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 4 阶导数。
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数{tan(x)}^{cot(x)} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( {tan(x)}^{cot(x)}\right)}{dx}\\=&({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))\\=&-{tan(x)}^{cot(x)}ln(tan(x))csc^{2}(x) + \frac{{tan(x)}^{cot(x)}cot(x)sec^{2}(x)}{tan(x)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( -{tan(x)}^{cot(x)}ln(tan(x))csc^{2}(x) + \frac{{tan(x)}^{cot(x)}cot(x)sec^{2}(x)}{tan(x)}\right)}{dx}\\=&-({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))ln(tan(x))csc^{2}(x) - \frac{{tan(x)}^{cot(x)}sec^{2}(x)(1)csc^{2}(x)}{(tan(x))} - {tan(x)}^{cot(x)}ln(tan(x))*-2csc^{2}(x)cot(x) + \frac{({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))cot(x)sec^{2}(x)}{tan(x)} + \frac{{tan(x)}^{cot(x)}*-sec^{2}(x)(1)cot(x)sec^{2}(x)}{tan^{2}(x)} + \frac{{tan(x)}^{cot(x)}*-csc^{2}(x)sec^{2}(x)}{tan(x)} + \frac{{tan(x)}^{cot(x)}cot(x)*2sec^{2}(x)tan(x)}{tan(x)}\\=&{tan(x)}^{cot(x)}ln^{2}(tan(x))csc^{4}(x) - \frac{2{tan(x)}^{cot(x)}ln(tan(x))cot(x)sec^{2}(x)csc^{2}(x)}{tan(x)} - \frac{2{tan(x)}^{cot(x)}sec^{2}(x)csc^{2}(x)}{tan(x)} + 2{tan(x)}^{cot(x)}ln(tan(x))cot(x)csc^{2}(x) + \frac{{tan(x)}^{cot(x)}cot^{2}(x)sec^{4}(x)}{tan^{2}(x)} - \frac{{tan(x)}^{cot(x)}cot(x)sec^{4}(x)}{tan^{2}(x)} + 2{tan(x)}^{cot(x)}cot(x)sec^{2}(x)\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( {tan(x)}^{cot(x)}ln^{2}(tan(x))csc^{4}(x) - \frac{2{tan(x)}^{cot(x)}ln(tan(x))cot(x)sec^{2}(x)csc^{2}(x)}{tan(x)} - \frac{2{tan(x)}^{cot(x)}sec^{2}(x)csc^{2}(x)}{tan(x)} + 2{tan(x)}^{cot(x)}ln(tan(x))cot(x)csc^{2}(x) + \frac{{tan(x)}^{cot(x)}cot^{2}(x)sec^{4}(x)}{tan^{2}(x)} - \frac{{tan(x)}^{cot(x)}cot(x)sec^{4}(x)}{tan^{2}(x)} + 2{tan(x)}^{cot(x)}cot(x)sec^{2}(x)\right)}{dx}\\=&({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))ln^{2}(tan(x))csc^{4}(x) + \frac{{tan(x)}^{cot(x)}*2ln(tan(x))sec^{2}(x)(1)csc^{4}(x)}{(tan(x))} + {tan(x)}^{cot(x)}ln^{2}(tan(x))*-4csc^{4}(x)cot(x) - \frac{2({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))ln(tan(x))cot(x)sec^{2}(x)csc^{2}(x)}{tan(x)} - \frac{2{tan(x)}^{cot(x)}sec^{2}(x)(1)cot(x)sec^{2}(x)csc^{2}(x)}{(tan(x))tan(x)} - \frac{2{tan(x)}^{cot(x)}ln(tan(x))*-sec^{2}(x)(1)cot(x)sec^{2}(x)csc^{2}(x)}{tan^{2}(x)} - \frac{2{tan(x)}^{cot(x)}ln(tan(x))*-csc^{2}(x)sec^{2}(x)csc^{2}(x)}{tan(x)} - \frac{2{tan(x)}^{cot(x)}ln(tan(x))cot(x)*2sec^{2}(x)tan(x)csc^{2}(x)}{tan(x)} - \frac{2{tan(x)}^{cot(x)}ln(tan(x))cot(x)sec^{2}(x)*-2csc^{2}(x)cot(x)}{tan(x)} - \frac{2({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))sec^{2}(x)csc^{2}(x)}{tan(x)} - \frac{2{tan(x)}^{cot(x)}*-sec^{2}(x)(1)sec^{2}(x)csc^{2}(x)}{tan^{2}(x)} - \frac{2{tan(x)}^{cot(x)}*2sec^{2}(x)tan(x)csc^{2}(x)}{tan(x)} - \frac{2{tan(x)}^{cot(x)}sec^{2}(x)*-2csc^{2}(x)cot(x)}{tan(x)} + 2({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))ln(tan(x))cot(x)csc^{2}(x) + \frac{2{tan(x)}^{cot(x)}sec^{2}(x)(1)cot(x)csc^{2}(x)}{(tan(x))} + 2{tan(x)}^{cot(x)}ln(tan(x))*-csc^{2}(x)csc^{2}(x) + 2{tan(x)}^{cot(x)}ln(tan(x))cot(x)*-2csc^{2}(x)cot(x) + \frac{({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))cot^{2}(x)sec^{4}(x)}{tan^{2}(x)} + \frac{{tan(x)}^{cot(x)}*-2sec^{2}(x)(1)cot^{2}(x)sec^{4}(x)}{tan^{3}(x)} + \frac{{tan(x)}^{cot(x)}*-2cot(x)csc^{2}(x)sec^{4}(x)}{tan^{2}(x)} + \frac{{tan(x)}^{cot(x)}cot^{2}(x)*4sec^{4}(x)tan(x)}{tan^{2}(x)} - \frac{({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))cot(x)sec^{4}(x)}{tan^{2}(x)} - \frac{{tan(x)}^{cot(x)}*-2sec^{2}(x)(1)cot(x)sec^{4}(x)}{tan^{3}(x)} - \frac{{tan(x)}^{cot(x)}*-csc^{2}(x)sec^{4}(x)}{tan^{2}(x)} - \frac{{tan(x)}^{cot(x)}cot(x)*4sec^{4}(x)tan(x)}{tan^{2}(x)} + 2({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))cot(x)sec^{2}(x) + 2{tan(x)}^{cot(x)}*-csc^{2}(x)sec^{2}(x) + 2{tan(x)}^{cot(x)}cot(x)*2sec^{2}(x)tan(x)\\=&-{tan(x)}^{cot(x)}ln^{3}(tan(x))csc^{6}(x) + \frac{3{tan(x)}^{cot(x)}ln^{2}(tan(x))cot(x)sec^{2}(x)csc^{4}(x)}{tan(x)} + \frac{6{tan(x)}^{cot(x)}ln(tan(x))sec^{2}(x)csc^{4}(x)}{tan(x)} - 