求导函数：
输入一个原函数（即需要求导的函数），然后设置需要求导的变量和求导的阶数，点击“下一步”按钮，即可获得该函数相应阶数的导函数。
注意，输入的函数支持数学函数和其它常量。
输入一个原函数（即需要求导的函数），然后设置需要求导的变量和求导的阶数，点击“下一步”按钮，即可获得该函数相应阶数的导函数。
注意，输入的函数支持数学函数和其它常量。
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。
\[ \begin{equation}\begin{split}【1/1】求函数xln(\frac{(x - sin(x))}{sin(x)}) 关于 x 的 3 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = xln(\frac{x}{sin(x)} - 1)\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( xln(\frac{x}{sin(x)} - 1)\right)}{dx}\\=&ln(\frac{x}{sin(x)} - 1) + \frac{x(\frac{1}{sin(x)} + \frac{x*-cos(x)}{sin^{2}(x)} + 0)}{(\frac{x}{sin(x)} - 1)}\\=&ln(\frac{x}{sin(x)} - 1) - \frac{x^{2}cos(x)}{(\frac{x}{sin(x)} - 1)sin^{2}(x)} + \frac{x}{(\frac{x}{sin(x)} - 1)sin(x)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( ln(\frac{x}{sin(x)} - 1) - \frac{x^{2}cos(x)}{(\frac{x}{sin(x)} - 1)sin^{2}(x)} + \frac{x}{(\frac{x}{sin(x)} - 1)sin(x)}\right)}{dx}\\=&\frac{(\frac{1}{sin(x)} + \frac{x*-cos(x)}{sin^{2}(x)} + 0)}{(\frac{x}{sin(x)} - 1)} - \frac{(\frac{-(\frac{1}{sin(x)} + \frac{x*-cos(x)}{sin^{2}(x)} + 0)}{(\frac{x}{sin(x)} - 1)^{2}})x^{2}cos(x)}{sin^{2}(x)} - \frac{2xcos(x)}{(\frac{x}{sin(x)} - 1)sin^{2}(x)} - \frac{x^{2}*-2cos(x)cos(x)}{(\frac{x}{sin(x)} - 1)sin^{3}(x)} - \frac{x^{2}*-sin(x)}{(\frac{x}{sin(x)} - 1)sin^{2}(x)} + \frac{(\frac{-(\frac{1}{sin(x)} + \frac{x*-cos(x)}{sin^{2}(x)} + 0)}{(\frac{x}{sin(x)} - 1)^{2}})x}{sin(x)} + \frac{1}{(\frac{x}{sin(x)} - 1)sin(x)} + \frac{x*-cos(x)}{(\frac{x}{sin(x)} - 1)sin^{2}(x)}\\=&\frac{2}{(\frac{x}{sin(x)} - 1)sin(x)} - \frac{4xcos(x)}{(\frac{x}{sin(x)} - 1)sin^{2}(x)} + \frac{2x^{2}cos(x)}{(\frac{x}{sin(x)} - 1)^{2}sin^{3}(x)} - \frac{x^{3}cos^{2}(x)}{(\frac{x}{sin(x)} - 1)^{2}sin^{4}(x)} + \frac{2x^{2}cos^{2}(x)}{(\frac{x}{sin(x)} - 1)sin^{3}(x)} + \frac{x^{2}}{(\frac{x}{sin(x)} - 1)sin(x)} - \frac{x}{(\frac{x}{sin(x)} - 1)^{2}sin^{2}(x)}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{2}{(\frac{x}{sin(x)} - 1)sin(x)} - \frac{4xcos(x)}{(\frac{x}{sin(x)} - 1)sin^{2}(x)} + \frac{2x^{2}cos(x)}{(\frac{x}{sin(x)} - 1)^{2}sin^{3}(x)} - \frac{x^{3}cos^{2}(x)}{(\frac{x}{sin(x)} - 1)^{2}sin^{4}(x)} + \frac{2x^{2}cos^{2}(x)}{(\frac{x}{sin(x)} - 1)sin^{3}(x)} + \frac{x^{2}}{(\frac{x}{sin(x)} - 1)sin(x)} - \frac{x}{(\frac{x}{sin(x)} - 1)^{2}sin^{2}(x)}\right)}{dx}\\=&\frac{2(\frac{-(\frac{1}{sin(x)} + \frac{x*-cos(x)}{sin^{2}(x)} + 0)}{(\frac{x}{sin(x)} - 1)^{2}})}{sin(x)} + \frac{2*-cos(x)}{(\frac{x}{sin(x)} - 