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.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{-{csc(x)}^{cot(x)}}{cos(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{-{csc(x)}^{cot(x)}}{cos(x)}\right)}{dx}\\=&\frac{-({csc(x)}^{cot(x)}((-csc^{2}(x))ln(csc(x)) + \frac{(cot(x))(-csc(x)cot(x))}{(csc(x))}))}{cos(x)} - \frac{{csc(x)}^{cot(x)}sin(x)}{cos^{2}(x)}\\=&\frac{{csc(x)}^{cot(x)}ln(csc(x))csc^{2}(x)}{cos(x)} + \frac{{csc(x)}^{cot(x)}cot^{2}(x)}{cos(x)} - \frac{{csc(x)}^{cot(x)}sin(x)}{cos^{2}(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{{csc(x)}^{cot(x)}ln(csc(x))csc^{2}(x)}{cos(x)} + \frac{{csc(x)}^{cot(x)}cot^{2}(x)}{cos(x)} - \frac{{csc(x)}^{cot(x)}sin(x)}{cos^{2}(x)}\right)}{dx}\\=&\frac{({csc(x)}^{cot(x)}((-csc^{2}(x))ln(csc(x)) + \frac{(cot(x))(-csc(x)cot(x))}{(csc(x))}))ln(csc(x))csc^{2}(x)}{cos(x)} + \frac{{csc(x)}^{cot(x)}*-csc(x)cot(x)csc^{2}(x)}{(csc(x))cos(x)} + \frac{{csc(x)}^{cot(x)}ln(csc(x))sin(x)csc^{2}(x)}{cos^{2}(x)} + \frac{{csc(x)}^{cot(x)}ln(csc(x))*-2csc^{2}(x)cot(x)}{cos(x)} + \frac{({csc(x)}^{cot(x)}((-csc^{2}(x))ln(csc(x)) + \frac{(cot(x))(-csc(x)cot(x))}{(csc(x))}))cot^{2}(x)}{cos(x)} + \frac{{csc(x)}^{cot(x)}sin(x)cot^{2}(x)}{cos^{2}(x)} + \frac{{csc(x)}^{cot(x)}*-2cot(x)csc^{2}(x)}{cos(x)} - \frac{({csc(x)}^{cot(x)}((-csc^{2}(x))ln(csc(x)) + \frac{(cot(x))(-csc(x)cot(x))}{(csc(x))}))sin(x)}{cos^{2}(x)} - \frac{{csc(x)}^{cot(x)}cos(x)}{cos^{2}(x)} - \frac{{csc(x)}^{cot(x)}sin(x)*2sin(x)}{cos^{3}(x)}\\=&\frac{-{csc(x)}^{cot(x)}ln^{2}(csc(x))csc^{4}(x)}{cos(x)} - \frac{2{csc(x)}^{cot(x)}ln(csc(x))cot^{2}(x)csc^{2}(x)}{cos(x)} - \frac{3{csc(x)}^{cot(x)}cot(x)csc^{2}(x)}{cos(x)} + \frac{2{csc(x)}^{cot(x)}ln(csc(x))sin(x)csc^{2}(x)}{cos^{2}(x)} - \frac{2{csc(x)}^{cot(x)}ln(csc(x))cot(x)csc^{2}(x)}{cos(x)} - \frac{{csc(x)}^{cot(x)}cot^{4}(x)}{cos(x)} + \frac{2{csc(x)}^{cot(x)}sin(x)cot^{2}(x)}{cos^{2}(x)} - \frac{{csc(x)}^{cot(x)}}{cos(x)} - \frac{2{csc(x)}^{cot(x)}sin^{2}(x)}{cos^{3}(x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-{csc(x)}^{cot(x)}ln^{2}(csc(x))csc^{4}(x)}{cos(x)} - \frac{2{csc(x)}^{cot(x)}ln(csc(x))cot^{2}(x)csc^{2}(x)}{cos(x)} - \frac{3{csc(x)}^{cot(x)}cot(x)csc^{2}(x)}{cos(x)} + \frac{2{csc(x)}^{cot(x)}ln(csc(x))sin(x)csc^{2}(x)}{cos^{2}(x)} - \frac{2{csc(x)}^{cot(x)}ln(csc(x))cot(x)csc^{2}(x)}{cos(x)} - \frac{{csc(x)}^{cot(x)}cot^{4}(x)}{cos(x)} + \frac{2{csc(x)}^{cot(x)}sin(x)cot^{2}(x)}{cos^{2}(x)} - \frac{{csc(x)}^{cot(x)}}{cos(x)} - \frac{2{csc(x)}^{cot(x)}sin^{2}(x)}{cos