6{tan(x)}^{cot(x)}ln^{2}(tan(x))cot(x)csc^{4}(x) - \frac{3{tan(x)}^{cot(x)}ln(tan(x))cot^{2}(x)sec^{4}(x)csc^{2}(x)}{tan^{2}(x)} - \frac{6{tan(x)}^{cot(x)}cot(x)sec^{4}(x)csc^{2}(x)}{tan^{2}(x)} + \frac{3{tan(x)}^{cot(x)}ln(tan(x))cot(x)sec^{4}(x)csc^{2}(x)}{tan^{2}(x)} - 6{tan(x)}^{cot(x)}ln(tan(x))cot(x)sec^{2}(x)csc^{2}(x) + \frac{6{tan(x)}^{cot(x)}ln(tan(x))cot^{2}(x)sec^{2}(x)csc^{2}(x)}{tan(x)} + \frac{3{tan(x)}^{cot(x)}sec^{4}(x)csc^{2}(x)}{tan^{2}(x)} - 6{tan(x)}^{cot(x)}sec^{2}(x)csc^{2}(x) + \frac{6{tan(x)}^{cot(x)}cot(x)sec^{2}(x)csc^{2}(x)}{tan(x)} - 2{tan(x)}^{cot(x)}ln(tan(x))csc^{4}(x) - 4{tan(x)}^{cot(x)}ln(tan(x))cot^{2}(x)csc^{2}(x) + \frac{{tan(x)}^{cot(x)}cot^{3}(x)sec^{6}(x)}{tan^{3}(x)} - \frac{3{tan(x)}^{cot(x)}cot^{2}(x)sec^{6}(x)}{tan^{3}(x)} + \frac{6{tan(x)}^{cot(x)}cot^{2}(x)sec^{4}(x)}{tan(x)} + \frac{2{tan(x)}^{cot(x)}cot(x)sec^{6}(x)}{tan^{3}(x)} - \frac{4{tan(x)}^{cot(x)}cot(x)sec^{4}(x)}{tan(x)} + 4{tan(x)}^{cot(x)}tan(x)cot(x)sec^{2}(x)\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( -{tan(x)}^{cot(x)}ln^{3}(tan(x))csc^{6}(x) + \frac{3{tan(x)}^{cot(x)}ln^{2}(tan(x))cot(x)sec^{2}(x)csc^{4}(x)}{tan(x)} + \frac{6{tan(x)}^{cot(x)}ln(tan(x))sec^{2}(x)csc^{4}(x)}{tan(x)} - 6{tan(x)}^{cot(x)}ln^{2}(tan(x))cot(x)csc^{4}(x) - \frac{3{tan(x)}^{cot(x)}ln(tan(x))cot^{2}(x)sec^{4}(x)csc^{2}(x)}{tan^{2}(x)} - \frac{6{tan(x)}^{cot(x)}cot(x)sec^{4}(x)csc^{2}(x)}{tan^{2}(x)} + \frac{3{tan(x)}^{cot(x)}ln(tan(x))cot(x)sec^{4}(x)csc^{2}(x)}{tan^{2}(x)} - 6{tan(x)}^{cot(x)}ln(tan(x))cot(x)sec^{2}(x)csc^{2}(x) + \frac{6{tan(x)}^{cot(x)}ln(tan(x))cot^{2}(x)sec^{2}(x)csc^{2}(x)}{tan(x)} + \frac{3{tan(x)}^{cot(x)}sec^{4}(x)csc^{2}(x)}{tan^{2}(x)} - 6{tan(x)}^{cot(x)}sec^{2}(x)csc^{2}(x) + \frac{6{tan(x)}^{cot(x)}cot(x)sec^{2}(x)csc^{2}(x)}{tan(x)} - 2{tan(x)}^{cot(x)}ln(tan(x))csc^{4}(x) - 4{tan(x)}^{cot(x)}ln(tan(x))cot^{2}(x)csc^{2}(x) + \frac{{tan(x)}^{cot(x)}cot^{3}(x)sec^{6}(x)}{tan^{3}(x)} - \frac{3{tan(x)}^{cot(x)}cot^{2}(x)sec^{6}(x)}{tan^{3}(x)} + \frac{6{tan(x)}^{cot(x)}cot^{2}(x)sec^{4}(x)}{tan(x)} + \frac{2{tan(x)}^{cot(x)}cot(x)sec^{6}(x)}{tan^{3}(x)} - \frac{4{tan(x)}^{cot(x)}cot(x)sec^{4}(x)}{tan(x)} + 4{tan(x)}^{cot(x)}tan(x)cot(x)sec^{2}(x)\right)}{dx}\\=&-({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))ln^{3}(tan(x))csc^{6}(x) - \frac{{tan(x)}^{cot(x)}*3ln^{2}(tan(x))sec^{2}(x)(1)csc^{6}(x)}{(tan(x))} - {tan(x)}^{cot(x)}ln^{3}(tan(x))*-6csc^{6}(x)cot(x) + \frac{3({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))ln^{2}(tan(x))cot(x)sec^{2}(x)csc^{4}(x)}{tan(x)} + \frac{3{tan(x)}^{cot(x)}*2ln(tan(x))sec^{2}(x)(1)cot(x)sec^{2}(x)csc^{4}(x)}{(tan(x))tan(x)} + \frac{3{tan(x)}^{cot(x)}ln^{2}(tan(x))*-sec^{2}(x)(1)cot(x)sec^{2}(x)csc^{4}(x)}{tan^{2}(x)} + \frac{3{tan(x)}^{cot(x)}ln^{2}(tan(x))*-csc^{2}(x)sec^{2}(x)csc^{4}(x)}{tan(x)} + \frac{3{tan(x)}^{cot(x)}ln^{2}(tan(x))cot(x)*2sec^{2}(x)tan(x)csc^{4}(x)}{tan(x)} + \frac{3{tan(x)}^{cot(x)}ln^{2}(tan(x))cot(x)sec^{2}(x)*-4csc^{4}(x)cot(x)}{tan(x)} + \frac{6({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))ln(tan(x))sec^{2}(x)csc^{4}(x)}{tan(x)} + \frac{6{tan(x)}^{cot(x)}sec^{2}(x)(1)sec^{2}(x)csc^{4}(x)}{(tan(x))tan(x)} + \frac{6{tan(x)}^{cot(x)}ln(tan(x))*-sec^{2}(x)(1)sec^{2}(x)csc^{4}(x)}{tan^{2}(x)} + \frac{6{tan(x)}^{cot(x)}ln(tan(x))*2sec^{2}(x)tan(x)csc^{4}(x)}{tan(x)} + \frac{6{tan(x)}^{cot(x)}ln(tan(x))sec^{2}(x)*-4csc^{4}(x)cot(x)}{tan(x)} - 6({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))ln^{2}(tan(x))cot(x)csc^{4}(x) - \frac{6{tan(x)}^{cot(x)}*2ln(tan(x))sec^{2}(x)(1)cot(x)csc^{4}(x)}{(tan(x))} - 6{tan(x)}^{cot(x)}ln^{2}(tan(x))*-csc^{2}(x)csc^{4}(x) - 