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 3 questions in this calculation: for each question, the 4 derivative of x is calculated.
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/3]Find\ the\ 4th\ derivative\ of\ function\ sec(x)sec(x) - sec(sec(x))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = sec^{2}(x) - sec(sec(x))\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( sec^{2}(x) - sec(sec(x))\right)}{dx}\\=&2sec^{2}(x)tan(x) - sec(sec(x))tan(sec(x))sec(x)tan(x)\\=&2tan(x)sec^{2}(x) - tan(x)tan(sec(x))sec(sec(x))sec(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( 2tan(x)sec^{2}(x) - tan(x)tan(sec(x))sec(sec(x))sec(x)\right)}{dx}\\=&2sec^{2}(x)(1)sec^{2}(x) + 2tan(x)*2sec^{2}(x)tan(x) - sec^{2}(x)(1)tan(sec(x))sec(sec(x))sec(x) - tan(x)sec^{2}(sec(x))(sec(x)tan(x))sec(sec(x))sec(x) - tan(x)tan(sec(x))sec(sec(x))tan(sec(x))sec(x)tan(x)sec(x) - tan(x)tan(sec(x))sec(sec(x))sec(x)tan(x)\\=&2sec^{4}(x) - tan^{2}(x)sec^{3}(sec(x))sec^{2}(x) - tan(sec(x))sec^{3}(x)sec(sec(x)) + 4tan^{2}(x)sec^{2}(x) - tan^{2}(sec(x))tan^{2}(x)sec(sec(x))sec^{2}(x) - tan(sec(x))tan^{2}(x)sec(sec(x))sec(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 2sec^{4}(x) - tan^{2}(x)sec^{3}(sec(x))sec^{2}(x) - tan(sec(x))sec^{3}(x)sec(sec(x)) + 4tan^{2}(x)sec^{2}(x) - tan^{2}(sec(x))tan^{2}(x)sec(sec(x))sec^{2}(x) - tan(sec(x))tan^{2}(x)sec(sec(x))sec(x)\right)}{dx}\\=&2*4sec^{4}(x)tan(x) - 2tan(x)sec^{2}(x)(1)sec^{3}(sec(x))sec^{2}(x) - tan^{2}(x)*3sec^{3}(sec(x))tan(sec(x))sec(x)tan(x)sec^{2}(x) - tan^{2}(x)sec^{3}(sec(x))*2sec^{2}(x)tan(x) - sec^{2}(sec(x))(sec(x)tan(x))sec^{3}(x)sec(sec(x)) - tan(sec(x))*3sec^{3}(x)tan(x)sec(sec(x)) - tan(sec(x))sec^{3}(x)sec(sec(x))tan(sec(x))sec(x)tan(x) + 4*2tan(x)sec^{2}(x)(1)sec^{2}(x) + 4tan^{2}(x)*2sec^{2}(x)tan(x) - 2tan(sec(x))sec^{2}(sec(x))(sec(x)tan(x))tan^{2}(x)sec(sec(x))sec^{2}(x) - tan^{2}(sec(x))*2tan(x)sec^{2}(x)(1)sec(sec(x))sec^{2}(x) - tan^{2}(sec(x))tan^{2}(x)sec(sec(x))tan(sec(x))sec(x)tan(x)sec^{2}(x) - tan^{2}(sec(x))tan^{2}(x)sec(sec(x))*2sec^{2}(x)tan(x) - sec^{2}(sec(x))(sec(x)tan(x))tan^{2}(x)sec(sec(x))sec(x) - tan(sec(x))*2tan(x)sec^{2}(x)(1)sec(sec(x))sec(x) - tan(sec(x))tan^{2}(x)sec(sec(x))tan(sec(x))sec(x)tan(x)sec(x) - tan(sec(x))tan^{2}(x)sec(sec(x))sec(x)tan(x)\\=& - tan(x)sec^{3}(sec(x))sec^{4}(x) - 