^{3}(x)}\right)}{dx}\\=&\frac{-({csc(x)}^{cot(x)}((-csc^{2}(x))ln(csc(x)) + \frac{(cot(x))(-csc(x)cot(x))}{(csc(x))}))ln^{2}(csc(x))csc^{4}(x)}{cos(x)} - \frac{{csc(x)}^{cot(x)}*2ln(csc(x))*-csc(x)cot(x)csc^{4}(x)}{(csc(x))cos(x)} - \frac{{csc(x)}^{cot(x)}ln^{2}(csc(x))sin(x)csc^{4}(x)}{cos^{2}(x)} - \frac{{csc(x)}^{cot(x)}ln^{2}(csc(x))*-4csc^{4}(x)cot(x)}{cos(x)} - \frac{2({csc(x)}^{cot(x)}((-csc^{2}(x))ln(csc(x)) + \frac{(cot(x))(-csc(x)cot(x))}{(csc(x))}))ln(csc(x))cot^{2}(x)csc^{2}(x)}{cos(x)} - \frac{2{csc(x)}^{cot(x)}*-csc(x)cot(x)cot^{2}(x)csc^{2}(x)}{(csc(x))cos(x)} - \frac{2{csc(x)}^{cot(x)}ln(csc(x))sin(x)cot^{2}(x)csc^{2}(x)}{cos^{2}(x)} - \frac{2{csc(x)}^{cot(x)}ln(csc(x))*-2cot(x)csc^{2}(x)csc^{2}(x)}{cos(x)} - \frac{2{csc(x)}^{cot(x)}ln(csc(x))cot^{2}(x)*-2csc^{2}(x)cot(x)}{cos(x)} - \frac{3({csc(x)}^{cot(x)}((-csc^{2}(x))ln(csc(x)) + \frac{(cot(x))(-csc(x)cot(x))}{(csc(x))}))cot(x)csc^{2}(x)}{cos(x)} - \frac{3{csc(x)}^{cot(x)}sin(x)cot(x)csc^{2}(x)}{cos^{2}(x)} - \frac{3{csc(x)}^{cot(x)}*-csc^{2}(x)csc^{2}(x)}{cos(x)} - \frac{3{csc(x)}^{cot(x)}cot(x)*-2csc^{2}(x)cot(x)}{cos(x)} + \frac{2({csc(x)}^{cot(x)}((-csc^{2}(x))ln(csc(x)) + \frac{(cot(x))(-csc(x)cot(x))}{(csc(x))}))ln(csc(x))sin(x)csc^{2}(x)}{cos^{2}(x)} + \frac{2{csc(x)}^{cot(x)}*-csc(x)cot(x)sin(x)csc^{2}(x)}{(csc(x))cos^{2}(x)} + \frac{2{csc(x)}^{cot(x)}ln(csc(x))cos(x)csc^{2}(x)}{cos^{2}(x)} + \frac{2{csc(x)}^{cot(x)}ln(csc(x))sin(x)*2sin(x)csc^{2}(x)}{cos^{3}(x)} + \frac{2{csc(x)}^{cot(x)}ln(csc(x))sin(x)*-2csc^{2}(x)cot(x)}{cos^{2}(x)} - \frac{2({csc(x)}^{cot(x)}((-csc^{2}(x))ln(csc(x)) + \frac{(cot(x))(-csc(x)cot(x))}{(csc(x))}))ln(csc(x))cot(x)csc^{2}(x)}{cos(x)} - \frac{2{csc(x)}^{cot(x)}*-csc(x)cot(x)cot(x)csc^{2}(x)}{(csc(x))cos(x)} - \frac{2{csc(x)}^{cot(x)}ln(csc(x))sin(x)cot(x)csc^{2}(x)}{cos^{2}(x)} - \frac{2{csc(x)}^{cot(x)}ln(csc(x))*-csc^{2}(x)csc^{2}(x)}{cos(x)} - \frac{2{csc(x)}^{cot(x)}ln(csc(x))cot(x)*-2csc^{2}(x)cot(x)}{cos(x)} - \frac{({csc(x)}^{cot(x)}((-csc^{2}(x))ln(csc(x)) + \frac{(cot(x))(-csc(x)cot(x))}{(csc(x))}))cot^{4}(x)}{cos(x)} - \frac{{csc(x)}^{cot(x)}sin(x)cot^{4}(x)}{cos^{2}(x)} - \frac{{csc(x)}^{cot(x)}*-4cot^{3}(x)csc^{2}(x)}{cos(x)} + \frac{2({csc(x)}^{cot(x)}((-csc^{2}(x))ln(csc(x)) + \frac{(cot(x))(-csc(x)cot(x))}{(csc(x))}))sin(x)cot^{2}(x)}{cos^{2}(x)} + \frac{2{csc(x)}^{cot(x)}cos(x)cot^{2}(x)}{cos^{2}(x)} + \frac{2{csc(x)}^{cot(x)}sin(x)*2sin(x)cot^{2}(x)}{cos^{3}(x)} + \frac{2{csc(x)}^{cot(x)}sin(x)*-2cot(x)csc^{2}(x)}{cos^{2}(x)} - \frac{({csc(x)}^{cot(x)}((-csc^{2}(x))ln(csc(x)) + \frac{(cot(x))(-csc(x)cot(x))}{(csc(x))}))}{cos(x)} - \frac{{csc(x)}^{cot(x)}sin(x)}{cos^{2}(x)} - \frac{2({csc(x)}^{cot(x)}((-csc^{2}(x))ln(csc(x)) + \frac{(cot(x))(-csc(x)cot(x))}{(csc(x))}))sin^{2}(x)}{cos^{3}(x)} - \frac{2{csc(x)}^{cot(x)}*2sin(x)cos(x)}{cos^{3}(x)} - \frac{2{csc(x)}^{cot(x)}sin^{2}(x)*3sin(x)}{cos^{4}(x)}\\=&\frac{{csc(x)}^{cot(x)}ln^{3}(csc(x))csc^{6}(x)}{cos(x)} + \frac{9{csc(x)}^{cot(x)}ln(csc(x))cot(x)csc^{4}(x)}{cos(x)} - \frac{3{csc(x)}^{cot(x)}ln^{2}(csc(x))sin(x)csc^{4}(x)}{cos^{2}(x)} + \frac{3{csc(x)}^{cot(x)}ln^{2}(csc(x))cot^{2}(x)csc^{4}(x)}{cos(x)} + \frac{3{csc(x)}^{cot(x)}ln(csc(x))cot^{4}(x)csc^{2}(x)}{cos(x)} + \frac{9{csc(x)}^{cot(x)}cot^{3}(x)csc^{2}(x)}{cos(x)} - \frac{6{csc(x)}^{cot(x)}ln(csc(x))sin(x)cot^{2}(x)csc^{2}(x)}{cos^{2}(x)} + \frac{6{csc(x)}^{cot(x)}ln^{2}(csc(x))cot(x)csc^{4}(x)}{cos(x)} + \frac{6{csc(x)}^{cot(x)}ln(csc(x))cot^{3}(x)csc^{2}(x)}{cos(x)} - \frac{9{csc(x)}^{cot(x)}sin(x)cot(x)csc^{2}(x)}{cos^{2}(x)} + \frac{3{csc(x)}^{cot(x)}csc^{4}(x)}{cos(x)} + \frac{8{csc(x)}^{cot(x)}cot^{2}(x)csc^{2}(x)}{cos(x)} + \frac{3{csc(x)}^{cot(x)}ln(csc(x))csc^{2}(x)}{cos(x)} + \frac{6{csc(x)}^{cot(x)}ln(csc(x))sin^{2}(x)csc^{2}(x)}{cos^{3}(x)} - \frac{6{csc(x)}^{cot(x)}ln(csc(x))sin(x)cot(x)csc^{2}(x)}{cos^{2}(x)} + \frac{2{csc(x)}^{cot(x)}ln(csc(x))csc^{4}(x)}{cos(x)} + \frac{4{csc(x)}^{cot(x)}ln(csc(x))cot^{2}(x)csc^{2}(x)}{cos(x)} + \frac{{csc(x)}^{cot(x)}cot^{6}(x)}{cos(x)} - \frac{3{csc(x)}^{cot(x)}sin(x)cot^{4}(x)}{cos^{2}(x)} + \frac{3{csc(x)}^{cot(x)}cot^{2}(x)}{cos(x)} + \frac{6{csc(x)}^{cot(x)}sin^{2}(x)cot^{2}(x)}{cos^{3}(x)} - \frac{5{csc(x)}^{cot(x)}sin(x)}{cos^{2}(x)} - \frac{6{csc(x)}^{cot(x)}sin^{3}(x)}{cos^{4}(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{{csc(x)}^{cot(x)}ln^{3}(csc(x))csc^{6}(x)}{cos(x)} + \frac{9{csc(x)}^{cot(x)}ln(csc(x))cot(x)csc^{4}(x)}{cos(x)} - \frac{3{csc(x)}^{cot(x)}ln^{2}(csc(x))sin(x)csc^{4}(x)}{cos^{2}(x)} + \frac{3{csc(x)}^{cot(x)}ln^{2}(csc(x))cot^{2}(x)csc^{4}(x)}{cos(x)} + \frac{3{csc(x)}^{cot(x)}ln(csc(x))cot^{4}(x)csc^{2}(x)}{cos(x)} + \frac{9{csc(x)}^{cot(x)}cot^{3}(x)csc^{2}(x)}{cos(x)} - \frac{6{csc(x)}^{cot(x)}ln(csc(x))sin(x)cot^{2}(x)csc^{2}(x)}{cos^{2}(x)} + \frac{6{csc(x)}^{cot(x)}ln^{2}(csc(x))cot(x)csc^{4}(x)}{cos(x)} + \frac{6{csc(x)}^{cot(x)}ln(csc(x))cot^{3}(x)csc^{2}(x)}{cos(x)} - \frac{9{csc(x)}^{cot(x)}sin(x)cot(x)csc^{2}(x)}{cos^{2}(x)} + \frac{3{csc(x)}^{cot(x)}csc^{4}(x)}{cos(x)} + \frac{8{csc(x)}^{cot(x)}cot^{2}(x)csc^{2}(x)}{cos(x)} + \frac{3{csc(x)}^{cot(x)}ln(csc(x))csc^{2}(x)}{cos(x)} + \frac{6{csc(x)}^{cot(x)}ln(csc(x))sin^{2}(x)csc^{2}(x)}{cos^{3}(x)} - \frac{6{csc(x)}^{cot(x)}ln(csc(x))sin(x)cot(x)csc^{2}(x)}{cos^{2}(x)} + \frac{2{csc(x)}^{cot(x)}ln(csc(x))csc^{4}(x)}{cos(x)} + \frac{4{csc(x)}^{cot(x)}ln(csc(x))cot^{2}(x)csc^{2}(x)}{cos(x)} + \frac{{csc(x)}^{cot(x)}cot^{6}(x)}{cos(x)} - \frac{3{csc(x)}^{cot(x)}sin(x)cot^{4}(x)}{cos^{2}(x)} + \frac{3{csc(x)}^{cot(x)}cot^{2}(x)}{cos(x)} + \frac{6{csc(x)}^{cot(x)}sin^{2}(x)cot^{2}(x)}{cos^{3}(x)} - \frac{5{csc(x)}^{cot(x)}sin(x)}{cos^{2}(x)} - \frac{6{csc(x)}^{cot(x)}sin^{3}(x)}{cos^{4}(x)}\right)}{dx}\\=&\frac{({csc(x)}^{cot(x)}((-csc^{2}(x))ln(csc(x)) + \frac{(cot(x))(-csc(x)cot(x))}{(csc(x))}))ln^{3}(csc(x))csc^{6}(x)}{cos(x)} + \frac{{csc(x)}^{cot(x)}*3ln^{2}(csc(x))*-csc(x)cot(x)csc^{6}(x)}{(csc(x))cos(x)} + \frac{{csc(x)}^{cot(x)}ln^{3}(csc(x))sin(x)csc^{6}(x)}{cos^{2}(x)} + \frac{{csc(x)}^{cot(x)}ln^{3}(csc(x))*-6csc^{6}(x)cot(x)}{cos(x)} + \frac{9({csc(x)}^{cot(x)}((-csc^{2}(x))ln(csc(x)) + \frac{(cot(x))(-csc(x)cot(x))}{(csc(x))}))ln(csc(x))cot(x)csc^{4}(x)}{cos(x)} + \frac{9{csc(x)}^{cot(x)}*-csc(x)cot(x)cot(x)csc^{4}(x)}{(csc(x))cos(x)} + \frac{9{csc(x)}^{cot(x)}ln(csc(x))sin(x)cot(x)csc^{4}(x)}{cos^{2}(x)} + \frac{9{csc(x)}^{cot(x)}ln(csc(x))*-csc^{2}(x)csc^{4}(x)}{cos(x)} + \frac{9{csc(x)}^{cot(x)}ln(csc(x))cot(x)*-4csc^{4}(x)cot(x)}{cos(x)} - \frac{3({csc(x)}^{cot(x)}((-csc^{2}(x))ln(csc(x)) + \frac{(cot(x))(-csc(x)cot(x))}{(csc(x))}))ln^{2}(csc(x))sin(x)csc^{4}(x)}{cos^{2}(x)} - \frac{3{csc(x)}^{cot(x)}*2ln(csc(x))*-csc(x)cot(x)sin(x)csc^{4}(x)}{(csc(x))cos^{2}(x)} - \frac{3{csc(x)}^{cot(x)}ln^{2}(csc(x))cos(x)csc^{4}(x)}{cos^{2}(x)} - \frac{3{csc(x)}^{cot(x)}ln^{2}(csc(x))sin(x)*2sin(x)csc^{4}(x)}{cos^{3}(x)} - \frac{3{csc(x)}^{cot(x)}ln^{2}(csc(x))sin(x)*-4csc^{4}(x)cot(x)}{cos^{2}(x)} + \frac{3({csc(x)}^{cot(x)}((-csc^{2}(x))ln(csc(x)) + \frac{(cot(x))(-csc(x)cot(x))}{(csc(x))}))ln^{2}(csc(x))cot^{2}(x)csc^{4}(x)}{cos(x)} + \frac{3{csc(x)}^{cot(x)}*2ln(csc(x))*-csc(x)cot(x)cot^{2}(x)csc^{4}(x)}{(csc(x))cos(x)} + \frac{3{csc(x)}^{cot(x)}ln^{2}(csc(x))sin(x)cot^{2}(x)csc^{4}(x)}{cos^{2}(x)} + \frac{3{csc(x)}^{cot(x)}ln^{2}(csc(x))*-2cot(x)csc^{2}(x)csc^{4}(x)}{cos(x)} + \frac{3{csc(x)}^{cot(x)}ln^{2}(csc(x))cot^{2}(x)*-4csc^{4}(x)cot(x)}{cos(x)} + \frac{3({csc(x)}^{cot(x)}((-csc^{2}(x))ln(csc(x)) + \frac{(cot(x))(-csc(x)cot(x))}{(csc(x))}))ln(csc(x))cot^{4}(x