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。\[ \begin{equation}\begin{split}【1/1】求函数\frac{(1 + x + {x}^{2})}{(1 - x + {x}^{2})} 关于 x 的 4 阶导数:\\\end{split}\end{equation} \]
\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{x}{(-x + x^{2} + 1)} + \frac{x^{2}}{(-x + x^{2} + 1)} + \frac{1}{(-x + x^{2} + 1)}\\&\color{blue}{函数的第 1 阶导数:}\\&\frac{d\left( \frac{x}{(-x + x^{2} + 1)} + \frac{x^{2}}{(-x + x^{2} + 1)} + \frac{1}{(-x + x^{2} + 1)}\right)}{dx}\\=&(\frac{-(-1 + 2x + 0)}{(-x + x^{2} + 1)^{2}})x + \frac{1}{(-x + x^{2} + 1)} + (\frac{-(-1 + 2x + 0)}{(-x + x^{2} + 1)^{2}})x^{2} + \frac{2x}{(-x + x^{2} + 1)} + (\frac{-(-1 + 2x + 0)}{(-x + x^{2} + 1)^{2}})\\=& - \frac{x^{2}}{(-x + x^{2} + 1)^{2}} - \frac{2x^{3}}{(-x + x^{2} + 1)^{2}} + \frac{2x}{(-x + x^{2} + 1)} - \frac{x}{(-x + x^{2} + 1)^{2}} + \frac{1}{(-x + x^{2} + 1)} + \frac{1}{(-x + x^{2} + 1)^{2}}\\\\ &\color{blue}{函数的第 2 阶导数:} \\&\frac{d\left( - \frac{x^{2}}{(-x + x^{2} + 1)^{2}} - \frac{2x^{3}}{(-x + x^{2} + 1)^{2}} + \frac{2x}{(-x + x^{2} + 1)} - \frac{x}{(-x + x^{2} + 1)^{2}} + \frac{1}{(-x + x^{2} + 1)} + \frac{1}{(-x + x^{2} + 1)^{2}}\right)}{dx}\\=& - (\frac{-2(-1 + 2x + 0)}{(-x + x^{2} + 1)^{3}})x^{2} - \frac{2x}{(-x + x^{2} + 1)^{2}} - 2(\frac{-2(-1 + 2x + 0)}{(-x + x^{2} + 1)^{3}})x^{3} - \frac{2*3x^{2}}{(-x + x^{2} + 1)^{2}} + 2(\frac{-(-1 + 2x + 0)}{(-x + x^{2} + 1)^{2}})x + \frac{2}{(-x + x^{2} + 1)} - (\frac{-2(-1 + 2x + 0)}{(-x + x^{2} + 1)^{3}})x - \frac{1}{(-x + x^{2} + 1)^{2}} + (\frac{-(-1 + 2x + 0)}{(-x + x^{2} + 1)^{2}}) + (\frac{-2(-1 + 2x + 0)}{(-x + x^{2} + 1)^{3}})\\=&\frac{8x^{4}}{(-x + x^{2} + 1)^{3}} - \frac{2x}{(-x + x^{2} + 1)^{2}} + \frac{2x^{2}}{(-x + x^{2} + 1)^{3}} - \frac{10x^{2}}{(-x + x^{2} + 1)^{2}} - \frac{6x}{(-x + x^{2} + 1)^{3}} + \frac{2}{(-x + x^{2} + 1)} + \frac{2}{(-x + x^{2} + 1)^{3}}\\\\ &\color{blue}{函数的第 3 阶导数:} \\&\frac{d\left( \frac{8x^{4}}{(-x + x^{2} + 1)^{3}} - \frac{2x}{(-x + x^{2} + 1)^{2}} + \frac{2x^{2}}{(-x + x^{2} + 1)^{3}} - \frac{10x^{2}}{(-x + x^{2} + 1)^{2}} - \frac{6x}{(-x + x^{2} + 1)^{3}} + \frac{2}{(-x + x^{2} + 1)} + \frac{2}{(-x + x^{2} + 1)^{3}}\right)}{dx}\\=&8(\frac{-3(-1 + 2x + 0)}{(-x + x^{2} + 1)^{4}})x^{4} + \frac{8*4x^{3}}{(-x + x^{2} + 1)^{3}} - 2(\frac{-2(-1 + 2x + 0)}{(-x + x^{2} + 1)^{3}})x - \frac{2}{(-x + x^{2} + 1)^{2}} + 2(\frac{-3(-1 + 2x + 0)}{(-x + x^{2} + 1)^{4}})x^{2} + \frac{2*2x}{(-x + x^{2} + 1)^{3}} - 10(\frac{-2(-1 + 2x + 0)}{(-x + x^{2} + 1)^{3}})x^{2} - \frac{10*2x}{(-x + x^{2} + 1)^{2}} - 6(\frac{-3(-1 + 2x + 0)}{(-x + x^{2} + 1)^{4}})x - \frac{6}{(-x + x^{2} + 1)^{3}} + 2(\frac{-(-1 + 2x + 0)}{(-x + x^{2} + 1)^{2}}) + 2(\frac{-3(-1 + 2x + 0)}{(-x + x^{2} + 1)^{4}})\\=&\frac{-48x^{5}}{(-x + x^{2} + 1)^{4}} - \frac{12x^{3}}{(-x + x^{2} + 1)^{4}} + \frac{72x^{3}}{(-x + x^{2} + 1)^{3}} - \frac{12x^{2}}{(-x + x^{2} + 1)^{3}} + \frac{42x^{2}}{(-x + x^{2} + 1)^{4}} - \frac{24x}{(-x + x^{2} + 1)^{2}} + \frac{24x^{4}}{(-x + x^{2} + 1)^{4}} - \frac{30x}{(-x + x^{2} + 1)^{4}} - \frac{6}{(-x + x^{2} + 1)^{3}} + \frac{6}{(-x + x^{2} + 1)^{4}}\\\\ &\color{blue}{函数的第 4 阶导数:} \\&\frac{d\left( \frac{-48x^{5}}{(-x + x^{2} + 1)^{4}} - \frac{12x^{3}}{(-x + x^{2} + 1)^{4}} + \frac{72x^{3}}{(-x + x^{2} + 1)^{3}} - \frac{12x^{2}}{(-x + x^{2} + 1)^{3}} + \frac{42x^{2}}{(-x + x^{2} + 1)^{4}} - \frac{24x}{(-x + x^{2} + 1)^{2}} + \frac{24x^{4}}{(-x + x^{2} + 1)^{4}} - \frac{30x}{(-x + x^{2} + 1)^{4}} - \frac{6}{(-x + x^{2} + 1)^{3}} + \frac{6}{(-x + x^{2} + 1)^{4}}\right)}{dx}\\=&-48(\frac{-4(-1 + 2x + 0)}{(-x + x^{2} + 1)^{5}})x^{5} - \frac{48*5x^{4}}{(-x + x^{2} + 1)^{4}} - 12(\frac{-4(-1 + 2x + 0)}{(-x + x^{2} + 1)^{5}})x^{3} - \frac{12*3x^{2}}{(-x + x^{2} + 1)^{4}} + 72(\frac{-3(-1 + 2x + 0)}{(-x + x^{2} + 1)^{4}})x^{3} + \frac{72*3x^{2}}{(-x + x^{2} + 1)^{3}} - 12(\frac{-3(-1 + 2x + 0)}{(-x + x^{2} + 1)^{4}})x^{2} - \frac{12*2x}{(-x + x^{2} + 1)^{3}} + 42(\frac{-4(-1 + 2x + 0)}{(-x + x^{2} + 1)^{5}})x^{2} + \frac{42*2x}{(-x + x^{2} + 1)^{4}} - 24(\frac{-2(-1 + 2x + 0)}{(-x + x^{2} + 1)^{3}})x - \frac{24}{(-x + x^{2} + 1)^{2}} + 24(\frac{-4(-1 + 2x + 0)}{(-x + x^{2} + 1)^{5}})x^{4} + \frac{24*4x^{3}}{(-x + x^{2} + 1)^{4}} - 30(\frac{-4(-1 + 2x + 0)}{(-x + x^{2} + 1)^{5}})x - \frac{30}{(-x + x^{2} + 1)^{4}} - 6(\frac{-3(-1 + 2x + 0)}{(-x + x^{2} + 1)^{4}}) + 6(\frac{-4(-1 + 2x + 0)}{(-x + x^{2} + 1)^{5}})\\=&\frac{384x^{6}}{(-x + x^{2} + 1)^{5}} + \frac{192x^{4}}{(-x + x^{2} + 1)^{5}} - \frac{672x^{4}}{(-x + x^{2} + 1)^{4}} - \frac{384x^{3}}{(-x + x^{2} + 1)^{5}} - \frac{384x^{5}}{(-x + x^{2} + 1)^{5}} - \frac{72x^{2}}{(-x + x^{2} + 1)^{4}} + \frac{384x^{3}}{(-x + x^{2} + 1)^{4}} + \frac{312x^{2}}{(-x + x^{2} + 1)^{3}} - \frac{72x}{(-x + x^{2} + 1)^{3}} + \frac{408x^{2}}{(-x + x^{2} + 1)^{5}} + \frac{120x}{(-x + x^{2} + 1)^{4}} - \frac{168x}{(-x + x^{2} + 1)^{5}} - \frac{48}{(-x + x^{2} + 1)^{4}} - \frac{24}{(-x + x^{2} + 1)^{2}} + \frac{24}{(-x + x^{2} + 1)^{5}}\\ \end{split}\end{equation} \]你的问题在这里没有得到解决?请到 热门难题 里面看看吧!