2tan(x)sec^{4}(x)sec^{3}(sec(x)) - 2tan^{3}(x)tan(sec(x))sec^{3}(x)sec^{3}(sec(x)) - tan^{3}(x)sec^{2}(x)sec^{3}(sec(x)) - 2tan^{3}(x)sec^{3}(sec(x))sec^{2}(x) - 2tan(x)tan^{2}(sec(x))sec^{4}(x)sec(sec(x)) - tan^{2}(sec(x))tan(x)sec^{4}(x)sec(sec(x)) - 5tan(x)tan(sec(x))sec^{3}(x)sec(sec(x)) - 3tan^{3}(x)tan(sec(x))sec^{3}(sec(x))sec^{3}(x) - tan^{3}(sec(x))tan^{3}(x)sec(sec(x))sec^{3}(x) - 3tan^{2}(sec(x))tan^{3}(x)sec(sec(x))sec^{2}(x) + 16tan(x)sec^{4}(x) + 8tan^{3}(x)sec^{2}(x) - tan^{3}(x)tan(sec(x))sec(sec(x))sec(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( - tan(x)sec^{3}(sec(x))sec^{4}(x) - 2tan(x)sec^{4}(x)sec^{3}(sec(x)) - 2tan^{3}(x)tan(sec(x))sec^{3}(x)sec^{3}(sec(x)) - tan^{3}(x)sec^{2}(x)sec^{3}(sec(x)) - 2tan^{3}(x)sec^{3}(sec(x))sec^{2}(x) - 2tan(x)tan^{2}(sec(x))sec^{4}(x)sec(sec(x)) - tan^{2}(sec(x))tan(x)sec^{4}(x)sec(sec(x)) - 5tan(x)tan(sec(x))sec^{3}(x)sec(sec(x)) - 3tan^{3}(x)tan(sec(x))sec^{3}(sec(x))sec^{3}(x) - tan^{3}(sec(x))tan^{3}(x)sec(sec(x))sec^{3}(x) - 3tan^{2}(sec(x))tan^{3}(x)sec(sec(x))sec^{2}(x) + 16tan(x)sec^{4}(x) + 8tan^{3}(x)sec^{2}(x) - tan^{3}(x)tan(sec(x))sec(sec(x))sec(x)\right)}{dx}\\=& - sec^{2}(x)(1)sec^{3}(sec(x))sec^{4}(x) - tan(x)*3sec^{3}(sec(x))tan(sec(x))sec(x)tan(x)sec^{4}(x) - tan(x)sec^{3}(sec(x))*4sec^{4}(x)tan(x) - 2sec^{2}(x)(1)sec^{4}(x)sec^{3}(sec(x)) - 2tan(x)*4sec^{4}(x)tan(x)sec^{3}(sec(x)) - 2tan(x)sec^{4}(x)*3sec^{3}(sec(x))tan(sec(x))sec(x)tan(x) - 2*3tan^{2}(x)sec^{2}(x)(1)tan(sec(x))sec^{3}(x)sec^{3}(sec(x)) - 2tan^{3}(x)sec^{2}(sec(x))(sec(x)tan(x))sec^{3}(x)sec^{3}(sec(x)) - 2tan^{3}(x)tan(sec(x))*3sec^{3}(x)tan(x)sec^{3}(sec(x)) - 2tan^{3}(x)tan(sec(x))sec^{3}(x)*3sec^{3}(sec(x))tan(sec(x))sec(x)tan(x) - 3tan^{2}(x)sec^{2}(x)(1)sec^{2}(x)sec^{3}(sec(x)) - tan^{3}(x)*2sec^{2}(x)tan(x)sec^{3}(sec(x)) - tan^{3}(x)sec^{2}(x)*3sec^{3}(sec(x))tan(sec(x))sec(x)tan(x) - 2*3tan^{2}(x)sec^{2}(x)(1)sec^{3}(sec(x))sec^{2}(x) - 2tan^{3}(x)*3sec^{3}(sec(x))tan(sec(x))sec(x)tan(x)sec^{2}(x) - 2tan^{3}(x)sec^{3}(sec(x))*2sec^{2}(x)tan(x) - 2sec^{2}(x)(1)tan^{2}(sec(x))sec^{4}(x)sec(sec(x)) - 2tan(x)*2tan(sec(x))sec^{2}(sec(x))(sec(x)tan(x))sec^{4}(x)sec(sec(x)) - 2tan(x)tan^{2}(sec(x))*4sec^{4}(x)tan(x)sec(sec(x)) - 