)csc^{2}(x)}{cos(x)} + \frac{3{csc(x)}^{cot(x)}*-csc(x)cot(x)cot^{4}(x)csc^{2}(x)}{(csc(x))cos(x)} + \frac{3{csc(x)}^{cot(x)}ln(csc(x))sin(x)cot^{4}(x)csc^{2}(x)}{cos^{2}(x)} + \frac{3{csc(x)}^{cot(x)}ln(csc(x))*-4cot^{3}(x)csc^{2}(x)csc^{2}(x)}{cos(x)} + \frac{3{csc(x)}^{cot(x)}ln(csc(x))cot^{4}(x)*-2csc^{2}(x)cot(x)}{cos(x)} + \frac{9({csc(x)}^{cot(x)}((-csc^{2}(x))ln(csc(x)) + \frac{(cot(x))(-csc(x)cot(x))}{(csc(x))}))cot^{3}(x)csc^{2}(x)}{cos(x)} + \frac{9{csc(x)}^{cot(x)}sin(x)cot^{3}(x)csc^{2}(x)}{cos^{2}(x)} + \frac{9{csc(x)}^{cot(x)}*-3cot^{2}(x)csc^{2}(x)csc^{2}(x)}{cos(x)} + \frac{9{csc(x)}^{cot(x)}cot^{3}(x)*-2csc^{2}(x)cot(x)}{cos(x)} - \frac{6({csc(x)}^{cot(x)}((-csc^{2}(x))ln(csc(x)) + \frac{(cot(x))(-csc(x)cot(x))}{(csc(x))}))ln(csc(x))sin(x)cot^{2}(x)csc^{2}(x)}{cos^{2}(x)} - \frac{6{csc(x)}^{cot(x)}*-csc(x)cot(x)sin(x)cot^{2}(x)csc^{2}(x)}{(csc(x))cos^{2}(x)} - \frac{6{csc(x)}^{cot(x)}ln(csc(x))cos(x)cot^{2}(x)csc^{2}(x)}{cos^{2}(x)} - \frac{6{csc(x)}^{cot(x)}ln(csc(x))sin(x)*2sin(x)cot^{2}(x)csc^{2}(x)}{cos^{3}(x)} - \frac{6{csc(x)}^{cot(x)}ln(csc(x))sin(x)*-2cot(x)csc^{2}(x)csc^{2}(x)}{cos^{2}(x)} - \frac{6{csc(x)}^{cot(x)}ln(csc(x))sin(x)cot^{2}(x)*-2csc^{2}(x)cot(x)}{cos^{2}(x)} + \frac{6({csc(x)}^{cot(x)}((-csc^{2}(x))ln(csc(x)) + \frac{(cot(x))(-csc(x)cot(x))}{(csc(x))}))ln^{2}(csc(x))cot(x)csc^{4}(x)}{cos(x)} + \frac{6{csc(x)}^{cot(x)}*2ln(csc(x))*-csc(x)cot(x)cot(x)csc^{4}(x)}{(csc(x))cos(x)} + \frac{6{csc(x)}^{cot(x)}ln^{2}(csc(x))sin(x)cot(x)csc^{4}(x)}{cos^{2}(x)} + \frac{6{csc(x)}^{cot(x)}ln^{2}(csc(x))*-csc^{2}(x)csc^{4}(x)}{cos(x)} + \frac{6{csc(x)}^{cot(x)}ln^{2}(csc(x))cot(x)*-4csc^{4}(x)cot(x)}{cos(x)} + \frac{6({csc(x)}^{cot(x)}((-csc^{2}(x))ln(csc(x)) + \frac{(cot(x))(-csc(x)cot(x))}{(csc(x))}))ln(csc(x))cot^{3}(x)csc^{2}(x)}{cos(x)} + \frac{6{csc(x)}^{cot(x)}*-csc(x)cot(x)cot^{3}(x)csc^{2}(x)}{(csc(x))cos(x)} + \frac{6{csc(x)}^{cot(x)}ln(csc(x))sin(x)cot^{3}(x)csc^{2}(x)}{cos^{2}(x)} + \frac{6{csc(x)}^{cot(x)}ln(csc(x))*-3cot^{2}(x)csc^{2}(x)csc^{2}(x)}{cos(x)} + \frac{6{csc(x)}^{cot(x)}ln(csc(x))cot^{3}(x)*-2csc^{2}(x)cot(x)}{cos(x)} - \frac{9({csc(x)}^{cot(x)}((-csc^{2}(x))ln(csc(x)) + \frac{(cot(x))(-csc(x)cot(x))}{(csc(x))}))sin(x)cot(x)csc^{2}(x)}{cos^{2}(x)} - \frac{9{csc(x)}^{cot(x)}cos(x)cot(x)csc^{2}(x)}{cos^{2}(x)} - \frac{9{csc(x)}^{cot(x)}sin(x)*2sin(x)cot(x)csc^{2}(x)}{cos^{3}(x)} - \frac{9{csc(x)}^{cot(x)}sin(x)*-csc^{2}(x)csc^{2}(x)}{cos^{2}(x)} - \frac{9{csc(x)}^{cot(x)}sin(x)cot(x)*-2csc^{2}(x)cot(x)}