本次共计算 1 个题目:每一题对 x 求 1 阶导数。
注意,变量是区分大小写的。\[ \begin{equation}\begin{split}【1/1】求函数({(1 + x)}^{\frac{1}{x}}{(1 + \frac{1}{x})}^{x} - 4){\frac{1}{(x - 1)}}^{2} 关于 x 的 1 阶导数:\\\end{split}\end{equation} \]
\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{(\frac{1}{x} + 1)^{x}(x + 1)^{\frac{1}{x}}}{(x - 1)^{2}} - \frac{4}{(x - 1)^{2}}\\&\color{blue}{函数的第 1 阶导数:}\\&\frac{d\left( \frac{(\frac{1}{x} + 1)^{x}(x + 1)^{\frac{1}{x}}}{(x - 1)^{2}} - \frac{4}{(x - 1)^{2}}\right)}{dx}\\=&(\frac{-2(1 + 0)}{(x - 1)^{3}})(\frac{1}{x} + 1)^{x}(x + 1)^{\frac{1}{x}} + \frac{((\frac{1}{x} + 1)^{x}((1)ln(\frac{1}{x} + 1) + \frac{(x)(\frac{-1}{x^{2}} + 0)}{(\frac{1}{x} + 1)}))(x + 1)^{\frac{1}{x}}}{(x - 1)^{2}} + \frac{(\frac{1}{x} + 1)^{x}((x + 1)^{\frac{1}{x}}((\frac{-1}{x^{2}})ln(x + 1) + \frac{(\frac{1}{x})(1 + 0)}{(x + 1)}))}{(x - 1)^{2}} - 4(\frac{-2(1 + 0)}{(x - 1)^{3}})\\=&\frac{(\frac{1}{x} + 1)^{x}(x + 1)^{\frac{1}{x}}ln(\frac{1}{x} + 1)}{(x - 1)^{2}} - \frac{2(\frac{1}{x} + 1)^{x}(x + 1)^{\frac{1}{x}}}{(x - 1)^{3}} - \frac{(\frac{1}{x} + 1)^{x}(x + 1)^{\frac{1}{x}}}{(x - 1)^{2}(\frac{1}{x} + 1)x} - \frac{(x + 1)^{\frac{1}{x}}(\frac{1}{x} + 1)^{x}ln(x + 1)}{(x - 1)^{2}x^{2}} + \frac{(x + 1)^{\frac{1}{x}}(\frac{1}{x} + 1)^{x}}{(x - 1)^{2}(x + 1)x} + \frac{8}{(x - 1)^{3}}\\ \end{split}\end{equation} \]你的问题在这里没有得到解决?请到 热门难题 里面看看吧!