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。\[ \begin{equation}\begin{split}【1/1】求函数\frac{({x}^{2} - 2x + 4)}{({x}^{2} + 2x + 4)} 关于 x 的 2 阶导数:\\\end{split}\end{equation} \]
\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{x^{2}}{(x^{2} + 2x + 4)} - \frac{2x}{(x^{2} + 2x + 4)} + \frac{4}{(x^{2} + 2x + 4)}\\&\color{blue}{函数的第 1 阶导数:}\\&\frac{d\left( \frac{x^{2}}{(x^{2} + 2x + 4)} - \frac{2x}{(x^{2} + 2x + 4)} + \frac{4}{(x^{2} + 2x + 4)}\right)}{dx}\\=&(\frac{-(2x + 2 + 0)}{(x^{2} + 2x + 4)^{2}})x^{2} + \frac{2x}{(x^{2} + 2x + 4)} - 2(\frac{-(2x + 2 + 0)}{(x^{2} + 2x + 4)^{2}})x - \frac{2}{(x^{2} + 2x + 4)} + 4(\frac{-(2x + 2 + 0)}{(x^{2} + 2x + 4)^{2}})\\=&\frac{-2x^{3}}{(x^{2} + 2x + 4)^{2}} + \frac{2x^{2}}{(x^{2} + 2x + 4)^{2}} + \frac{2x}{(x^{2} + 2x + 4)} - \frac{4x}{(x^{2} + 2x + 4)^{2}} - \frac{2}{(x^{2} + 2x + 4)} - \frac{8}{(x^{2} + 2x + 4)^{2}}\\\\ &\color{blue}{函数的第 2 阶导数:} \\&\frac{d\left( \frac{-2x^{3}}{(x^{2} + 2x + 4)^{2}} + \frac{2x^{2}}{(x^{2} + 2x + 4)^{2}} + \frac{2x}{(x^{2} + 2x + 4)} - \frac{4x}{(x^{2} + 2x + 4)^{2}} - \frac{2}{(x^{2} + 2x + 4)} - \frac{8}{(x^{2} + 2x + 4)^{2}}\right)}{dx}\\=&-2(\frac{-2(2x + 2 + 0)}{(x^{2} + 2x + 4)^{3}})x^{3} - \frac{2*3x^{2}}{(x^{2} + 2x + 4)^{2}} + 2(\frac{-2(2x + 2 + 0)}{(x^{2} + 2x + 4)^{3}})x^{2} + \frac{2*2x}{(x^{2} + 2x + 4)^{2}} + 2(\frac{-(2x + 2 + 0)}{(x^{2} + 2x + 4)^{2}})x + \frac{2}{(x^{2} + 2x + 4)} - 4(\frac{-2(2x + 2 + 0)}{(x^{2} + 2x + 4)^{3}})x - \frac{4}{(x^{2} + 2x + 4)^{2}} - 2(\frac{-(2x + 2 + 0)}{(x^{2} + 2x + 4)^{2}}) - 8(\frac{-2(2x + 2 + 0)}{(x^{2} + 2x + 4)^{3}})\\=&\frac{8x^{4}}{(x^{2} + 2x + 4)^{3}} - \frac{10x^{2}}{(x^{2} + 2x + 4)^{2}} + \frac{8x^{2}}{(x^{2} + 2x + 4)^{3}} + \frac{4x}{(x^{2} + 2x + 4)^{2}} + \frac{48x}{(x^{2} + 2x + 4)^{3}} + \frac{2}{(x^{2} + 2x + 4)} + \frac{32}{(x^{2} + 2x + 4)^{3}}\\ \end{split}\end{equation} \]你的问题在这里没有得到解决?请到 热门难题 里面看看吧!