{cos^{2}(x)} + \frac{3({csc(x)}^{cot(x)}((-csc^{2}(x))ln(csc(x)) + \frac{(cot(x))(-csc(x)cot(x))}{(csc(x))}))csc^{4}(x)}{cos(x)} + \frac{3{csc(x)}^{cot(x)}sin(x)csc^{4}(x)}{cos^{2}(x)} + \frac{3{csc(x)}^{cot(x)}*-4csc^{4}(x)cot(x)}{cos(x)} + \frac{8({csc(x)}^{cot(x)}((-csc^{2}(x))ln(csc(x)) + \frac{(cot(x))(-csc(x)cot(x))}{(csc(x))}))cot^{2}(x)csc^{2}(x)}{cos(x)} + \frac{8{csc(x)}^{cot(x)}sin(x)cot^{2}(x)csc^{2}(x)}{cos^{2}(x)} + \frac{8{csc(x)}^{cot(x)}*-2cot(x)csc^{2}(x)csc^{2}(x)}{cos(x)} + \frac{8{csc(x)}^{cot(x)}cot^{2}(x)*-2csc^{2}(x)cot(x)}{cos(x)} + \frac{3({csc(x)}^{cot(x)}((-csc^{2}(x))ln(csc(x)) + \frac{(cot(x))(-csc(x)cot(x))}{(csc(x))}))ln(csc(x))csc^{2}(x)}{cos(x)} + \frac{3{csc(x)}^{cot(x)}*-csc(x)cot(x)csc^{2}(x)}{(csc(x))cos(x)} + \frac{3{csc(x)}^{cot(x)}ln(csc(x))sin(x)csc^{2}(x)}{cos^{2}(x)} + \frac{3{csc(x)}^{cot(x)}ln(csc(x))*-2csc^{2}(x)cot(x)}{cos(x)} + \frac{6({csc(x)}^{cot(x)}((-csc^{2}(x))ln(csc(x)) + \frac{(cot(x))(-csc(x)cot(x))}{(csc(x))}))ln(csc(x))sin^{2}(x)csc^{2}(x)}{cos^{3}(x)} + \frac{6{csc(x)}^{cot(x)}*-csc(x)cot(x)sin^{2}(x)csc^{2}(x)}{(csc(x))cos^{3}(x)} + \frac{6{csc(x)}^{cot(x)}ln(csc(x))*2sin(x)cos(x)csc^{2}(x)}{cos^{3}(x)} + \frac{6{csc(x)}^{cot(x)}ln(csc(x))sin^{2}(x)*3sin(x)csc^{2}(x)}{cos^{4}(x)} + \frac{6{csc(x)}^{cot(x)}ln(csc(x))sin^{2}(x)*-2csc^{2}(x)cot(x)}{cos^{3}(x)} - \frac{6({csc(x)}^{cot(x)}((-csc^{2}(x))ln(csc(x)) + \frac{(cot(x))(-csc(x)cot(x))}{(csc(x))}))ln(csc(x))sin(x)cot(x)csc^{2}(x)}{cos^{2}(x)} - \frac{6{csc(x)}^{cot(x)}*-csc(x)cot(x)sin(x)cot(x)csc^{2}(x)}{(csc(x))cos^{2}(x)} - \frac{6{csc(x)}^{cot(x)}ln(csc(x))cos(x)cot(x)csc^{2}(x)}{cos^{2}(x)} - \frac{6{csc(x)}^{cot(x)}ln(csc(x))sin(x)*2sin(x)cot(x)csc^{2}(x)}{cos^{3}(x)} - \frac{6{csc(x)}^{cot(x)}ln(csc(x))sin(x)*-csc^{2}(x)csc^{2}(x)}{cos^{2}(x)} - \frac{6{csc(x)}^{cot(x)}ln(csc(x))sin(x)cot(x)*-2csc^{2}(x)cot(x)}{cos^{2}(x)} + \frac{2({csc(x)}^{cot(x)}((-csc^{2}(x))ln(csc(x)) + \frac{(cot(x))(-csc(x)cot(x))}{(csc(x))}))ln(csc(x))csc^{4}(x)}{cos(x)} + \frac{2{csc(x)}^{cot(x)}*-csc(x)cot(x)csc^{4}(x)}{(csc(x))cos(x)} + \frac{2{csc(x)}^{cot(x)}ln(csc(x))sin(x)csc^{4}(x)}{cos^{2}(x)} + \frac{2{csc(x)}^{cot(x)}ln(csc(x))*-4csc^{4}(x)cot(x)}{cos(x)} + \frac{4({csc(x)}^{cot(x)}((-csc^{2}(x))ln(csc(x)) + \frac{(cot(x))(-csc(x)cot(x))}{(csc(x))}))ln(csc(x))cot^{2}(x)csc^{2}(x)}{cos(x)} + \frac{4{csc(x)}^{cot(x)}*-csc(x)cot(x)cot^{2}(x)csc^{2}(x)}{(csc(x))cos(x)} + \frac{4{csc(x)}^{cot(x)}ln(csc(x))sin(x)cot^{2}(x)csc^{2}(x)}{cos^{2}(x)} + \frac