6{tan(x)}^{cot(x)}ln^{2}(tan(x))cot(x)*-4csc^{4}(x)cot(x) - \frac{3({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))ln(tan(x))cot^{2}(x)sec^{4}(x)csc^{2}(x)}{tan^{2}(x)} - \frac{3{tan(x)}^{cot(x)}sec^{2}(x)(1)cot^{2}(x)sec^{4}(x)csc^{2}(x)}{(tan(x))tan^{2}(x)} - \frac{3{tan(x)}^{cot(x)}ln(tan(x))*-2sec^{2}(x)(1)cot^{2}(x)sec^{4}(x)csc^{2}(x)}{tan^{3}(x)} - \frac{3{tan(x)}^{cot(x)}ln(tan(x))*-2cot(x)csc^{2}(x)sec^{4}(x)csc^{2}(x)}{tan^{2}(x)} - \frac{3{tan(x)}^{cot(x)}ln(tan(x))cot^{2}(x)*4sec^{4}(x)tan(x)csc^{2}(x)}{tan^{2}(x)} - \frac{3{tan(x)}^{cot(x)}ln(tan(x))cot^{2}(x)sec^{4}(x)*-2csc^{2}(x)cot(x)}{tan^{2}(x)} - \frac{6({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))cot(x)sec^{4}(x)csc^{2}(x)}{tan^{2}(x)} - \frac{6{tan(x)}^{cot(x)}*-2sec^{2}(x)(1)cot(x)sec^{4}(x)csc^{2}(x)}{tan^{3}(x)} - \frac{6{tan(x)}^{cot(x)}*-csc^{2}(x)sec^{4}(x)csc^{2}(x)}{tan^{2}(x)} - \frac{6{tan(x)}^{cot(x)}cot(x)*4sec^{4}(x)tan(x)csc^{2}(x)}{tan^{2}(x)} - \frac{6{tan(x)}^{cot(x)}cot(x)sec^{4}(x)*-2csc^{2}(x)cot(x)}{tan^{2}(x)} + \frac{3({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))ln(tan(x))cot(x)sec^{4}(x)csc^{2}(x)}{tan^{2}(x)} + \frac{3{tan(x)}^{cot(x)}sec^{2}(x)(1)cot(x)sec^{4}(x)csc^{2}(x)}{(tan(x))tan^{2}(x)} + \frac{3{tan(x)}^{cot(x)}ln(tan(x))*-2sec^{2}(x)(1)cot(x)sec^{4}(x)csc^{2}(x)}{tan^{3}(x)} + \frac{3{tan(x)}^{cot(x)}ln(tan(x))*-csc^{2}(x)sec^{4}(x)csc^{2}(x)}{tan^{2}(x)} + \frac{3{tan(x)}^{cot(x)}ln(tan(x))cot(x)*4sec^{4}(x)tan(x)csc^{2}(x)}{tan^{2}(x)} + \frac{3{tan(x)}^{cot(x)}ln(tan(x))cot(x)sec^{4}(x)*-2csc^{2}(x)cot(x)}{tan^{2}(x)} - 6({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))ln(tan(x))cot(x)sec^{2}(x)csc^{2}(x) - \frac{6{tan(x)}^{cot(x)}sec^{2}(x)(1)cot(x)sec^{2}(x)csc^{2}(x)}{(tan(x))} - 6{tan(x)}^{cot(x)}ln(tan(x))*-csc^{2}(x)sec^{2}(x)csc^{2}(x) - 6{tan(x)}^{cot(x)}ln(tan(x))cot(x)*2sec^{2}(x)tan(x)csc^{2}(x) - 6{tan(x)}^{cot(x)}ln(tan(x))cot(x)sec^{2}(x)*-2csc^{2}(x)cot(x) + \frac{6({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))ln(tan(x))cot^{2}(x)sec^{2}(x)csc^{2}(x)}{tan(x)} + \frac{6{tan(x)}^{cot(x)}sec^{2}(x)(1)cot^{2}(x)sec^{2}(x)csc^{2}(x)}{(tan(x))tan(x)} + \frac{6{tan(x)}^{cot(x)}ln(tan(x))*-sec^{2}(x)(1)cot^{2}(x)sec^{2}(x)csc^{2}(x)}{tan^{2}(x)} + \frac{6{tan(x)}^{cot(x)}ln(tan(x))*-2cot(x)csc^{2}(x)sec^{2}(x)csc^{2}(x)}{tan(x)} + \frac{6{tan(x)}^{cot(x)}ln(tan(x))cot^{2}(x)*2sec^{2}(x)tan(x)csc^{2}(x)}{tan(x)} + \frac{6{tan(x)}^{cot(x)}ln(tan(x))cot^{2}(x)sec^{2}(x)*-2csc^{2}(x)cot(x)}{tan(x)} + \frac{3({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))sec^{4}(x)csc^{2}(x)}{tan^{2}(x)} + \frac{3{tan(x)}^{cot(x)}*-2sec^{2}(x)(1)sec^{4}(x)csc^{2}(x)}{tan^{3}(x)} + \frac{3{tan(x)}^{cot(x)}*4sec^{4}(x)tan(x)csc^{2}(x)}{tan^{2}(x)} + \frac{3{tan(x)}^{cot(x)}sec^{4}(x)*-2csc^{2}(x)cot(x)}{tan^{2}(x)} - 6({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))sec^{2}(x)csc^{2}(x) - 6{tan(x)}^{cot(x)}*2sec^{2}(x)tan(x)csc^{2}(x) - 6{tan(x)}^{cot(x)}sec^{2}(x)*-2csc^{2}(x)cot(x) + \frac{6({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))cot(x)sec^{2}(x)csc^{2}(x)}{tan(x)} + \frac{6{tan(x)}^{cot(x)}*-sec^{2}(x)(1)cot(x)sec^{2}(x)csc^{2}(x)}{tan^{2}(x)} + \frac{6{tan(x)}^{cot(x)}*-csc^{2}(x)sec^{2}(x)csc^{2}(x)}{tan(x)} + \frac{6{tan(x)}^{cot(x)}cot(x)*2sec^{2}(x)tan(x)csc^{2}(x)}{tan(x)} + \frac{6{tan(x)}^{cot(x)}cot(x)sec^{2}(x)*-2csc^{2}(x)cot(x)}{tan(x)} - 2({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))ln(tan(x))csc^{4}(x) - \frac{2{tan(x)}^{cot(x)}sec^{2}(x)(1)csc^{4}(x)}{(tan(x))} - 2{tan(x)}^{cot(x)}ln(tan(x))*-4csc^{4}(x)cot(x) - 4({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))ln(tan(x))cot^{2}(x)csc^{2}(x) - \frac{4{tan(x)}^{cot(x)}sec^{2}(x)(1)cot^{2}(x)csc^{2}(x)}{(tan(x))} - 4{tan(x)}^{cot(x)}ln(tan(x))*-2cot(x)csc^{2}(x)csc^{2}(x) - 4{tan(x)}^{cot(x)}ln(tan(x))cot^{2}(x)*-2csc^{2}(x)cot(x) + \frac{({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))cot^{3}(x)sec^{6}(x)}{tan^{3}(x)} + \frac{{tan(x)}^{cot(x)}*-3sec^{2}(x)(1)cot^{3}(x)sec^{6}(x)}{tan^{4}(x)} + \frac{{tan(x)}^{cot(x)}*-3cot^{2}(x)csc^{2}(x)sec^{6}(x)}{tan^{3}(x)} + \frac{{tan(x)}^{cot(x)}cot^{3}(x)*6sec^{6}(x)tan(x)}{tan^{3}(x)} - \frac{3({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))cot^{2}(x)sec^{6}(x)}{tan^{3}(x)} - \frac{3{tan(x)}^{cot(x)}*-3sec^{2}(x)(1)cot^{2}(x)sec^{6}(x)}{tan^{4}(x)} - \frac{3{tan(x)}^{cot(x)}*-2cot(x)csc^{2}(x)sec^{6}(x)}{tan^{3}(x)} - \frac{3{tan(x)}^{cot(x)}cot^{2}(x)*6sec^{6}(x)tan(x)}{tan^{3}(x)} + \frac{6({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))cot^{2}(x)sec^{4}(x)}{tan(x)} + \frac{6{tan(x)}^{cot(x)}*-sec^{2}(x)(1)cot^{2}(x)sec^{4}(x)}{tan^{2}(x)} + \frac{6{tan(x)}^{cot(x)}*-2cot(x)csc^{2}(x)sec^{4}(x)}{tan(x)} + \frac{6{tan(x)}^{cot(x)}cot^{2}(x)*4sec^{4}(x)tan(x)}{tan(x)} + \frac{2({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))cot(x)sec^{6}(x)}{tan^{3}(x)} + \frac{2{tan(x)}^{cot(x)}*-3sec^{2}(x)(1)cot(x)sec^{6}(x)}{tan^{4}(x)} + \frac{2{tan(x)}^{cot(x)}*-csc^{2}(x)sec^{6}(x)}{tan^{3}(x)} + \frac{2{tan(x)}^{cot(x)}cot(x)*6sec^{6}(x)tan(x)}{tan^{3}(x)} - \frac{4({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))cot(x)sec^{4}(x)}{tan(x)} - \frac{4{tan(x)}^{cot(x)}*-sec^{2}(x)(1)cot(x)sec^{4}(x)}{tan^{2}(x)} - \frac{4{tan(x)}^{cot(x)}*-csc^{2}(x)sec^{4}(x)}{tan(x)} - \frac{4{tan(x)}^{cot(x)}cot(x)*4sec^{4}(x)tan(x)}{tan(x)} + 4({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))tan(x)cot(x)sec^{2}(x) + 4{tan(x)}^{cot(x)}sec^{2}(x)(1)cot(x)sec^{2}(x) + 4{tan(x)}^{cot(x)}tan(x)*-csc^{2}(x)sec^{2}(x) + 4{tan(x)}^{cot(x)}tan(x)cot(x)*2sec^{2}(x)tan(x)\\=&{tan(x)}^{cot(x)}ln^{4}(tan(x))csc^{8}(x) + \frac{24{tan(x)}^{cot(x)}ln(tan(x))cot(x)sec^{4}(x)csc^{4}(x)}{tan^{2}(x)} - \frac{12{tan(x)}^{cot(x)}ln^{2}(tan(x))sec^{2}(x)csc^{6}(x)}{tan(x)} + 16{tan(x)}^{cot(x)}ln(tan(x))cot(x)csc^{4}(x) - \frac{4{tan(x)}^{cot(x)}ln^{3}(tan(x))cot(x)sec^{2}(x)csc^{6}(x)}{tan(x)} + \frac{6{tan(x)}^{cot(x)}ln^{2}(tan(x))cot^{2}(x)sec^{4}(x)csc^{4}(x)}{tan^{2}(x)} - \frac{6{tan(x)}^{cot(x)}ln^{2}(tan(x))cot(x)sec^{4}(x)csc^{4}(x)}{tan^{2}(x)} + 12{tan(x)}^{cot(x)}ln^{2}(tan(x))cot(x)sec^{2}(x)csc^{4}(x) - \frac{24{tan(x)}^{cot(x)}ln^{2}(tan(x))cot^{2}(x)sec^{2}(x)csc^{4}(x)}{tan(x)} + \frac{12{tan(x)}^{cot(x)}sec^{4}(x)csc^{4}(x)}{tan^{2}(x)} - \frac{12{tan(x)}^{cot(x)}ln(tan(x))sec^{4}(x)csc^{4}(x)}{tan^{2}(x)} + 24{tan(x)}^{cot(x)}ln(tan(x))sec^{2}(x)csc^{4}(x) + 12{tan(x)}^{cot(x)}ln^{3}(tan(x))cot(x)csc^{6}(x) - \frac{56{tan(x)}^{cot(x)}ln(tan(x))cot(x)sec^{2}(x)csc^{4}(x)}{tan(x)} + 8{tan(x)}^{cot(x)}ln^{2}(tan(x))csc^{6}(x) + 28{tan(x)}^{cot(x)}ln^{2}(tan(x))cot^{2}(x)csc^{4}(x) - \frac{4{tan(x)}^{cot(x)}ln(tan(x))cot^{3}(x)sec^{6}(x)csc^{2}(x)}{tan^{3}(x)} - \frac{12{tan(x)}^{cot(x)}cot^{2}(x)sec^{6}(x)csc^{2}(x)}{tan^{3}(x)} + \frac{12{tan(x)}^{cot(x)}ln(tan(x))cot^{2}(x)sec^{6}(x)csc^{2}(x)}{tan^{3}(x)} - \frac{24{tan(x)}^{cot(x)}ln(tan(x))cot^{2}(x)sec^{4}(x)csc^{2}(x)}{tan(x)} + \frac{12{tan(x)}^{cot(x)}ln(tan(x))cot^{3}(x)sec^{4}(x)csc^{2}(x)}{tan^{2}(x)} + \frac{24{tan(x)}^{cot(x)}cot(x)sec^{6}(x)csc^{2}(x)}{tan^{3}(x)} - \frac{48{tan(x)}^{cot(x)}cot(x)sec^{4}(x)csc^{2}(x)}{tan(x)} + \frac{24{tan(x)}^{cot(x)}cot^{2}(x)sec^{4}(x)csc^{2}(x)}{tan^{2}(x)} - \frac{8{tan(x)}^{cot(x)}ln(tan(x))cot(x)sec^{6}(x)csc^{2}(x)}{tan^{3}(x)} + \frac{16{tan(x)}^{cot(x)}ln(tan(x))cot(x)sec^{4}(x)csc^{2}(x)}{tan(x)} - \frac{12{tan(x)}^{cot(x)}ln(tan(x))cot^{2}(x)sec^{4}(x)csc^{2}(x)}{tan^{2}(x)} - 16{tan(x)}^{cot(x)}ln(tan(x))tan(x)cot(x)sec^{2}(x)csc^{2}(x) + 24{tan(x)}^{cot(x)}ln(tan(x))cot^{2}(x)sec^{2}(x)csc^{2}(x) - \frac{16{tan(x)}^{cot(x)}ln(tan(x))cot^{3}(x)sec^{2}(x)csc^{2}(x)}{tan(x)} - \frac{8{tan(x)}^{cot(x)}sec^{6}(x)csc^{2}(x)}{tan^{3}(x)} + \frac{16{tan(x)}^{cot(x)}sec^{4}(x)csc^{2}(x)}{tan(x)} - \frac{12{tan(x)}^{cot(x)}cot(x)sec^{4}(x)csc^{2}(x)}{tan^{2}(x)} - \frac{8{tan(x)}^{cot(x)}sec^{2}(x)csc^{4}(x)}{tan(x)} + 24{tan(x)}^{cot(x)}cot(x)sec^{2}(x)csc^{2}(x) - 16{tan(x)}^{cot(x)}tan(x)sec^{2}(x)csc^{2}(x) - \frac{16{tan(x)}^{cot(x)}cot^{2}(x)sec^{2}(x)csc^{2}(x)}{tan(x)} + 8{tan(x)}^{cot(x)}ln(tan(x))cot^{3}(x)csc^{2}(x) + \frac{{tan(x)}^{cot(x)}cot^{4}(x)sec^{8}(x)}{tan^{4}(x)} - \frac{6{tan(x)}^{cot(x)}cot^{3}(x)sec^{8}(x)}{tan^{4}(x)} + \frac{12{tan(x)}^{cot(x)}cot^{3}(x)sec^{6}(x)}{tan^{2}(x)} + \frac{11{tan(x)}^{cot(x)}cot^{2}(x)sec^{8}(x)}{tan^{4}(x)} - \frac{28{tan(x)}^{cot(x)}cot^{2}(x)sec^{6}(x)}{tan^{2}(x)} + 28{tan(x)}^{cot(x)}cot^{2}(x)sec^{4}(x) - \frac{6{tan(x)}^{cot(x)}cot(x)sec^{8}(x)}{tan^{4}(x)} + \frac{16{tan(x)}^{cot(x)}cot(x)sec^{6}(x)}{tan^{2}(x)} - 12{tan(x)}^{cot(x)}cot(x)sec^{4}(x) + 8{tan(x)}^{cot(x)}tan^{2}(x)cot(x)sec^{2}(x)\\ \end{split}\end{equation} \]
