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