Derivative function：
Enter an original function (that is, the function to be derived), then set the variable to be derived and the order of the derivative, and click the "Next" button to obtain the derivative function of the corresponding order of the function.
Note that the input function supports mathematical functions and other constants.
Enter an original function (that is, the function to be derived), then set the variable to be derived and the order of the derivative, and click the "Next" button to obtain the derivative function of the corresponding order of the function.
Note that the input function supports mathematical functions and other constants.
Current location：Derivative function > Derivative function calculation history > Answer
There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ {tan(x)}^{cot(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\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}{The\ second\ derivative\ of\ function:} \\&\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}{The\ third\ derivative\ of\ function:} \\&\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}{The\ 4th\ derivative\ of\ function:} \\&\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} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ {tan(x)}^{cot(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\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}{The\ second\ derivative\ of\ function:} \\&\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}{The\ third\ derivative\ of\ function:} \\&\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}{The\ 4th\ derivative\ of\ function:} \\&\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} \]