1)sin^{2}(x)} - \frac{4(\frac{-(\frac{1}{sin(x)} + \frac{x*-cos(x)}{sin^{2}(x)} + 0)}{(\frac{x}{sin(x)} - 1)^{2}})xcos(x)}{sin^{2}(x)} - \frac{4cos(x)}{(\frac{x}{sin(x)} - 1)sin^{2}(x)} - \frac{4x*-2cos(x)cos(x)}{(\frac{x}{sin(x)} - 1)sin^{3}(x)} - \frac{4x*-sin(x)}{(\frac{x}{sin(x)} - 1)sin^{2}(x)} + \frac{2(\frac{-2(\frac{1}{sin(x)} + \frac{x*-cos(x)}{sin^{2}(x)} + 0)}{(\frac{x}{sin(x)} - 1)^{3}})x^{2}cos(x)}{sin^{3}(x)} + \frac{2*2xcos(x)}{(\frac{x}{sin(x)} - 1)^{2}sin^{3}(x)} + \frac{2x^{2}*-3cos(x)cos(x)}{(\frac{x}{sin(x)} - 1)^{2}sin^{4}(x)} + \frac{2x^{2}*-sin(x)}{(\frac{x}{sin(x)} - 1)^{2}sin^{3}(x)} - \frac{(\frac{-2(\frac{1}{sin(x)} + \frac{x*-cos(x)}{sin^{2}(x)} + 0)}{(\frac{x}{sin(x)} - 1)^{3}})x^{3}cos^{2}(x)}{sin^{4}(x)} - \frac{3x^{2}cos^{2}(x)}{(\frac{x}{sin(x)} - 1)^{2}sin^{4}(x)} - \frac{x^{3}*-4cos(x)cos^{2}(x)}{(\frac{x}{sin(x)} - 1)^{2}sin^{5}(x)} - \frac{x^{3}*-2cos(x)sin(x)}{(\frac{x}{sin(x)} - 1)^{2}sin^{4}(x)} + \frac{2(\frac{-(\frac{1}{sin(x)} + \frac{x*-cos(x)}{sin^{2}(x)} + 0)}{(\frac{x}{sin(x)} - 1)^{2}})x^{2}cos^{2}(x)}{sin^{3}(x)} + \frac{2*2xcos^{2}(x)}{(\frac{x}{sin(x)} - 1)sin^{3}(x)} + \frac{2x^{2}*-3cos(x)cos^{2}(x)}{(\frac{x}{sin(x)} - 1)sin^{4}(x)} + \frac{2x^{2}*-2cos(x)sin(x)}{(\frac{x}{sin(x)} - 1)sin^{3}(x)} + \frac{(\frac{-(\frac{1}{sin(x)} + \frac{x*-cos(x)}{sin^{2}(x)} + 0)}{(\frac{x}{sin(x)} - 1)^{2}})x^{2}}{sin(x)} + \frac{2x}{(\frac{x}{sin(x)} - 1)sin(x)} + \frac{x^{2}*-cos(x)}{(\frac{x}{sin(x)} - 1)sin^{2}(x)} - \frac{(\frac{-2(\frac{1}{sin(x)} + \frac{x*-cos(x)}{sin^{2}(x)} + 0)}{(\frac{x}{sin(x)} - 1)^{3}})x}{sin^{2}(x)} - \frac{1}{(\frac{x}{sin(x)} - 1)^{2}sin^{2}(x)} - \frac{x*-2cos(x)}{(\frac{x}{sin(x)} - 1)^{2}sin^{3}(x)}\\=&\frac{-6cos(x)}{(\frac{x}{sin(x)} - 1)sin^{2}(x)} + \frac{12xcos(x)}{(\frac{x}{sin(x)} - 1)^{2}sin^{3}(x)} - \frac{3}{(\frac{x}{sin(x)} - 1)^{2}sin^{2}(x)} - \frac{15x^{2}cos^{2}(x)}{(\frac{x}{sin(x)} - 1)^{2}sin^{4}(x)} + \frac{12xcos^{2}(x)}{(\frac{x}{sin(x)} - 1)sin^{3}(x)} + \frac{3x^{3}cos(x)}{(\frac{x}{sin(x)} - 1)^{2}sin^{3}(x)} - \frac{6x^{2}cos(x)}{(\frac{x}{sin(x)} - 1)^{3}sin^{4}(x)} + \frac{6x^{3}cos^{2}(x)}{(\frac{x}{sin(x)} - 1)^{3}sin^{5}(x)} - \frac{5x^{2}cos(x)}{(\frac{x}{sin(x)} - 1)sin^{2}(x)} - \frac{2x^{4}cos^{3}(x)}{(\frac{x}{sin(x)} - 1)^{3}sin^{6}(x)} + \frac{6x^{3}cos^{3}(x)}{(\frac{x}{sin(x)} - 1)^{2}sin^{5}(x)} - \frac{6x^{2}cos^{3}(x)}{(\frac{x}{sin(x)} - 1)sin^{4}(x)} + \frac{6x}{(\frac{x}{sin(x)} - 1)sin(x)} - \frac{3x^{2}}{(\frac{x}{sin(x)} - 1)^{2}sin^{2}(x)} + \frac{2x}{(\frac{x}{sin(x)} - 1)^{3}sin^{3}(x)}\\ \end{split}\end{equation} \]