2tan(x)tan^{2}(sec(x))sec^{4}(x)sec(sec(x))tan(sec(x))sec(x)tan(x) - 2tan(sec(x))sec^{2}(sec(x))(sec(x)tan(x))tan(x)sec^{4}(x)sec(sec(x)) - tan^{2}(sec(x))sec^{2}(x)(1)sec^{4}(x)sec(sec(x)) - tan^{2}(sec(x))tan(x)*4sec^{4}(x)tan(x)sec(sec(x)) - tan^{2}(sec(x))tan(x)sec^{4}(x)sec(sec(x))tan(sec(x))sec(x)tan(x) - 5sec^{2}(x)(1)tan(sec(x))sec^{3}(x)sec(sec(x)) - 5tan(x)sec^{2}(sec(x))(sec(x)tan(x))sec^{3}(x)sec(sec(x)) - 5tan(x)tan(sec(x))*3sec^{3}(x)tan(x)sec(sec(x)) - 5tan(x)tan(sec(x))sec^{3}(x)sec(sec(x))tan(sec(x))sec(x)tan(x) - 3*3tan^{2}(x)sec^{2}(x)(1)tan(sec(x))sec^{3}(sec(x))sec^{3}(x) - 3tan^{3}(x)sec^{2}(sec(x))(sec(x)tan(x))sec^{3}(sec(x))sec^{3}(x) - 3tan^{3}(x)tan(sec(x))*3sec^{3}(sec(x))tan(sec(x))sec(x)tan(x)sec^{3}(x) - 3tan^{3}(x)tan(sec(x))sec^{3}(sec(x))*3sec^{3}(x)tan(x) - 3tan^{2}(sec(x))sec^{2}(sec(x))(sec(x)tan(x))tan^{3}(x)sec(sec(x))sec^{3}(x) - tan^{3}(sec(x))*3tan^{2}(x)sec^{2}(x)(1)sec(sec(x))sec^{3}(x) - tan^{3}(sec(x))tan^{3}(x)sec(sec(x))tan(sec(x))sec(x)tan(x)sec^{3}(x) - tan^{3}(sec(x))tan^{3}(x)sec(sec(x))*3sec^{3}(x)tan(x) - 3*2tan(sec(x))sec^{2}(sec(x))(sec(x)tan(x))tan^{3}(x)sec(sec(x))sec^{2}(x) - 3tan^{2}(sec(x))*3tan^{2}(x)sec^{2}(x)(1)sec(sec(x))sec^{2}(x) - 3tan^{2}(sec(x))tan^{3}(x)sec(sec(x))tan(sec(x))sec(x)tan(x)sec^{2}(x) - 3tan^{2}(sec(x))tan^{3}(x)sec(sec(x))*2sec^{2}(x)tan(x) + 16sec^{2}(x)(1)sec^{4}(x) + 16tan(x)*4sec^{4}(x)tan(x) + 8*3tan^{2}(x)sec^{2}(x)(1)sec^{2}(x) + 8tan^{3}(x)*2sec^{2}(x)tan(x) - 3tan^{2}(x)sec^{2}(x)(1)tan(sec(x))sec(sec(x))sec(x) - tan^{3}(x)sec^{2}(sec(x))(sec(x)tan(x))sec(sec(x))sec(x) - tan^{3}(x)tan(sec(x))sec(sec(x))tan(sec(x))sec(x)tan(x)sec(x) - tan^{3}(x)tan(sec(x))sec(sec(x))sec(x)tan(x)\\=& - 3sec^{6}(x)sec^{3}(sec(x)) - 7tan^{2}(x)tan(sec(x))sec^{3}(sec(x))sec^{5}(x) - 5tan^{4}(x)sec^{5}(sec(x))sec^{4}(x) - 9tan^{2}(x)sec^{3}(sec(x))sec^{4}(x) - 23tan^{2}(x)tan(sec(x))sec^{5}(x)sec^{3}(sec(x)) - 5tan^{4}(x)sec^{3}(sec(x))sec^{2}(x) - 15tan^{4}(x)tan(sec(x))sec^{3}(x)sec^{3}(sec(x)) - 6tan^{2}(sec(x))tan^{4}(x)sec^{4}(x)sec^{3}(sec(x)) - 17tan^{2}(x)sec^{4}(x)sec^{3}(sec(x)) - 3tan^{4}(x)tan^{2}(sec(x))sec^{4}(x)sec^{3}(sec(x)) - 3tan^{2}(x)tan^{3}(sec(x))sec^{5}(x)sec(sec(x)) - 3tan^{2}(sec(x))sec^{6}(x)sec(sec(x)) - 17tan^{2}(x)tan^{2}(sec(x))sec^{4}(x)sec(sec(x)) - 