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