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数xln(\frac{(x - sin(x))}{sin(x)}) 关于 x 的 3 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = xln(\frac{x}{sin(x)} - 1)\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( xln(\frac{x}{sin(x)} - 1)\right)}{dx}\\=&ln(\frac{x}{sin(x)} - 1) + \frac{x(\frac{1}{sin(x)} + \frac{x*-cos(x)}{sin^{2}(x)} + 0)}{(\frac{x}{sin(x)} - 1)}\\=&ln(\frac{x}{sin(x)} - 1) - \frac{x^{2}cos(x)}{(\frac{x}{sin(x)} - 1)sin^{2}(x)} + \frac{x}{(\frac{x}{sin(x)} - 1)sin(x)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( ln(\frac{x}{sin(x)} - 1) - \frac{x^{2}cos(x)}{(\frac{x}{sin(x)} - 1)sin^{2}(x)} + \frac{x}{(\frac{x}{sin(x)} - 1)sin(x)}\right)}{dx}\\=&\frac{(\frac{1}{sin(x)} + \frac{x*-cos(x)}{sin^{2}(x)} + 0)}{(\frac{x}{sin(x)} - 1)} - \frac{(\frac{-(\frac{1}{sin(x)} + \frac{x*-cos(x)}{sin^{2}(x)} + 0)}{(\frac{x}{sin(x)} - 1)^{2}})x^{2}cos(x)}{sin^{2}(x)} - \frac{2xcos(x)}{(\frac{x}{sin(x)} - 1)sin^{2}(x)} - \frac{x^{2}*-2cos(x)cos(x)}{(\frac{x}{sin(x)} - 1)sin^{3}(x)} - \frac{x^{2}*-sin(x)}{(\frac{x}{sin(x)} - 1)sin^{2}(x)} + \frac{(\frac{-(\frac{1}{sin(x)} + \frac{x*-cos(x)}{sin^{2}(x)} + 0)}{(\frac{x}{sin(x)} - 1)^{2}})x}{sin(x)} + \frac{1}{(\frac{x}{sin(x)} - 1)sin(x)} + \frac{x*-cos(x)}{(\frac{x}{sin(x)} - 1)sin^{2}(x)}\\=&\frac{2}{(\frac{x}{sin(x)} - 1)sin(x)} - \frac{4xcos(x)}{(\frac{x}{sin(x)} - 1)sin^{2}(x)} + \frac{2x^{2}cos(x)}{(\frac{x}{sin(x)} - 1)^{2}sin^{3}(x)} - \frac{x^{3}cos^{2}(x)}{(\frac{x}{sin(x)} - 1)^{2}sin^{4}(x)} + \frac{2x^{2}cos^{2}(x)}{(\frac{x}{sin(x)} - 1)sin^{3}(x)} + \frac{x^{2}}{(\frac{x}{sin(x)} - 1)sin(x)} - \frac{x}{(\frac{x}{sin(x)} - 1)^{2}sin^{2}(x)}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{2}{(\frac{x}{sin(x)} - 1)sin(x)} - \frac{4xcos(x)}{(\frac{x}{sin(x)} - 1)sin^{2}(x)} + \frac{2x^{2}cos(x)}{(\frac{x}{sin(x)} - 1)^{2}sin^{3}(x)} - \frac{x^{3}cos^{2}(x)}{(\frac{x}{sin(x)} - 1)^{2}sin^{4}(x)} + \frac{2x^{2}cos^{2}(x)}{(\frac{x}{sin(x)} - 1)sin^{3}(x)} + \frac{x^{2}}{(\frac{x}{sin(x)} - 1)sin(x)} - \frac{x}{(\frac{x}{sin(x)} - 1)^{2}sin^{2}(x)}\right)}{dx}\\=&\frac{2(\frac{-(\frac{1}{sin(x)} + \frac{x*-cos(x)}{sin^{2}(x)} + 0)}{(\frac{x}{sin(x)} - 1)^{2}})}{sin(x)} + \frac{2*-cos(x)}{(\frac{x}{sin(x)} - 1)sin^{2}(x)} - \frac{4(\frac{-(\frac{1}{sin(x)} + \frac{x*-cos(x)}{sin^{2}(x)} + 0)}{(\frac{x}{sin(x)} - 1)^{2}})xcos(x)}{sin^{2}(x)} - \frac{4cos(x)}{(\frac{x}{sin(x)} - 