{4{csc(x)}^{cot(x)}ln(csc(x))*-2cot(x)csc^{2}(x)csc^{2}(x)}{cos(x)} + \frac{4{csc(x)}^{cot(x)}ln(csc(x))cot^{2}(x)*-2csc^{2}(x)cot(x)}{cos(x)} + \frac{({csc(x)}^{cot(x)}((-csc^{2}(x))ln(csc(x)) + \frac{(cot(x))(-csc(x)cot(x))}{(csc(x))}))cot^{6}(x)}{cos(x)} + \frac{{csc(x)}^{cot(x)}sin(x)cot^{6}(x)}{cos^{2}(x)} + \frac{{csc(x)}^{cot(x)}*-6cot^{5}(x)csc^{2}(x)}{cos(x)} - \frac{3({csc(x)}^{cot(x)}((-csc^{2}(x))ln(csc(x)) + \frac{(cot(x))(-csc(x)cot(x))}{(csc(x))}))sin(x)cot^{4}(x)}{cos^{2}(x)} - \frac{3{csc(x)}^{cot(x)}cos(x)cot^{4}(x)}{cos^{2}(x)} - \frac{3{csc(x)}^{cot(x)}sin(x)*2sin(x)cot^{4}(x)}{cos^{3}(x)} - \frac{3{csc(x)}^{cot(x)}sin(x)*-4cot^{3}(x)csc^{2}(x)}{cos^{2}(x)} + \frac{3({csc(x)}^{cot(x)}((-csc^{2}(x))ln(csc(x)) + \frac{(cot(x))(-csc(x)cot(x))}{(csc(x))}))cot^{2}(x)}{cos(x)} + \frac{3{csc(x)}^{cot(x)}sin(x)cot^{2}(x)}{cos^{2}(x)} + \frac{3{csc(x)}^{cot(x)}*-2cot(x)csc^{2}(x)}{cos(x)} + \frac{6({csc(x)}^{cot(x)}((-csc^{2}(x))ln(csc(x)) + \frac{(cot(x))(-csc(x)cot(x))}{(csc(x))}))sin^{2}(x)cot^{2}(x)}{cos^{3}(x)} + \frac{6{csc(x)}^{cot(x)}*2sin(x)cos(x)cot^{2}(x)}{cos^{3}(x)} + \frac{6{csc(x)}^{cot(x)}sin^{2}(x)*3sin(x)cot^{2}(x)}{cos^{4}(x)} + \frac{6{csc(x)}^{cot(x)}sin^{2}(x)*-2cot(x)csc^{2}(x)}{cos^{3}(x)} - \frac{5({csc(x)}^{cot(x)}((-csc^{2}(x))ln(csc(x)) + \frac{(cot(x))(-csc(x)cot(x))}{(csc(x))}))sin(x)}{cos^{2}(x)} - \frac{5{csc(x)}^{cot(x)}cos(x)}{cos^{2}(x)} - \frac{5{csc(x)}^{cot(x)}sin(x)*2sin(x)}{cos^{3}(x)} - \frac{6({csc(x)}^{cot(x)}((-csc^{2}(x))ln(csc(x)) + \frac{(cot(x))(-csc(x)cot(x))}{(csc(x))}))sin^{3}(x)}{cos^{4}(x)} - \frac{6{csc(x)}^{cot(x)}*3sin^{2}(x)cos(x)}{cos^{4}(x)} - \frac{6{csc(x)}^{cot(x)}sin^{3}(x)*4sin(x)}{cos^{5}(x)}\\=&\frac{-{csc(x)}^{cot(x)}ln^{4}(csc(x))csc^{8}(x)}{cos(x)} - \frac{36{csc(x)}^{cot(x)}ln(csc(x))cot^{3}(x)csc^{4}(x)}{cos(x)} - \frac{18{csc(x)}^{cot(x)}ln^{2}(csc(x))cot(x)csc^{6}(x)}{cos(x)} + \frac{8{csc(x)}^{cot(x)}ln(csc(x))sin(x)csc^{4}(x)}{cos^{2}(x)} - \frac{39{csc(x)}^{cot(x)}cot^{2}(x)csc^{4}(x)}{cos(x)} + \frac{36{csc(x)}^{cot(x)}ln(csc(x))sin(x)cot(x)csc^{4}(x)}{cos^{2}(x)} - \frac{12{csc(x)}^{cot(x)}ln(csc(x))csc^{6}(x)}{cos(x)} - \frac{76{csc(x)}^{cot(x)}ln(csc(x))cot^{2}(x)csc^{4}(x)}{cos(x)} + \frac{4{csc(x)}^{cot(x)}ln^{3}(csc(x))sin(x)csc^{6}(x)}{cos^{2}(x)} - \frac{6{csc(x)}^{cot(x)}ln^{2}(csc(x))csc^{4}(x)}{cos(x)} - \frac{12{csc(x)}^{cot(x)}ln^{2}(csc(x))sin^{2}(x)csc^{4}(x)}{cos^{3}(x)} + \frac{24{csc(x)}^{cot(x)}ln^{2}(csc(x))sin(x)cot(x)csc^{4}(x)}{cos^{2}(x)} - \frac