3tan^{3}(sec(x))tan^{2}(x)sec^{5}(x)sec(sec(x)) - 9tan^{2}(sec(x))tan^{2}(x)sec^{4}(x)sec(sec(x)) - 5tan(sec(x))sec^{5}(x)sec(sec(x)) - 18tan^{2}(x)tan(sec(x))sec^{3}(x)sec(sec(x)) - 9tan^{2}(sec(x))tan^{4}(x)sec^{3}(sec(x))sec^{4}(x) - 15tan^{4}(x)tan(sec(x))sec^{3}(sec(x))sec^{3}(x) - tan^{4}(sec(x))tan^{4}(x)sec(sec(x))sec^{4}(x) - 6tan^{3}(sec(x))tan^{4}(x)sec(sec(x))sec^{3}(x) - 7tan^{2}(sec(x))tan^{4}(x)sec(sec(x))sec^{2}(x) + 16sec^{6}(x) - 2tan^{4}(x)sec^{2}(x)sec^{3}(sec(x)) + 88tan^{2}(x)sec^{4}(x) + 16tan^{4}(x)sec^{2}(x) - tan(sec(x))tan^{4}(x)sec(sec(x))sec(x)\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/3]Find\ the\ 4th\ derivative\ of\ function\ csc(x)csc(x) - csc(csc(x))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = csc^{2}(x) - csc(csc(x))\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( csc^{2}(x) - csc(csc(x))\right)}{dx}\\=&-2csc^{2}(x)cot(x) - -csc(csc(x))cot(csc(x))*-csc(x)cot(x)\\=&-2cot(x)csc^{2}(x) - cot(x)cot(csc(x))csc(csc(x))csc(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( -2cot(x)csc^{2}(x) - cot(x)cot(csc(x))csc(csc(x))csc(x)\right)}{dx}\\=&-2*-csc^{2}(x)csc^{2}(x) - 2cot(x)*-2csc^{2}(x)cot(x) - -csc^{2}(x)cot(csc(x))csc(csc(x))csc(x) - cot(x)*-csc^{2}(csc(x))*-csc(x)cot(x)csc(csc(x))csc(x) - cot(x)cot(csc(x))*-csc(csc(x))cot(csc(x))*-csc(x)cot(x)csc(x) - cot(x)cot(csc(x))csc(csc(x))*-csc(x)cot(x)\\=&2csc^{4}(x) + cot(csc(x))csc^{3}(x)csc(csc(x)) - cot^{2}(x)csc^{3}(csc(x))csc^{2}(x) + 4cot^{2}(x)csc^{2}(x) - cot^{2}(csc(x))cot^{2}(x)csc(csc(x))csc^{2}(x) + cot^{2}(x)cot(csc(x))csc(csc(x))csc(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 2csc^{4}(x) + cot(csc(x))csc^{3}(x)csc(csc(x)) - cot^{2}(x)csc^{3}(csc(x))csc^{2}(x) + 4cot^{2}(x)csc^{2}(x) - cot^{2}(csc(x))cot^{2}(x)csc(csc(x))csc^{2}(x) + cot^{2}(x)cot(csc(x))csc(csc(x))csc(x)\right)}{dx}\\=&2*-4csc^{4}(x)cot(x) + -csc^{2}(csc(x))*-csc(x)cot(x)csc^{3}(x)csc(csc(x)) + cot(csc(x))*-3csc^{3}(x)cot(x)csc(csc(x)) + cot(csc(x))csc^{3}(x)*-csc(csc(x))cot(csc(x))*-csc(x)cot(x) - -2cot(x)csc^{2}(x)csc^{3}(csc(x))csc^{2}(x) - cot^{2}(x)*-3csc^{3}(csc(x))cot(csc(x))*-csc(x)cot(x)csc^{2}(x) - cot^{2}(x)csc^{3}(csc(x))*-2csc^{2}(x)cot(x) + 4*-2cot(x)csc^{2}(x)csc^{2}(x) + 