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