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数{tan(x)}^{cot(x)} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( {tan(x)}^{cot(x)}\right)}{dx}\\=&({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))\\=&-{tan(x)}^{cot(x)}ln(tan(x))csc^{2}(x) + \frac{{tan(x)}^{cot(x)}cot(x)sec^{2}(x)}{tan(x)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( -{tan(x)}^{cot(x)}ln(tan(x))csc^{2}(x) + \frac{{tan(x)}^{cot(x)}cot(x)sec^{2}(x)}{tan(x)}\right)}{dx}\\=&-({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))ln(tan(x))csc^{2}(x) - \frac{{tan(x)}^{cot(x)}sec^{2}(x)(1)csc^{2}(x)}{(tan(x))} - {tan(x)}^{cot(x)}ln(tan(x))*-2csc^{2}(x)cot(x) + \frac{({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))cot(x)sec^{2}(x)}{tan(x)} + \frac{{tan(x)}^{cot(x)}*-sec^{2}(x)(1)cot(x)sec^{2}(x)}{tan^{2}(x)} + \frac{{tan(x)}^{cot(x)}*-csc^{2}(x)sec^{2}(x)}{tan(x)} + \frac{{tan(x)}^{cot(x)}cot(x)*2sec^{2}(x)tan(x)}{tan(x)}\\=&{tan(x)}^{cot(x)}ln^{2}(tan(x))csc^{4}(x) - \frac{2{tan(x)}^{cot(x)}ln(tan(x))cot(x)sec^{2}(x)csc^{2}(x)}{tan(x)} - \frac{2{tan(x)}^{cot(x)}sec^{2}(x)csc^{2}(x)}{tan(x)} + 2{tan(x)}^{cot(x)}ln(tan(x))cot(x)csc^{2}(x) + \frac{{tan(x)}^{cot(x)}cot^{2}(x)sec^{4}(x)}{tan^{2}(x)} - \frac{{tan(x)}^{cot(x)}cot(x)sec^{4}(x)}{tan^{2}(x)} + 2{tan(x)}^{cot(x)}cot(x)sec^{2}(x)\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( {tan(x)}^{cot(x)}ln^{2}(tan(x))csc^{4}(x) - \frac{2{tan(x)}^{cot(x)}ln(tan(x))cot(x)sec^{2}(x)csc^{2}(x)}{tan(x)} - \frac{2{tan(x)}^{cot(x)}sec^{2}(x)csc^{2}(x)}{tan(x)} + 2{tan(x)}^{cot(x)}ln(tan(x))cot(x)csc^{2}(x) + \frac{{tan(x)}^{cot(x)}cot^{2}(x)sec^{4}(x)}{tan^{2}(x)} - \frac{{tan(x)}^{cot(x)}cot(x)sec^{4}(x)}{tan^{2}(x)} + 2{tan(x)}^{cot(x)}cot(x)sec^{2}(x)\right)}{dx}\\=&({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))ln^{2}(tan(x))csc^{4}(x) + \frac{{tan(x)}^{cot(x)}*2ln(tan(x))sec^{2}(x)(1)csc^{4}(x)}{(tan(x))} + {tan(x)}^{cot(x)}ln^{2}(tan(x))*-4csc^{4}(x)cot(x) - \frac{2({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))ln(tan(x))cot(x)sec^{2}(x)csc^{2}(x)}{tan(x)} - \frac{2{tan(x)}^{cot(x)}sec^{2}(x)(1)cot(x)sec^{2}(x)csc^{2}(x)}{(tan(x))tan(x)} - \frac{2{tan(x)}^{cot(x)}ln(tan(x))*-sec^{2}(x)(1)cot(x)sec^{2}(x)csc^{2}(x)}{tan^{2}(x)} - \frac{2{tan(x)}^{cot(x)}ln(tan(x))*-csc^{2}(x)sec^{2}(x)csc^{2}(x)}{tan(x)} - \frac{2{tan(x)}^{cot(x)}ln(tan(x))cot(x)*2sec^{2}(x)tan(x)csc^{2}(x)}{tan(x)} - \frac{2{tan(x)}^{cot(x)}ln(tan(x))cot(x)sec^{2}(x)*-2csc^{2}(x)cot(x)}{tan(x)} - \frac{2({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))sec^{2}(x)csc^{2}(x)}{tan(x)} - \frac{2{tan(x)}^{cot(x)}*-sec^{2}(x)(1)sec^{2}(x)csc^{2}(x)}{tan^{2}(x)} - \frac{2{tan(x)}^{cot(x)}*2sec^{2}(x)tan(x)csc^{2}(x)}{tan(x)} - \frac{2{tan(x)}^{cot(x)}sec^{2}(x)*-2csc^{2}(x)cot(x)}{tan(x)} + 2({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))ln(tan(x))cot(x)csc^{2}(x) + \frac{2{tan(x)}^{cot(x)}sec^{2}(x)(1)cot(x)csc^{2}(x)}{(tan(x))} + 2{tan(x)}^{cot(x)}ln(tan(x))*-csc^{2}(x)csc^{2}(x) + 2{tan(x)}^{cot(x)}ln(tan(x))cot(x)*-2csc^{2}(x)cot(x) + \frac{({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))cot^{2}(x)sec^{4}(x)}{tan^{2}(x)} + \frac{{tan(x)}^{cot(x)}*-2sec^{2}(x)(1)cot^{2}(x)sec^{4}(x)}{tan^{3}(x)} + \frac{{tan(x)}^{cot(x)}*-2cot(x)csc^{2}(x)sec^{4}(x)}{tan^{2}(x)} + \frac{{tan(x)}^{cot(x)}cot^{2}(x)*4sec^{4}(x)tan(x)}{tan^{2}(x)} - \frac{({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))cot(x)sec^{4}(x)}{tan^{2}(x)} - \frac{{tan(x)}^{cot(x)}*-2sec^{2}(x)(1)cot(x)sec^{4}(x)}{tan^{3}(x)} - \frac{{tan(x)}^{cot(x)}*-csc^{2}(x)sec^{4}(x)}{tan^{2}(x)} - \frac{{tan(x)}^{cot(x)}cot(x)*4sec^{4}(x)tan(x)}{tan^{2}(x)} + 2({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))cot(x)sec^{2}(x) + 2{tan(x)}^{cot(x)}*-csc^{2}(x)sec^{2}(x) + 2{tan(x)}^{cot(x)}cot(x)*2sec^{2}(x)tan(x)\\=&-{tan(x)}^{cot(x)}ln^{3}(tan(x))csc^{6}(x) + \frac{3{tan(x)}^{cot(x)}ln^{2}(tan(x))cot(x)sec^{2}(x)csc^{4}(x)}{tan(x)} + \frac{6{tan(x)}^{cot(x)}ln(tan(x))sec^{2}(x)csc^{4}(x)}{tan(x)} - 