nx^{3}}{(nx - x^{2})^{3}(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})} - \frac{n^{2}x^{4}}{(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})^{2}(nx - x^{2})^{4}} + \frac{4nx^{5}}{(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})^{2}(nx - x^{2})^{4}} + \frac{2nx^{3}}{(nx - x^{2})^{3}(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})^{2}} + \frac{2n^{2}x^{2}}{(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})^{2}(nx - x^{2})^{4}} - \frac{8nx^{3}}{(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})^{2}(nx - x^{2})^{4}} + \frac{2nx}{(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})(nx - x^{2})^{2}} - \frac{8x^{4}}{(nx - x^{2})^{3}(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})} - \frac{4x^{6}}{(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})^{2}(nx - x^{2})^{4}} - \frac{4x^{4}}{(nx - x^{2})^{3}(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})^{2}} + \frac{8x^{4}}{(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})^{2}(nx - x^{2})^{4}} - \frac{6x^{2}}{(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})(nx - x^{2})^{2}} + \frac{2nx}{(nx - x^{2})^{2}(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})} - \frac{4x^{2}}{(nx - x^{2})^{2}(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})} + \frac{2nx^{3}}{(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})^{2}(nx - x^{2})^{3}} - \frac{4x^{4}}{(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})^{2}(nx - x^{2})^{3}} - \frac{4x^{2}}{(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})^{2}(nx - x^{2})^{2}} - \frac{2nx}{(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})^{2}(nx - x^{2})^{3}} + \frac{4x^{2}}{(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})^{2}(nx - x^{2})^{3}} - \frac{8nx}{(nx - x^{2})^{3}(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})} - \frac{2nx}{(nx - x^{2})^{3}(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})^{2}} + \frac{4nx}{(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})^{2}(nx - x^{2})^{4}} - \frac{n^{2}}{(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})^{2}(nx - x^{2})^{4}} + \frac{2n^{2}}{(nx - x^{2})^{3}(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})} + \frac{8x^{2}}{(nx - x^{2})^{3}(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})} + \frac{4x^{2}}{(nx - x^{2})^{3}(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})^{2}} - \frac{4x^{2}}{(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})^{2}(nx - x^{2})^{4}} + \frac{2}{(nx - x^{2})^{2}(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})} - \frac{2}{(nx - x^{2})(\frac{-x^{2}}{(nx - x^{2})} + \frac{1}{(nx - x^{2})})}\\ \end{split}\end{equation} \]

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