4cot^{2}(x)*-2csc^{2}(x)cot(x) - -2cot(csc(x))csc^{2}(csc(x))*-csc(x)cot(x)cot^{2}(x)csc(csc(x))csc^{2}(x) - cot^{2}(csc(x))*-2cot(x)csc^{2}(x)csc(csc(x))csc^{2}(x) - cot^{2}(csc(x))cot^{2}(x)*-csc(csc(x))cot(csc(x))*-csc(x)cot(x)csc^{2}(x) - cot^{2}(csc(x))cot^{2}(x)csc(csc(x))*-2csc^{2}(x)cot(x) + -2cot(x)csc^{2}(x)cot(csc(x))csc(csc(x))csc(x) + cot^{2}(x)*-csc^{2}(csc(x))*-csc(x)cot(x)csc(csc(x))csc(x) + cot^{2}(x)cot(csc(x))*-csc(csc(x))cot(csc(x))*-csc(x)cot(x)csc(x) + cot^{2}(x)cot(csc(x))csc(csc(x))*-csc(x)cot(x)\\=&2cot(x)csc^{4}(x)csc^{3}(csc(x)) + cot(x)csc^{3}(csc(x))csc^{4}(x) + 2cot(x)cot^{2}(csc(x))csc^{4}(x)csc(csc(x)) - 2cot^{3}(x)cot(csc(x))csc^{3}(x)csc^{3}(csc(x)) + 3cot^{3}(x)csc^{3}(csc(x))csc^{2}(x) - 3cot(csc(x))cot^{3}(x)csc^{3}(csc(x))csc^{3}(x) - 5cot(x)cot(csc(x))csc^{3}(x)csc(csc(x)) + cot^{2}(csc(x))cot(x)csc^{4}(x)csc(csc(x)) - 16cot(x)csc^{4}(x) - cot^{3}(x)cot^{3}(csc(x))csc(csc(x))csc^{3}(x) + 3cot^{2}(csc(x))cot^{3}(x)csc(csc(x))csc^{2}(x) - 8cot^{3}(x)csc^{2}(x) - cot^{3}(x)cot(csc(x))csc(csc(x))csc(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 2cot(x)csc^{4}(x)csc^{3}(csc(x)) + cot(x)csc^{3}(csc(x))csc^{4}(x) + 2cot(x)cot^{2}(csc(x))csc^{4}(x)csc(csc(x)) - 2cot^{3}(x)cot(csc(x))csc^{3}(x)csc^{3}(csc(x)) + 3cot^{3}(x)csc^{3}(csc(x))csc^{2}(x) - 3cot(csc(x))cot^{3}(x)csc^{3}(csc(x))csc^{3}(x) - 5cot(x)cot(csc(x))csc^{3}(x)csc(csc(x)) + cot^{2}(csc(x))cot(x)csc^{4}(x)csc(csc(x)) - 16cot(x)csc^{4}(x) - cot^{3}(x)cot^{3}(csc(x))csc(csc(x))csc^{3}(x) + 3cot^{2}(csc(x))cot^{3}(x)csc(csc(x))csc^{2}(x) - 8cot^{3}(x)csc^{2}(x) - cot^{3}(x)cot(csc(x))csc(csc(x))csc(x)\right)}{dx}\\=&2*-csc^{2}(x)csc^{4}(x)csc^{3}(csc(x)) + 2cot(x)*-4csc^{4}(x)cot(x)csc^{3}(csc(x)) + 2cot(x)csc^{4}(x)*-3csc^{3}(csc(x))cot(csc(x))*-csc(x)cot(x) + -csc^{2}(x)csc^{3}(csc(x))csc^{4}(x) + cot(x)*-3csc^{3}(csc(x))cot(csc(x))*-csc(x)cot(x)csc^{4}(x) + cot(x)csc^{3}(csc(x))*-4csc^{4}(x)cot(x) + 2*-csc^{2}(x)cot^{2}(csc(x))csc^{4}(x)csc(csc(x)) + 2cot(x)*-2cot(csc(x))csc^{2}(csc(x))*-csc(x)cot(x)csc^{4}(x)csc(csc(x)) + 2cot(x)cot^{2}(csc(x))*-4csc^{4}(x)cot(x)csc(csc(x)) + 2cot(x)cot^{2}(csc(x))csc^{4}(x)*-csc(csc(x))cot(csc(x))*-csc(x)cot(x) - 2*-3cot^{2}(x)csc^{2}(x)cot(csc(x))csc^{3}(x)csc^{3}(csc(x)) - 2cot^{3}(x)*-csc^{2}(csc(x))*-csc(x)cot(x)csc^{3}(x)csc^{3}(csc(x)) - 2cot^{3}(x)cot(csc(x))*-3csc^{3}(x)cot(x)csc^{3}(csc(x)) - 