6{tan(x)}^{cot(x)}ln^{2}(tan(x))cot(x)csc^{4}(x) - \frac{3{tan(x)}^{cot(x)}ln(tan(x))cot^{2}(x)sec^{4}(x)csc^{2}(x)}{tan^{2}(x)} - \frac{6{tan(x)}^{cot(x)}cot(x)sec^{4}(x)csc^{2}(x)}{tan^{2}(x)} + \frac{3{tan(x)}^{cot(x)}ln(tan(x))cot(x)sec^{4}(x)csc^{2}(x)}{tan^{2}(x)} - 6{tan(x)}^{cot(x)}ln(tan(x))cot(x)sec^{2}(x)csc^{2}(x) + \frac{6{tan(x)}^{cot(x)}ln(tan(x))cot^{2}(x)sec^{2}(x)csc^{2}(x)}{tan(x)} + \frac{3{tan(x)}^{cot(x)}sec^{4}(x)csc^{2}(x)}{tan^{2}(x)} - 6{tan(x)}^{cot(x)}sec^{2}(x)csc^{2}(x) + \frac{6{tan(x)}^{cot(x)}cot(x)sec^{2}(x)csc^{2}(x)}{tan(x)} - 2{tan(x)}^{cot(x)}ln(tan(x))csc^{4}(x) - 4{tan(x)}^{cot(x)}ln(tan(x))cot^{2}(x)csc^{2}(x) + \frac{{tan(x)}^{cot(x)}cot^{3}(x)sec^{6}(x)}{tan^{3}(x)} - \frac{3{tan(x)}^{cot(x)}cot^{2}(x)sec^{6}(x)}{tan^{3}(x)} + \frac{6{tan(x)}^{cot(x)}cot^{2}(x)sec^{4}(x)}{tan(x)} + \frac{2{tan(x)}^{cot(x)}cot(x)sec^{6}(x)}{tan^{3}(x)} - \frac{4{tan(x)}^{cot(x)}cot(x)sec^{4}(x)}{tan(x)} + 4{tan(x)}^{cot(x)}tan(x)cot(x)sec^{2}(x)\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( -{tan(x)}^{cot(x)}ln^{3}(tan(x))csc^{6}(x) + \frac{3{tan(x)}^{cot(x)}ln^{2}(tan(x))cot(x)sec^{2}(x)csc^{4}(x)}{tan(x)} + \frac{6{tan(x)}^{cot(x)}ln(tan(x))sec^{2}(x)csc^{4}(x)}{tan(x)} - 6{tan(x)}^{cot(x)}ln^{2}(tan(x))cot(x)csc^{4}(x) - \frac{3{tan(x)}^{cot(x)}ln(tan(x))cot^{2}(x)sec^{4}(x)csc^{2}(x)}{tan^{2}(x)} - \frac{6{tan(x)}^{cot(x)}cot(x)sec^{4}(x)csc^{2}(x)}{tan^{2}(x)} + \frac{3{tan(x)}^{cot(x)}ln(tan(x))cot(x)sec^{4}(x)csc^{2}(x)}{tan^{2}(x)} - 6{tan(x)}^{cot(x)}ln(tan(x))cot(x)sec^{2}(x)csc^{2}(x) + \frac{6{tan(x)}^{cot(x)}ln(tan(x))cot^{2}(x)sec^{2}(x)csc^{2}(x)}{tan(x)} + \frac{3{tan(x)}^{cot(x)}sec^{4}(x)csc^{2}(x)}{tan^{2}(x)} - 6{tan(x)}^{cot(x)}sec^{2}(x)csc^{2}(x) + \frac{6{tan(x)}^{cot(x)}cot(x)sec^{2}(x)csc^{2}(x)}{tan(x)} - 2{tan(x)}^{cot(x)}ln(tan(x))csc^{4}(x) - 4{tan(x)}^{cot(x)}ln(tan(x))cot^{2}(x)csc^{2}(x) + \frac{{tan(x)}^{cot(x)}cot^{3}(x)sec^{6}(x)}{tan^{3}(x)} - \frac{3{tan(x)}^{cot(x)}cot^{2}(x)sec^{6}(x)}{tan^{3}(x)} + \frac{6{tan(x)}^{cot(x)}cot^{2}(x)sec^{4}(x)}{tan(x)} + \frac{2{tan(x)}^{cot(x)}cot(x)sec^{6}(x)}{tan^{3}(x)} - \frac{4{tan(x)}^{cot(x)}cot(x)sec^{4}(x)}{tan(x)} + 4{tan(x)}^{cot(x)}tan(x)cot(x)sec^{2}(x)\right)}{dx}\\=&-({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))ln^{3}(tan(x))csc^{6}(x) - \frac{{tan(x)}^{cot(x)}*3ln^{2}(tan(x))sec^{2}(x)(1)csc^{6}(x)}{(tan(x))} - {tan(x)}^{cot(x)}ln^{3}(tan(x))*-6csc^{6}(x)cot(x) + \frac{3({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))ln^{2}(tan(x))cot(x)sec^{2}(x)csc^{4}(x)}{tan(x)} + \frac{3{tan(x)}^{cot(x)}*2ln(tan(x))sec^{2}(x)(1)cot(x)sec^{2}(x)csc^{4}(x)}{(tan(x))tan(x)} + \frac{3{tan(x)}^{cot(x)}ln^{2}(tan(x))*-sec^{2}(x)(1)cot(x)sec^{2}(x)csc^{4}(x)}{tan^{2}(x)} + \frac{3{tan(x)}^{cot(x)}ln^{2}(tan(x))*-csc^{2}(x)sec^{2}(x)csc^{4}(x)}{tan(x)} + \frac{3{tan(x)}^{cot(x)}ln^{2}(tan(x))cot(x)*2sec^{2}(x)tan(x)csc^{4}(x)}{tan(x)} + \frac{3{tan(x)}^{cot(x)}ln^{2}(tan(x))cot(x)sec^{2}(x)*-4csc^{4}(x)cot(x)}{tan(x)} + \frac{6({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))ln(tan(x))sec^{2}(x)csc^{4}(x)}{tan(x)} + \frac{6{tan(x)}^{cot(x)}sec^{2}(x)(1)sec^{2}(x)csc^{4}(x)}{(tan(x))tan(x)} + \frac{6{tan(x)}^{cot(x)}ln(tan(x))*-sec^{2}(x)(1)sec^{2}(x)csc^{4}(x)}{tan^{2}(x)} + \frac{6{tan(x)}^{cot(x)}ln(tan(x))*2sec^{2}(x)tan(x)csc^{4}(x)}{tan(x)} + \frac{6{tan(x)}^{cot(x)}ln(tan(x))sec^{2}(x)*-4csc^{4}(x)cot(x)}{tan(x)} - 6({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))ln^{2}(tan(x))cot(x)csc^{4}(x) - \frac{6{tan(x)}^{cot(x)}*2ln(tan(x))sec^{2}(x)(1)cot(x)csc^{4}(x)}{(tan(x))} - 6{tan(x)}^{cot(x)}ln^{2}(tan(x))*-csc^{2}(x)csc^{4}(x) - 