本次共计算 1 个题目:每一题对 x 求 3 阶导数。
注意,变量是区分大小写的。\[ \begin{equation}\begin{split}【1/1】求函数ln(x + \frac{{(1 + {x}^{2})}^{1}}{2}) 关于 x 的 3 阶导数:\\\end{split}\end{equation} \]
\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = ln(x + \frac{1}{2}x^{2} + \frac{1}{2})\\&\color{blue}{函数的第 1 阶导数:}\\&\frac{d\left( ln(x + \frac{1}{2}x^{2} + \frac{1}{2})\right)}{dx}\\=&\frac{(1 + \frac{1}{2}*2x + 0)}{(x + \frac{1}{2}x^{2} + \frac{1}{2})}\\=&\frac{x}{(x + \frac{1}{2}x^{2} + \frac{1}{2})} + \frac{1}{(x + \frac{1}{2}x^{2} + \frac{1}{2})}\\\\ &\color{blue}{函数的第 2 阶导数:} \\&\frac{d\left( \frac{x}{(x + \frac{1}{2}x^{2} + \frac{1}{2})} + \frac{1}{(x + \frac{1}{2}x^{2} + \frac{1}{2})}\right)}{dx}\\=&(\frac{-(1 + \frac{1}{2}*2x + 0)}{(x + \frac{1}{2}x^{2} + \frac{1}{2})^{2}})x + \frac{1}{(x + \frac{1}{2}x^{2} + \frac{1}{2})} + (\frac{-(1 + \frac{1}{2}*2x + 0)}{(x + \frac{1}{2}x^{2} + \frac{1}{2})^{2}})\\=&\frac{-x^{2}}{(x + \frac{1}{2}x^{2} + \frac{1}{2})^{2}} - \frac{2x}{(x + \frac{1}{2}x^{2} + \frac{1}{2})^{2}} + \frac{1}{(x + \frac{1}{2}x^{2} + \frac{1}{2})} - \frac{1}{(x + \frac{1}{2}x^{2} + \frac{1}{2})^{2}}\\\\ &\color{blue}{函数的第 3 阶导数:} \\&\frac{d\left( \frac{-x^{2}}{(x + \frac{1}{2}x^{2} + \frac{1}{2})^{2}} - \frac{2x}{(x + \frac{1}{2}x^{2} + \frac{1}{2})^{2}} + \frac{1}{(x + \frac{1}{2}x^{2} + \frac{1}{2})} - \frac{1}{(x + \frac{1}{2}x^{2} + \frac{1}{2})^{2}}\right)}{dx}\\=&-(\frac{-2(1 + \frac{1}{2}*2x + 0)}{(x + \frac{1}{2}x^{2} + \frac{1}{2})^{3}})x^{2} - \frac{2x}{(x + \frac{1}{2}x^{2} + \frac{1}{2})^{2}} - 2(\frac{-2(1 + \frac{1}{2}*2x + 0)}{(x + \frac{1}{2}x^{2} + \frac{1}{2})^{3}})x - \frac{2}{(x + \frac{1}{2}x^{2} + \frac{1}{2})^{2}} + (\frac{-(1 + \frac{1}{2}*2x + 0)}{(x + \frac{1}{2}x^{2} + \frac{1}{2})^{2}}) - (\frac{-2(1 + \frac{1}{2}*2x + 0)}{(x + \frac{1}{2}x^{2} + \frac{1}{2})^{3}})\\=&\frac{2x^{3}}{(x + \frac{1}{2}x^{2} + \frac{1}{2})^{3}} + \frac{6x^{2}}{(x + \frac{1}{2}x^{2} + \frac{1}{2})^{3}} - \frac{3x}{(x + \frac{1}{2}x^{2} + \frac{1}{2})^{2}} + \frac{6x}{(x + \frac{1}{2}x^{2} + \frac{1}{2})^{3}} - \frac{3}{(x + \frac{1}{2}x^{2} + \frac{1}{2})^{2}} + \frac{2}{(x + \frac{1}{2}x^{2} + \frac{1}{2})^{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!