2cot^{3}(x)cot(csc(x))csc^{3}(x)*-3csc^{3}(csc(x))cot(csc(x))*-csc(x)cot(x) + 3*-3cot^{2}(x)csc^{2}(x)csc^{3}(csc(x))csc^{2}(x) + 3cot^{3}(x)*-3csc^{3}(csc(x))cot(csc(x))*-csc(x)cot(x)csc^{2}(x) + 3cot^{3}(x)csc^{3}(csc(x))*-2csc^{2}(x)cot(x) - 3*-csc^{2}(csc(x))*-csc(x)cot(x)cot^{3}(x)csc^{3}(csc(x))csc^{3}(x) - 3cot(csc(x))*-3cot^{2}(x)csc^{2}(x)csc^{3}(csc(x))csc^{3}(x) - 3cot(csc(x))cot^{3}(x)*-3csc^{3}(csc(x))cot(csc(x))*-csc(x)cot(x)csc^{3}(x) - 3cot(csc(x))cot^{3}(x)csc^{3}(csc(x))*-3csc^{3}(x)cot(x) - 5*-csc^{2}(x)cot(csc(x))csc^{3}(x)csc(csc(x)) - 5cot(x)*-csc^{2}(csc(x))*-csc(x)cot(x)csc^{3}(x)csc(csc(x)) - 5cot(x)cot(csc(x))*-3csc^{3}(x)cot(x)csc(csc(x)) - 5cot(x)cot(csc(x))csc^{3}(x)*-csc(csc(x))cot(csc(x))*-csc(x)cot(x) + -2cot(csc(x))csc^{2}(csc(x))*-csc(x)cot(x)cot(x)csc^{4}(x)csc(csc(x)) + cot^{2}(csc(x))*-csc^{2}(x)csc^{4}(x)csc(csc(x)) + cot^{2}(csc(x))cot(x)*-4csc^{4}(x)cot(x)csc(csc(x)) + cot^{2}(csc(x))cot(x)csc^{4}(x)*-csc(csc(x))cot(csc(x))*-csc(x)cot(x) - 16*-csc^{2}(x)csc^{4}(x) - 16cot(x)*-4csc^{4}(x)cot(x) - -3cot^{2}(x)csc^{2}(x)cot^{3}(csc(x))csc(csc(x))csc^{3}(x) - cot^{3}(x)*-3cot^{2}(csc(x))csc^{2}(csc(x))*-csc(x)cot(x)csc(csc(x))csc^{3}(x) - cot^{3}(x)cot^{3}(csc(x))*-csc(csc(x))cot(csc(x))*-csc(x)cot(x)csc^{3}(x) - cot^{3}(x)cot^{3}(csc(x))csc(csc(x))*-3csc^{3}(x)cot(x) + 3*-2cot(csc(x))csc^{2}(csc(x))*-csc(x)cot(x)cot^{3}(x)csc(csc(x))csc^{2}(x) + 3cot^{2}(csc(x))*-3cot^{2}(x)csc^{2}(x)csc(csc(x))csc^{2}(x) + 3cot^{2}(csc(x))cot^{3}(x)*-csc(csc(x))cot(csc(x))*-csc(x)cot(x)csc^{2}(x) + 3cot^{2}(csc(x))cot^{3}(x)csc(csc(x))*-2csc^{2}(x)cot(x) - 8*-3cot^{2}(x)csc^{2}(x)csc^{2}(x) - 8cot^{3}(x)*-2csc^{2}(x)cot(x) - -3cot^{2}(x)csc^{2}(x)cot(csc(x))csc(csc(x))csc(x) - cot^{3}(x)*-csc^{2}(csc(x))*-csc(x)cot(x)csc(csc(x))csc(x) - cot^{3}(x)cot(csc(x))*-csc(csc(x))cot(csc(x))*-csc(x)cot(x)csc(x) - cot^{3}(x)cot(csc(x))csc(csc(x))*-csc(x)cot(x)\\=& - 3csc^{6}(x)csc^{3}(csc(x)) + 17cot^{2}(x)cot(csc(x))csc^{5}(x)csc^{3}(csc(x)) + 7cot(csc(x))cot^{2}(x)csc^{3}(csc(x))csc^{5}(x) - 3cot^{2}(csc(x))csc^{6}(x)csc(csc(x)) - 3cot^{2}(csc(x))cot^{4}(x)csc^{3}(csc(x))csc^{4}(x) - 17cot^{2}(x)cot^{2}(csc(x))csc^{4}(x)csc(csc(x)) + 2cot^{3}(csc(x))cot^{2}(x)csc^{5}(x)csc(csc(x)) + 4cot^{2}(x)cot^{3}(csc(x))csc^{5}(x)csc(csc(x)) - 