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.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ tan(tan(tan(x)))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( tan(tan(tan(x)))\right)}{dx}\\=&sec^{2}(tan(tan(x)))(sec^{2}(tan(x))(sec^{2}(x)(1)))\\=&sec^{2}(tan(tan(x)))sec^{2}(x)sec^{2}(tan(x))\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( sec^{2}(tan(tan(x)))sec^{2}(x)sec^{2}(tan(x))\right)}{dx}\\=&2sec^{2}(tan(tan(x)))tan(tan(tan(x)))sec^{2}(tan(x))(sec^{2}(x)(1))sec^{2}(x)sec^{2}(tan(x)) + sec^{2}(tan(tan(x)))*2sec^{2}(x)tan(x)sec^{2}(tan(x)) + sec^{2}(tan(tan(x)))sec^{2}(x)*2sec^{2}(tan(x))tan(tan(x))sec^{2}(x)(1)\\=&2tan(tan(tan(x)))sec^{2}(tan(tan(x)))sec^{4}(x)sec^{4}(tan(x)) + 2tan(x)sec^{2}(tan(tan(x)))sec^{2}(x)sec^{2}(tan(x)) + 2tan(tan(x))sec^{4}(x)sec^{2}(tan(tan(x)))sec^{2}(tan(x))\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 2tan(tan(tan(x)))sec^{2}(tan(tan(x)))sec^{4}(x)sec^{4}(tan(x)) + 2tan(x)sec^{2}(tan(tan(x)))sec^{2}(x)sec^{2}(tan(x)) + 2tan(tan(x))sec^{4}(x)sec^{2}(tan(tan(x)))sec^{2}(tan(x))\right)}{dx}\\=&2sec^{2}(tan(tan(x)))(sec^{2}(tan(x))(sec^{2}(x)(1)))sec^{2}(tan(tan(x)))sec^{4}(x)sec^{4}(tan(x)) + 2tan(tan(tan(x)))*2sec^{2}(tan(tan(x)))tan(tan(tan(x)))sec^{2}(tan(x))(sec^{2}(x)(1))sec^{4}(x)sec^{4}(tan(x)) + 2tan(tan(tan(x)))sec^{2}(tan(tan(x)))*4sec^{4}(x)tan(x)sec^{4}(tan(x)) + 2tan(tan(tan(x)))sec^{2}(tan(tan(x)))sec^{4}(x)*4sec^{4}(tan(x))tan(tan(x))sec^{2}(x)(1) + 2sec^{2}(x)(1)sec^{2}(tan(tan(x)))sec^{2}(x)sec^{2}(tan(x)) + 2tan(x)*2sec^{2}(tan(tan(x)))tan(tan(tan(x)))sec^{2}(tan(x))(sec^{2}(x)(1))sec^{2}(x)sec^{2}(tan(x)) + 2tan(x)sec^{2}(tan(tan(x)))*2sec^{2}(x)tan(x)sec^{2}(tan(x)) + 2tan(x)sec^{2}(tan(tan(x)))sec^{2}(x)*2sec^{2}(tan(x))tan(tan(x))sec^{2}(x)(1) + 2sec^{2}(tan(x))(sec^{2}(x)(1))sec^{4}(x)sec^{2}(tan(tan(x)))sec^{2}(tan(x)) + 2tan(tan(x))*4sec^{4}(x)tan(x)sec^{2}(tan(tan(x)))sec^{2}(tan(x)) + 2tan(tan(x))sec^{4}(x)*2sec^{2}(tan(tan(x)))tan(tan(tan(x)))sec^{2}(tan(x))(sec^{2}(x)(1))sec^{2}(tan(x)) + 2tan(tan(x))sec^{4}(x)sec^{2}(tan(tan(x)))*2sec^{2}(tan(x))tan(tan(x))sec^{2}(x)(1)\\=&2sec^{4}(tan(tan(x)))sec^{6}(x)sec^{6}(tan(x)) + 4tan^{2}(tan(tan(x)))sec^{2}(tan(tan(x)))sec^{6}(x)sec^{6}(tan(x)) + 8tan(x)tan(tan(tan(x)))sec^{2}(tan(tan(x)))sec^{4}(x)sec^{4}(tan(x)) + 8tan(tan(tan(x)))tan(tan(x))sec^{6}(x)sec^{2}(tan(tan(x)))sec^{4}(tan(x)) + 