6{tan(x)}^{cot(x)}ln^{2}(tan(x))cot(x)*-4csc^{4}(x)cot(x) - \frac{3({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))ln(tan(x))cot^{2}(x)sec^{4}(x)csc^{2}(x)}{tan^{2}(x)} - \frac{3{tan(x)}^{cot(x)}sec^{2}(x)(1)cot^{2}(x)sec^{4}(x)csc^{2}(x)}{(tan(x))tan^{2}(x)} - \frac{3{tan(x)}^{cot(x)}ln(tan(x))*-2sec^{2}(x)(1)cot^{2}(x)sec^{4}(x)csc^{2}(x)}{tan^{3}(x)} - \frac{3{tan(x)}^{cot(x)}ln(tan(x))*-2cot(x)csc^{2}(x)sec^{4}(x)csc^{2}(x)}{tan^{2}(x)} - \frac{3{tan(x)}^{cot(x)}ln(tan(x))cot^{2}(x)*4sec^{4}(x)tan(x)csc^{2}(x)}{tan^{2}(x)} - \frac{3{tan(x)}^{cot(x)}ln(tan(x))cot^{2}(x)sec^{4}(x)*-2csc^{2}(x)cot(x)}{tan^{2}(x)} - \frac{6({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))cot(x)sec^{4}(x)csc^{2}(x)}{tan^{2}(x)} - \frac{6{tan(x)}^{cot(x)}*-2sec^{2}(x)(1)cot(x)sec^{4}(x)csc^{2}(x)}{tan^{3}(x)} - \frac{6{tan(x)}^{cot(x)}*-csc^{2}(x)sec^{4}(x)csc^{2}(x)}{tan^{2}(x)} - \frac{6{tan(x)}^{cot(x)}cot(x)*4sec^{4}(x)tan(x)csc^{2}(x)}{tan^{2}(x)} - \frac{6{tan(x)}^{cot(x)}cot(x)sec^{4}(x)*-2csc^{2}(x)cot(x)}{tan^{2}(x)} + \frac{3({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))ln(tan(x))cot(x)sec^{4}(x)csc^{2}(x)}{tan^{2}(x)} + \frac{3{tan(x)}^{cot(x)}sec^{2}(x)(1)cot(x)sec^{4}(x)csc^{2}(x)}{(tan(x))tan^{2}(x)} + \frac{3{tan(x)}^{cot(x)}ln(tan(x))*-2sec^{2}(x)(1)cot(x)sec^{4}(x)csc^{2}(x)}{tan^{3}(x)} + \frac{3{tan(x)}^{cot(x)}ln(tan(x))*-csc^{2}(x)sec^{4}(x)csc^{2}(x)}{tan^{2}(x)} + \frac{3{tan(x)}^{cot(x)}ln(tan(x))cot(x)*4sec^{4}(x)tan(x)csc^{2}(x)}{tan^{2}(x)} + \frac{3{tan(x)}^{cot(x)}ln(tan(x))cot(x)sec^{4}(x)*-2csc^{2}(x)cot(x)}{tan^{2}(x)} - 6({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))ln(tan(x))cot(x)sec^{2}(x)csc^{2}(x) - \frac{6{tan(x)}^{cot(x)}sec^{2}(x)(1)cot(x)sec^{2}(x)csc^{2}(x)}{(tan(x))} - 6{tan(x)}^{cot(x)}ln(tan(x))*-csc^{2}(x)sec^{2}(x)csc^{2}(x) - 6{tan(x)}^{cot(x)}ln(tan(x))cot(x)*2sec^{2}(x)tan(x)csc^{2}(x) - 6{tan(x)}^{cot(x)}ln(tan(x))cot(x)sec^{2}(x)*-2csc^{2}(x)cot(x) + \frac{6({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))ln(tan(x))cot^{2}(x)sec^{2}(x)csc^{2}(x)}{tan(x)} + \frac{6{tan(x)}^{cot(x)}sec^{2}(x)(1)cot^{2}(x)sec^{2}(x)csc^{2}(x)}{(tan(x))tan(x)} + \frac{6{tan(x)}^{cot(x)}ln(tan(x))*-sec^{2}(x)(1)cot^{2}(x)sec^{2}(x)csc^{2}(x)}{tan^{2}(x)} + \frac{6{tan(x)}^{cot(x)}ln(tan(x))*-2cot(x)csc^{2}(x)sec^{2}(x)csc^{2}(x)}{tan(x)} + \frac{6{tan(x)}^{cot(x)}ln(tan(x))cot^{2}(x)*2sec^{2}(x)tan(x)csc^{2}(x)}{tan(x)} + \frac{6{tan(x)}^{cot(x)}ln(tan(x))cot^{2}(x)sec^{2}(x)*-2csc^{2}(x)cot(x)}{tan(x)} + \frac{3({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))sec^{4}(x)csc^{2}(x)}{tan^{2}(x)} + \frac{3{tan(x)}^{cot(x)}*-2sec^{2}(x)(1)sec^{4}(x)csc^{2}(x)}{tan^{3}(x)} + \frac{3{tan(x)}^{cot(x)}*4sec^{4}(x)tan(x)csc^{2}(x)}{tan^{2}(x)} + \frac{3{tan(x)}^{cot(x)}sec^{4}(x)*-2csc^{2}(x)cot(x)}{tan^{2}(x)} - 6({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))sec^{2}(x)csc^{2}(x) - 6{tan(x)}^{cot(x)}*2sec^{2}(x)tan(x)csc^{2}(x) - 6{tan(x)}^{cot(x)}sec^{2}(x)*-2csc^{2}(x)cot(x) + \frac{6({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))cot(x)sec^{2}(x)csc^{2}(x)}{tan(x)} + \frac{6{tan(x)}^{cot(x)}*-sec^{2}(x)(1)cot(x)sec^{2}(x)csc^{2}(x)}{tan^{2}(x)} + \frac{6{tan(x)}^{cot(x)}*-csc^{2}(x)sec^{2}(x)csc^{2}(x)}{tan(x)} + \frac{6{tan(x)}^{cot(x)}cot(x)*2sec^{2}(x)tan(x)csc^{2}(x)}{tan(x)} + \frac{6{tan(x)}^{cot(x)}cot(x)sec^{2}(x)*-2csc^{2}(x)cot(x)}{tan(x)} - 2({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))ln(tan(x))csc^{4}(x) - \frac{2{tan(x)}^{cot(x)}sec^{2}(x)(1)csc^{4}(x)}{(tan(x))} - 2{tan(x)}^{cot(x)}ln(tan(x))*-4csc^{4}(x)cot(x) - 4({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))ln(tan(x))cot^{2}(x)csc^{2}(x) - \frac{4{tan(x)}^{cot(x)}sec^{2}(x)(1)cot^{2}(x)csc^{2}(x)}{(tan(x))} - 4{tan(x)}^{cot(x)}ln(tan(x))*-2cot(x)csc^{2}(x)csc^{2}(x) - 4{tan(x)}^{cot(x)}ln(tan(x))cot^{2}(x)*-2csc^{2}(x)cot(x) + \frac{({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))cot^{3}(x)sec^{6}(x)}{tan^{3}(x)} + \frac{{tan(x)}^{cot(x)}*-3sec^{2}(x)(1)cot^{3}(x)sec^{6}(x)}{tan^{4}(x)} + \frac{{tan(x)}^{cot(x)}*-3cot^{2}(x)csc^{2}(x)sec^{6}(x)}{tan^{3}(x)} + \frac{{tan(x)}^{cot(x)}cot^{3}(x)*6sec^{6}(x)tan(x)}{tan^{3}(x)} - \frac{3({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))cot^{2}(x)sec^{6}(x)}{tan^{3}(x)} - \frac{3{tan(x)}^{cot(x)}*-3sec^{2}(x)(1)cot^{2}(x)sec^{6}(x)}{tan^{4}(x)} - \frac{3{tan(x)}^{cot(x)}*-2cot(x)csc^{2}(x)sec^{6}(x)}{tan^{3}(x)} - \frac{3{tan(x)}^{cot(x)}cot^{2}(x)*6sec^{6}(x)tan(x)}{tan^{3}(x)} + \frac{6({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))cot^{2}(x)sec^{4}(x)}{tan(x)} + \frac{6{tan(x)}^{cot(x)}*-sec^{2}(x)(1)cot^{2}(x)sec^{4}(x)}{tan^{2}(x)} + \frac{