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 sin 求 4 阶导数。
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数({sin(x)}^{x}) 关于 sin 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = {sin(x)}^{x}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( {sin(x)}^{x}\right)}{dsin}\\=&({sin(x)}^{x}((0)ln(sin(x)) + \frac{(x)(1)}{(sin(x))}))\\=&\frac{x{sin(x)}^{x}}{sin(x)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{x{sin(x)}^{x}}{sin(x)}\right)}{dsin}\\=&\frac{x*-{sin(x)}^{x}}{sin^{2}} + \frac{x({sin(x)}^{x}((0)ln(sin(x)) + \frac{(x)(1)}{(sin(x))}))}{sin(x)}\\=&\frac{-x{sin(x)}^{x}}{sin^{2}} + \frac{x^{2}{sin(x)}^{x}}{sin(x)^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-x{sin(x)}^{x}}{sin^{2}} + \frac{x^{2}{sin(x)}^{x}}{sin(x)^{2}}\right)}{dsin}\\=&\frac{-x*-2{sin(x)}^{x}}{sin^{3}} - \frac{x({sin(x)}^{x}((0)ln(sin(x)) + \frac{(x)(1)}{(sin(x))}))}{sin^{2}} + \frac{x^{2}*-2{sin(x)}^{x}}{sin^{3}} + \frac{x^{2}({sin(x)}^{x}((0)ln(sin(x)) + \frac{(x)(1)}{(sin(x))}))}{sin(x)^{2}}\\=&\frac{2x{sin(x)}^{x}}{sin^{3}} - \frac{x^{2}{sin(x)}^{x}}{sin(x)sin^{2}} - \frac{2x^{2}{sin(x)}^{x}}{sin^{3}} + \frac{x^{3}{sin(x)}^{x}}{sin(x)^{3}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{2x{sin(x)}^{x}}{sin^{3}} - \frac{x^{2}{sin(x)}^{x}}{sin(x)sin^{2}} - \frac{2x^{2}{sin(x)}^{x}}{sin^{3}} + \frac{x^{3}{sin(x)}^{x}}{sin(x)^{3}}\right)}{dsin}\\=&\frac{2x*-3{sin(x)}^{x}}{sin^{4}} + \frac{2x({sin(x)}^{x}((0)ln(sin(x)) + \frac{(x)(1)}{(sin(x))}))}{sin^{3}} - \frac{x^{2}*-{sin(x)}^{x}}{sin^{2}sin^{2}} - \frac{x^{2}*-2{sin(x)}^{x}}{sin(x)sin^{3}} - \frac{x^{2}({sin(x)}^{x}((0)ln(sin(x)) + \frac{(x)(1)}{(sin(x))}))}{sin(x)sin^{2}} - \frac{2x^{2}*-3{sin(x)}^{x}}{sin^{4}} - \frac{2x^{2}({sin(x)}^{x}((0)ln(sin(x)) + \frac{(x)(1)}{(sin(x))}))}{sin^{3}} + \frac{x^{3}*-3{sin(x)}^{x}}{sin^{4}} + \frac{x^{3}({sin(x)}^{x}((0)ln(sin(x)) + \frac{(x)(1)}{(sin(x))}))}{sin(x)^{3}}\\=&\frac{-6x{sin(x)}^{x}}{sin^{4}} + \frac{4x^{2}{sin(x)}^{x}}{sin(x)sin^{3}} + \frac{7x^{2}{sin(x)}^{x}}{sin^{4}} - \frac{x^{3}{sin(x)}^{x}}{sin(x)^{2}sin^{2}} - \frac{2x^{3}{sin(x)}^{x}}{sin(x)sin^{3}} - \frac{3x^{3}{sin(x)}^{x}}{sin^{4}} + \frac{x^{4}{sin(x)}^{x}}{sin(x)^{4}}\\ \end{split}\end{equation} \]
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数({sin(x)}^{x}) 关于 sin 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = {sin(x)}^{x}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( {sin(x)}^{x}\right)}{dsin}\\=&({sin(x)}^{x}((0)ln(sin(x)) + \frac{(x)(1)}{(sin(x))}))\\=&\frac{x{sin(x)}^{x}}{sin(x)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{x{sin(x)}^{x}}{sin(x)}\right)}{dsin}\\=&\frac{x*-{sin(x)}^{x}}{sin^{2}} + \frac{x({sin(x)}^{x}((0)ln(sin(x)) + \frac{(x)(1)}{(sin(x))}))}{sin(x)}\\=&\frac{-x{sin(x)}^{x}}{sin^{2}} + \frac{x^{2}{sin(x)}^{x}}{sin(x)^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-x{sin(x)}^{x}}{sin^{2}} + \frac{x^{2}{sin(x)}^{x}}{sin(x)^{2}}\right)}{dsin}\\=&\frac{-x*-2{sin(x)}^{x}}{sin^{3}} - \frac{x({sin(x)}^{x}((0)ln(sin(x)) + \frac{(x)(1)}{(sin(x))}))}{sin^{2}} + \frac{x^{2}*-2{sin(x)}^{x}}{sin^{3}} + \frac{x^{2}({sin(x)}^{x}((0)ln(sin(x)) + \frac{(x)(1)}{(sin(x))}))}{sin(x)^{2}}\\=&\frac{2x{sin(x)}^{x}}{sin^{3}} - \frac{x^{2}{sin(x)}^{x}}{sin(x)sin^{2}} - \frac{2x^{2}{sin(x)}^{x}}{sin^{3}} + \frac{x^{3}{sin(x)}^{x}}{sin(x)^{3}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{2x{sin(x)}^{x}}{sin^{3}} - \frac{x^{2}{sin(x)}^{x}}{sin(x)sin^{2}} - \frac{2x^{2}{sin(x)}^{x}}{sin^{3}} + \frac{x^{3}{sin(x)}^{x}}{sin(x)^{3}}\right)}{dsin}\\=&\frac{2x*-3{sin(x)}^{x}}{sin^{4}} + \frac{2x({sin(x)}^{x}((0)ln(sin(x)) + \frac{(x)(1)}{(sin(x))}))}{sin^{3}} - \frac{x^{2}*-{sin(x)}^{x}}{sin^{2}sin^{2}} - \frac{x^{2}*-2{sin(x)}^{x}}{sin(x)sin^{3}} - \frac{x^{2}({sin(x)}^{x}((0)ln(sin(x)) + \frac{(x)(1)}{(sin(x))}))}{sin(x)sin^{2}} - \frac{2x^{2}*-3{sin(x)}^{x}}{sin^{4}} - \frac{2x^{2}({sin(x)}^{x}((0)ln(sin(x)) + \frac{(x)(1)}{(sin(x))}))}{sin^{3}} + \frac{x^{3}*-3{sin(x)}^{x}}{sin^{4}} + \frac{x^{3}({sin(x)}^{x}((0)ln(sin(x)) + \frac{(x)(1)}{(sin(x))}))}{sin(x)^{3}}\\=&\frac{-6x{sin(x)}^{x}}{sin^{4}} + \frac{4x^{2}{sin(x)}^{x}}{sin(x)sin^{3}} + \frac{7x^{2}{sin(x)}^{x}}{sin^{4}} - \frac{x^{3}{sin(x)}^{x}}{sin(x)^{2}sin^{2}} - \frac{2x^{3}{sin(x)}^{x}}{sin(x)sin^{3}} - \frac{3x^{3}{sin(x)}^{x}}{sin^{4}} + \frac{x^{4}{sin(x)}^{x}}{sin(x)^{4}}\\ \end{split}\end{equation} \]
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