2cot^{4}(x)csc^{5}(csc(x))csc^{4}(x) + 12cot^{4}(x)cot(csc(x))csc^{3}(x)csc^{3}(csc(x)) - 6cot^{2}(csc(x))cot^{4}(x)csc^{4}(x)csc^{3}(csc(x)) - 17cot^{2}(x)csc^{4}(x)csc^{3}(csc(x)) - 3cot^{4}(x)csc^{4}(x)csc^{5}(csc(x)) + 18cot^{2}(x)cot(csc(x))csc^{3}(x)csc(csc(x)) - 9cot^{4}(x)cot^{2}(csc(x))csc^{3}(csc(x))csc^{4}(x) + 18cot(csc(x))cot^{4}(x)csc^{3}(csc(x))csc^{3}(x) + 5cot(csc(x))csc^{5}(x)csc(csc(x)) - 9cot^{2}(x)csc^{3}(csc(x))csc^{4}(x) - 9cot^{2}(csc(x))cot^{2}(x)csc^{4}(x)csc(csc(x)) + 16csc^{6}(x) - 7cot^{4}(x)csc^{3}(csc(x))csc^{2}(x) + 6cot(csc(x))cot^{2}(x)csc^{5}(x)csc^{3}(csc(x)) - cot^{4}(csc(x))cot^{4}(x)csc(csc(x))csc^{4}(x) + 6cot^{4}(x)cot^{3}(csc(x))csc(csc(x))csc^{3}(x) - 7cot^{2}(csc(x))cot^{4}(x)csc(csc(x))csc^{2}(x) + 88cot^{2}(x)csc^{4}(x) + 16cot^{4}(x)csc^{2}(x) + cot^{4}(x)cot(csc(x))csc(csc(x))csc(x)\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[3/3]Find\ the\ 4th\ derivative\ of\ function\ cot(x)cot(x) - cot(cot(x))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = cot^{2}(x) - cot(cot(x))\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( cot^{2}(x) - cot(cot(x))\right)}{dx}\\=&-2cot(x)csc^{2}(x) - -csc^{2}(cot(x))*-csc^{2}(x)\\=&-2cot(x)csc^{2}(x) - csc^{2}(x)csc^{2}(cot(x))\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( -2cot(x)csc^{2}(x) - csc^{2}(x)csc^{2}(cot(x))\right)}{dx}\\=&-2*-csc^{2}(x)csc^{2}(x) - 2cot(x)*-2csc^{2}(x)cot(x) - -2csc^{2}(x)cot(x)csc^{2}(cot(x)) - csc^{2}(x)*-2csc^{2}(cot(x))cot(cot(x))*-csc^{2}(x)\\=&2csc^{4}(x) + 2cot(x)csc^{2}(x)csc^{2}(cot(x)) - 2cot(cot(x))csc^{4}(x)csc^{2}(cot(x)) + 4cot^{2}(x)csc^{2}(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 2csc^{4}(x) + 2cot(x)csc^{2}(x)csc^{2}(cot(x)) - 2cot(cot(x))csc^{4}(x)csc^{2}(cot(x)) + 4cot^{2}(x)csc^{2}(x)\right)}{dx}\\=&2*-4csc^{4}(x)cot(x) + 2*-csc^{2}(x)csc^{2}(x)csc^{2}(cot(x)) + 2cot(x)*-2csc^{2}(x)cot(x)csc^{2}(cot(x)) + 2cot(x)csc^{2}(x)*-2csc^{2}(cot(x))cot(cot(x))*-csc^{2}(x) - 2*-csc^{2}(cot(x))*-csc^{2}(x)csc^{4}(x)csc^{2}(cot(x)) - 2cot(cot(x))*-4csc^{4}(x)cot(x)csc^{2}(cot(x)) - 2cot(cot(x))csc^{4}(x)*-2csc^{2}(cot(x))cot(cot(x))*-csc^{2}(x) + 4*-2cot(x)csc^{2}(x)csc^{2}(x) + 4cot^{2}(x)*-2csc^{2}(x)cot(x)\\=& - 4cot^{2}(x)csc^{2}(x)csc^{2}(cot(x)) - 2csc^{4}(x)csc^{2}(cot(x)) - 