{4{csc(x)}^{cot(x)}ln^{3}(csc(x))cot^{2}(x)csc^{6}(x)}{cos(x)} - \frac{6{csc(x)}^{cot(x)}ln^{2}(csc(x))cot^{4}(x)csc^{4}(x)}{cos(x)} + \frac{12{csc(x)}^{cot(x)}ln^{2}(csc(x))sin(x)cot^{2}(x)csc^{4}(x)}{cos^{2}(x)} - \frac{24{csc(x)}^{cot(x)}ln^{2}(csc(x))cot^{3}(x)csc^{4}(x)}{cos(x)} - \frac{4{csc(x)}^{cot(x)}ln(csc(x))cot^{6}(x)csc^{2}(x)}{cos(x)} - \frac{18{csc(x)}^{cot(x)}cot^{5}(x)csc^{2}(x)}{cos(x)} + \frac{12{csc(x)}^{cot(x)}ln(csc(x))sin(x)cot^{4}(x)csc^{2}(x)}{cos^{2}(x)} - \frac{12{csc(x)}^{cot(x)}ln(csc(x))cot^{5}(x)csc^{2}(x)}{cos(x)} + \frac{36{csc(x)}^{cot(x)}sin(x)cot^{3}(x)csc^{2}(x)}{cos^{2}(x)} - \frac{32{csc(x)}^{cot(x)}cot^{4}(x)csc^{2}(x)}{cos(x)} - \frac{12{csc(x)}^{cot(x)}ln(csc(x))cot^{2}(x)csc^{2}(x)}{cos(x)} - \frac{24{csc(x)}^{cot(x)}ln(csc(x))sin^{2}(x)cot^{2}(x)csc^{2}(x)}{cos^{3}(x)} + \frac{24{csc(x)}^{cot(x)}ln(csc(x))sin(x)cot^{3}(x)csc^{2}(x)}{cos^{2}(x)} - \frac{12{csc(x)}^{cot(x)}ln^{3}(csc(x))cot(x)csc^{6}(x)}{cos(x)} - \frac{8{csc(x)}^{cot(x)}ln^{2}(csc(x))csc^{6}(x)}{cos(x)} - \frac{28{csc(x)}^{cot(x)}ln^{2}(csc(x))cot^{2}(x)csc^{4}(x)}{cos(x)} - \frac{16{csc(x)}^{cot(x)}ln(csc(x))cot(x)csc^{4}(x)}{cos(x)} - \frac{16{csc(x)}^{cot(x)}ln(csc(x))cot^{4}(x)csc^{2}(x)}{cos(x)} - \frac{18{csc(x)}^{cot(x)}cot(x)csc^{2}(x)}{cos(x)} - \frac{36{csc(x)}^{cot(x)}sin^{2}(x)cot(x)csc^{2}(x)}{cos^{3}(x)} + \frac{12{csc(x)}^{cot(x)}sin(x)csc^{4}(x)}{cos^{2}(x)} + \frac{32{csc(x)}^{cot(x)}sin(x)cot^{2}(x)csc^{2}(x)}{cos^{2}(x)} - \frac{30{csc(x)}^{cot(x)}cot(x)csc^{4}(x)}{cos(x)} - \frac{20{csc(x)}^{cot(x)}cot^{3}(x)csc^{2}(x)}{cos(x)} + \frac{20{csc(x)}^{cot(x)}ln(csc(x))sin(x)csc^{2}(x)}{cos^{2}(x)} - \frac{12{csc(x)}^{cot(x)}ln(csc(x))cot(x)csc^{2}(x)}{cos(x)} + \frac{24{csc(x)}^{cot(x)}ln(csc(x))sin^{3}(x)csc^{2}(x)}{cos^{4}(x)} - \frac{24{csc(x)}^{cot(x)}ln(csc(x))sin^{2}(x)cot(x)csc^{2}(x)}{cos^{3}(x)} + \frac{16{csc(x)}^{cot(x)}ln(csc(x))sin(x)cot^{2}(x)csc^{2}(x)}{cos^{2}(x)} - \frac{8{csc(x)}^{cot(x)}ln(csc(x))cot^{3}(x)csc^{2}(x)}{cos(x)} - \frac{{csc(x)}^{cot(x)}cot^{8}(x)}{cos(x)} + \frac{4{csc(x)}^{cot(x)}sin(x)cot^{6}(x)}{cos^{2}(x)} - \frac{6{csc(x)}^{cot(x)}cot^{4}(x)}{cos(x)} - \frac{12{csc(x)}^{cot(x)}sin^{2}(x)cot^{4}(x)}{cos^{3}(x)} + \frac{20{csc(x)}^{cot(x)}sin(x)cot^{2}(x)}{cos^{2}(x)} + \frac{24{csc(x)}^{cot(x)}sin^{3}(x)cot^{2}(x)}{cos^{4}(x)} - \frac{5{csc(x)}^{cot(x)}}{cos(x)} - \frac{28{csc(x)}^{cot(x)}sin^{2}(x)}{cos^{3}(x)} - \frac{24{csc(x)}^{cot(x)}sin^{4}(x)}{cos^{5}(x)}\\ \end{split}\end{equation} \]