6{tan(x)}^{cot(x)}*-2cot(x)csc^{2}(x)sec^{4}(x)}{tan(x)} + \frac{6{tan(x)}^{cot(x)}cot^{2}(x)*4sec^{4}(x)tan(x)}{tan(x)} + \frac{2({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))cot(x)sec^{6}(x)}{tan^{3}(x)} + \frac{2{tan(x)}^{cot(x)}*-3sec^{2}(x)(1)cot(x)sec^{6}(x)}{tan^{4}(x)} + \frac{2{tan(x)}^{cot(x)}*-csc^{2}(x)sec^{6}(x)}{tan^{3}(x)} + \frac{2{tan(x)}^{cot(x)}cot(x)*6sec^{6}(x)tan(x)}{tan^{3}(x)} - \frac{4({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))cot(x)sec^{4}(x)}{tan(x)} - \frac{4{tan(x)}^{cot(x)}*-sec^{2}(x)(1)cot(x)sec^{4}(x)}{tan^{2}(x)} - \frac{4{tan(x)}^{cot(x)}*-csc^{2}(x)sec^{4}(x)}{tan(x)} - \frac{4{tan(x)}^{cot(x)}cot(x)*4sec^{4}(x)tan(x)}{tan(x)} + 4({tan(x)}^{cot(x)}((-csc^{2}(x))ln(tan(x)) + \frac{(cot(x))(sec^{2}(x)(1))}{(tan(x))}))tan(x)cot(x)sec^{2}(x) + 4{tan(x)}^{cot(x)}sec^{2}(x)(1)cot(x)sec^{2}(x) + 4{tan(x)}^{cot(x)}tan(x)*-csc^{2}(x)sec^{2}(x) + 4{tan(x)}^{cot(x)}tan(x)cot(x)*2sec^{2}(x)tan(x)\\=&{tan(x)}^{cot(x)}ln^{4}(tan(x))csc^{8}(x) + \frac{24{tan(x)}^{cot(x)}ln(tan(x))cot(x)sec^{4}(x)csc^{4}(x)}{tan^{2}(x)} - \frac{12{tan(x)}^{cot(x)}ln^{2}(tan(x))sec^{2}(x)csc^{6}(x)}{tan(x)} + 16{tan(x)}^{cot(x)}ln(tan(x))cot(x)csc^{4}(x) - \frac{4{tan(x)}^{cot(x)}ln^{3}(tan(x))cot(x)sec^{2}(x)csc^{6}(x)}{tan(x)} + \frac{6{tan(x)}^{cot(x)}ln^{2}(tan(x))cot^{2}(x)sec^{4}(x)csc^{4}(x)}{tan^{2}(x)} - \frac{6{tan(x)}^{cot(x)}ln^{2}(tan(x))cot(x)sec^{4}(x)csc^{4}(x)}{tan^{2}(x)} + 12{tan(x)}^{cot(x)}ln^{2}(tan(x))cot(x)sec^{2}(x)csc^{4}(x) - \frac{24{tan(x)}^{cot(x)}ln^{2}(tan(x))cot^{2}(x)sec^{2}(x)csc^{4}(x)}{tan(x)} + \frac{12{tan(x)}^{cot(x)}sec^{4}(x)csc^{4}(x)}{tan^{2}(x)} - \frac{12{tan(x)}^{cot(x)}ln(tan(x))sec^{4}(x)csc^{4}(x)}{tan^{2}(x)} + 24{tan(x)}^{cot(x)}ln(tan(x))sec^{2}(x)csc^{4}(x) + 12{tan(x)}^{cot(x)}ln^{3}(tan(x))cot(x)csc^{6}(x) - \frac{56{tan(x)}^{cot(x)}ln(tan(x))cot(x)sec^{2}(x)csc^{4}(x)}{tan(x)} + 8{tan(x)}^{cot(x)}ln^{2}(tan(x))csc^{6}(x) + 28{tan(x)}^{cot(x)}ln^{2}(tan(x))cot^{2}(x)csc^{4}(x) - \frac{4{tan(x)}^{cot(x)}ln(tan(x))cot^{3}(x)sec^{6}(x)csc^{2}(x)}{tan^{3}(x)} - \frac{12{tan(x)}^{cot(x)}cot^{2}(x)sec^{6}(x)csc^{2}(x)}{tan^{3}(x)} + \frac{12{tan(x)}^{cot(x)}ln(tan(x))cot^{2}(x)sec^{6}(x)csc^{2}(x)}{tan^{3}(x)} - \frac{24{tan(x)}^{cot(x)}ln(tan(x))cot^{2}(x)sec^{4}(x)csc^{2}(x)}{tan(x)} + \frac{12{tan(x)}^{cot(x)}ln(tan(x))cot^{3}(x)sec^{4}(x)csc^{2}(x)}{tan^{2}(x)} + \frac{24{tan(x)}^{cot(x)}cot(x)sec^{6}(x)csc^{2}(x)}{tan^{3}(x)} - \frac{48{tan(x)}^{cot(x)}cot(x)sec^{4}(x)csc^{2}(x)}{tan(x)} + \frac{24{tan(x)}^{cot(x)}cot^{2}(x)sec^{4}(x)csc^{2}(x)}{tan^{2}(x)} - \frac{8{tan(x)}^{cot(x)}ln(tan(x))cot(x)sec^{6}(x)csc^{2}(x)}{tan^{3}(x)} + \frac{16{tan(x)}^{cot(x)}ln(tan(x))cot(x)sec^{4}(x)csc^{2}(x)}{tan(x)} - \frac{12{tan(x)}^{cot(x)}ln(tan(x))cot^{2}(x)sec^{4}(x)csc^{2}(x)}{tan^{2}(x)} - 16{tan(x)}^{cot(x)}ln(tan(x))tan(x)cot(x)sec^{2}(x)csc^{2}(x) + 24{tan(x)}^{cot(x)}ln(tan(x))cot^{2}(x)sec^{2}(x)csc^{2}(x) - \frac{16{tan(x)}^{cot(x)}ln(tan(x))cot^{3}(x)sec^{2}(x)csc^{2}(x)}{tan(x)} - \frac{8{tan(x)}^{cot(x)}sec^{6}(x)csc^{2}(x)}{tan^{3}(x)} + \frac{16{tan(x)}^{cot(x)}sec^{4}(x)csc^{2}(x)}{tan(x)} - \frac{12{tan(x)}^{cot(x)}cot(x)sec^{4}(x)csc^{2}(x)}{tan^{2}(x)} - \frac{8{tan(x)}^{cot(x)}sec^{2}(x)csc^{4}(x)}{tan(x)} + 24{tan(x)}^{cot(x)}cot(x)sec^{2}(x)csc^{2}(x) - 16{tan(x)}^{cot(x)}tan(x)sec^{2}(x)csc^{2}(x) - \frac{16{tan(x)}^{cot(x)}cot^{2}(x)sec^{2}(x)csc^{2}(x)}{tan(x)} + 8{tan(x)}^{cot(x)}ln(tan(x))cot^{3}(x)csc^{2}(x) + \frac{{tan(x)}^{cot(x)}cot^{4}(x)sec^{8}(x)}{tan^{4}(x)} - \frac{6{tan(x)}^{cot(x)}cot^{3}(x)sec^{8}(x)}{tan^{4}(x)} + \frac{12{tan(x)}^{cot(x)}cot^{3}(x)sec^{6}(x)}{tan^{2}(x)} + \frac{11{tan(x)}^{cot(x)}cot^{2}(x)sec^{8}(x)}{tan^{4}(x)} - \frac{28{tan(x)}^{cot(x)}cot^{2}(x)sec^{6}(x)}{tan^{2}(x)} + 28{tan(x)}^{cot(x)}cot^{2}(x)sec^{4}(x) - \frac{6{tan(x)}^{cot(x)}cot(x)sec^{8}(x)}{tan^{4}(x)} + \frac{16{tan(x)}^{cot(x)}cot(x)sec^{6}(x)}{tan^{2}(x)} - 12{tan(x)}^{cot(x)}cot(x)sec^{4}(x) + 8{tan(x)}^{cot(x)}tan^{2}(x)cot(x)sec^{2}(x)\\ \end{split}\end{equation} \]
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