1)sin^{2}(x)} - \frac{4x*-2cos(x)cos(x)}{(\frac{x}{sin(x)} - 1)sin^{3}(x)} - \frac{4x*-sin(x)}{(\frac{x}{sin(x)} - 1)sin^{2}(x)} + \frac{2(\frac{-2(\frac{1}{sin(x)} + \frac{x*-cos(x)}{sin^{2}(x)} + 0)}{(\frac{x}{sin(x)} - 1)^{3}})x^{2}cos(x)}{sin^{3}(x)} + \frac{2*2xcos(x)}{(\frac{x}{sin(x)} - 1)^{2}sin^{3}(x)} + \frac{2x^{2}*-3cos(x)cos(x)}{(\frac{x}{sin(x)} - 1)^{2}sin^{4}(x)} + \frac{2x^{2}*-sin(x)}{(\frac{x}{sin(x)} - 1)^{2}sin^{3}(x)} - \frac{(\frac{-2(\frac{1}{sin(x)} + \frac{x*-cos(x)}{sin^{2}(x)} + 0)}{(\frac{x}{sin(x)} - 1)^{3}})x^{3}cos^{2}(x)}{sin^{4}(x)} - \frac{3x^{2}cos^{2}(x)}{(\frac{x}{sin(x)} - 1)^{2}sin^{4}(x)} - \frac{x^{3}*-4cos(x)cos^{2}(x)}{(\frac{x}{sin(x)} - 1)^{2}sin^{5}(x)} - \frac{x^{3}*-2cos(x)sin(x)}{(\frac{x}{sin(x)} - 1)^{2}sin^{4}(x)} + \frac{2(\frac{-(\frac{1}{sin(x)} + \frac{x*-cos(x)}{sin^{2}(x)} + 0)}{(\frac{x}{sin(x)} - 1)^{2}})x^{2}cos^{2}(x)}{sin^{3}(x)} + \frac{2*2xcos^{2}(x)}{(\frac{x}{sin(x)} - 1)sin^{3}(x)} + \frac{2x^{2}*-3cos(x)cos^{2}(x)}{(\frac{x}{sin(x)} - 1)sin^{4}(x)} + \frac{2x^{2}*-2cos(x)sin(x)}{(\frac{x}{sin(x)} - 1)sin^{3}(x)} + \frac{(\frac{-(\frac{1}{sin(x)} + \frac{x*-cos(x)}{sin^{2}(x)} + 0)}{(\frac{x}{sin(x)} - 1)^{2}})x^{2}}{sin(x)} + \frac{2x}{(\frac{x}{sin(x)} - 1)sin(x)} + \frac{x^{2}*-cos(x)}{(\frac{x}{sin(x)} - 1)sin^{2}(x)} - \frac{(\frac{-2(\frac{1}{sin(x)} + \frac{x*-cos(x)}{sin^{2}(x)} + 0)}{(\frac{x}{sin(x)} - 1)^{3}})x}{sin^{2}(x)} - \frac{1}{(\frac{x}{sin(x)} - 1)^{2}sin^{2}(x)} - \frac{x*-2cos(x)}{(\frac{x}{sin(x)} - 1)^{2}sin^{3}(x)}\\=&\frac{-6cos(x)}{(\frac{x}{sin(x)} - 1)sin^{2}(x)} + \frac{12xcos(x)}{(\frac{x}{sin(x)} - 1)^{2}sin^{3}(x)} - \frac{3}{(\frac{x}{sin(x)} - 1)^{2}sin^{2}(x)} - \frac{15x^{2}cos^{2}(x)}{(\frac{x}{sin(x)} - 1)^{2}sin^{4}(x)} + \frac{12xcos^{2}(x)}{(\frac{x}{sin(x)} - 1)sin^{3}(x)} + \frac{3x^{3}cos(x)}{(\frac{x}{sin(x)} - 1)^{2}sin^{3}(x)} - \frac{6x^{2}cos(x)}{(\frac{x}{sin(x)} - 1)^{3}sin^{4}(x)} + \frac{6x^{3}cos^{2}(x)}{(\frac{x}{sin(x)} - 1)^{3}sin^{5}(x)} - \frac{5x^{2}cos(x)}{(\frac{x}{sin(x)} - 1)sin^{2}(x)} - \frac{2x^{4}cos^{3}(x)}{(\frac{x}{sin(x)} - 1)^{3}sin^{6}(x)} + \frac{6x^{3}cos^{3}(x)}{(\frac{x}{sin(x)} - 1)^{2}sin^{5}(x)} - \frac{6x^{2}cos^{3}(x)}{(\frac{x}{sin(x)} - 1)sin^{4}(x)} + \frac{6x}{(\frac{x}{sin(x)} - 1)sin(x)} - \frac{3x^{2}}{(\frac{x}{sin(x)} - 1)^{2}sin^{2}(x)} + \frac{2x}{(\frac{x}{sin(x)} - 1)^{3}sin^{3}(x)}\\ \end{split}\end{equation} \]
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