4cot^{2}(cot(x))csc^{6}(x)csc^{2}(cot(x)) + 4cot(cot(x))cot(x)csc^{4}(x)csc^{2}(cot(x)) - 2csc^{6}(x)csc^{4}(cot(x)) + 8cot(x)cot(cot(x))csc^{4}(x)csc^{2}(cot(x)) - 16cot(x)csc^{4}(x) - 8cot^{3}(x)csc^{2}(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( - 4cot^{2}(x)csc^{2}(x)csc^{2}(cot(x)) - 2csc^{4}(x)csc^{2}(cot(x)) - 4cot^{2}(cot(x))csc^{6}(x)csc^{2}(cot(x)) + 4cot(cot(x))cot(x)csc^{4}(x)csc^{2}(cot(x)) - 2csc^{6}(x)csc^{4}(cot(x)) + 8cot(x)cot(cot(x))csc^{4}(x)csc^{2}(cot(x)) - 16cot(x)csc^{4}(x) - 8cot^{3}(x)csc^{2}(x)\right)}{dx}\\=& - 4*-2cot(x)csc^{2}(x)csc^{2}(x)csc^{2}(cot(x)) - 4cot^{2}(x)*-2csc^{2}(x)cot(x)csc^{2}(cot(x)) - 4cot^{2}(x)csc^{2}(x)*-2csc^{2}(cot(x))cot(cot(x))*-csc^{2}(x) - 2*-4csc^{4}(x)cot(x)csc^{2}(cot(x)) - 2csc^{4}(x)*-2csc^{2}(cot(x))cot(cot(x))*-csc^{2}(x) - 4*-2cot(cot(x))csc^{2}(cot(x))*-csc^{2}(x)csc^{6}(x)csc^{2}(cot(x)) - 4cot^{2}(cot(x))*-6csc^{6}(x)cot(x)csc^{2}(cot(x)) - 4cot^{2}(cot(x))csc^{6}(x)*-2csc^{2}(cot(x))cot(cot(x))*-csc^{2}(x) + 4*-csc^{2}(cot(x))*-csc^{2}(x)cot(x)csc^{4}(x)csc^{2}(cot(x)) + 4cot(cot(x))*-csc^{2}(x)csc^{4}(x)csc^{2}(cot(x)) + 4cot(cot(x))cot(x)*-4csc^{4}(x)cot(x)csc^{2}(cot(x)) + 4cot(cot(x))cot(x)csc^{4}(x)*-2csc^{2}(cot(x))cot(cot(x))*-csc^{2}(x) - 2*-6csc^{6}(x)cot(x)csc^{4}(cot(x)) - 2csc^{6}(x)*-4csc^{4}(cot(x))cot(cot(x))*-csc^{2}(x) + 8*-csc^{2}(x)cot(cot(x))csc^{4}(x)csc^{2}(cot(x)) + 8cot(x)*-csc^{2}(cot(x))*-csc^{2}(x)csc^{4}(x)csc^{2}(cot(x)) + 8cot(x)cot(cot(x))*-4csc^{4}(x)cot(x)csc^{2}(cot(x)) + 8cot(x)cot(cot(x))csc^{4}(x)*-2csc^{2}(cot(x))cot(cot(x))*-csc^{2}(x) - 16*-csc^{2}(x)csc^{4}(x) - 16cot(x)*-4csc^{4}(x)cot(x) - 8*-3cot^{2}(x)csc^{2}(x)csc^{2}(x) - 8cot^{3}(x)*-2csc^{2}(x)cot(x)\\=&16cot(x)csc^{4}(x)csc^{2}(cot(x)) + 8cot^{3}(x)csc^{2}(x)csc^{2}(cot(x)) - 24cot(cot(x))cot^{2}(x)csc^{4}(x)csc^{2}(cot(x)) - 16cot(cot(x))csc^{6}(x)csc^{2}(cot(x)) - 16cot(cot(x))csc^{8}(x)csc^{4}(cot(x)) + 4cot(x)csc^{4}(cot(x))csc^{6}(x) + 32cot(x)cot^{2}(cot(x))csc^{6}(x)csc^{2}(cot(x)) - 8cot^{3}(cot(x))csc^{8}(x)csc^{2}(cot(x)) + 20cot(x)csc^{6}(x)csc^{4}(cot(x)) - 32cot^{2}(x)cot(cot(x))csc^{4}(x)csc^{2}(cot(x)) + 16cot^{2}(cot(x))cot(x)csc^{6}(x)csc^{2}(cot(x)) + 16csc^{6}(x) + 88cot^{2}(x)csc^{4}(x) + 16cot^{4}(x)csc^{2}(x)\\ \end{split}\end{equation} \]