2sec^{4}(x)sec^{2}(tan(tan(x)))sec^{2}(tan(x)) + 4tan(tan(tan(x)))tan(x)sec^{2}(tan(tan(x)))sec^{4}(x)sec^{4}(tan(x)) + 4tan^{2}(x)sec^{2}(tan(tan(x)))sec^{2}(x)sec^{2}(tan(x)) + 12tan(x)tan(tan(x))sec^{4}(x)sec^{2}(tan(tan(x)))sec^{2}(tan(x)) + 2sec^{6}(x)sec^{4}(tan(x))sec^{2}(tan(tan(x))) + 4tan(tan(tan(x)))tan(tan(x))sec^{4}(tan(x))sec^{6}(x)sec^{2}(tan(tan(x))) + 4tan^{2}(tan(x))sec^{2}(tan(tan(x)))sec^{6}(x)sec^{2}(tan(x))\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 2sec^{4}(tan(tan(x)))sec^{6}(x)sec^{6}(tan(x)) + 4tan^{2}(tan(tan(x)))sec^{2}(tan(tan(x)))sec^{6}(x)sec^{6}(tan(x)) + 8tan(x)tan(tan(tan(x)))sec^{2}(tan(tan(x)))sec^{4}(x)sec^{4}(tan(x)) + 8tan(tan(tan(x)))tan(tan(x))sec^{6}(x)sec^{2}(tan(tan(x)))sec^{4}(tan(x)) + 2sec^{4}(x)sec^{2}(tan(tan(x)))sec^{2}(tan(x)) + 4tan(tan(tan(x)))tan(x)sec^{2}(tan(tan(x)))sec^{4}(x)sec^{4}(tan(x)) + 4tan^{2}(x)sec^{2}(tan(tan(x)))sec^{2}(x)sec^{2}(tan(x)) + 12tan(x)tan(tan(x))sec^{4}(x)sec^{2}(tan(tan(x)))sec^{2}(tan(x)) + 2sec^{6}(x)sec^{4}(tan(x))sec^{2}(tan(tan(x))) + 4tan(tan(tan(x)))tan(tan(x))sec^{4}(tan(x))sec^{6}(x)sec^{2}(tan(tan(x))) + 4tan^{2}(tan(x))sec^{2}(tan(tan(x)))sec^{6}(x)sec^{2}(tan(x))\right)}{dx}\\=&2*4sec^{4}(tan(tan(x)))tan(tan(tan(x)))sec^{2}(tan(x))(sec^{2}(x)(1))sec^{6}(x)sec^{6}(tan(x)) + 2sec^{4}(tan(tan(x)))*6sec^{6}(x)tan(x)sec^{6}(tan(x)) + 2sec^{4}(tan(tan(x)))sec^{6}(x)*6sec^{6}(tan(x))tan(tan(x))sec^{2}(x)(1) + 4*2tan(tan(tan(x)))sec^{2}(tan(tan(x)))(sec^{2}(tan(x))(sec^{2}(x)(1)))sec^{2}(tan(tan(x)))sec^{6}(x)sec^{6}(tan(x)) + 4tan^{2}(tan(tan(x)))*2sec^{2}(tan(tan(x)))tan(tan(tan(x)))sec^{2}(tan(x))(sec^{2}(x)(1))sec^{6}(x)sec^{6}(tan(x)) + 4tan^{2}(tan(tan(x)))sec^{2}(tan(tan(x)))*6sec^{6}(x)tan(x)sec^{6}(tan(x)) + 4tan^{2}(tan(tan(x)))sec^{2}(tan(tan(x)))sec^{6}(x)*6sec^{6}(tan(x))tan(tan(x))sec^{2}(x)(1) + 8sec^{2}(x)(1)tan(tan(tan(x)))sec^{2}(tan(tan(x)))sec^{4}(x)sec^{4}(tan(x)) + 8tan(x)sec^{2}(tan(tan(x)))(sec^{2}(tan(x))(sec^{2}(x)(1)))sec^{2}(tan(tan(x)))sec^{4}(x)sec^{4}(tan(x)) + 8tan(x)tan(tan(tan(x)))*2sec^{2}(tan(tan(x)))tan(tan(tan(x)))sec^{2}(tan(x))(sec^{2}(x)(1))sec^{4}(x)sec^{4}(tan(x)) + 8tan(x)tan(tan(tan(x)))sec^{2}(tan(tan(x)))*4sec^{4}(x)tan(x)sec^{4}(tan(x)) + 8tan(x)tan(tan(tan(x)))sec^{2}(tan(tan(x)))sec^{4}(x)*4sec^{4}(tan(x))tan(tan(x))sec^{2}(x)(1) + 8sec^{2}(tan(tan(x)))(sec^{2}(tan(x))(sec^{2}(x)(1)))tan(tan(x))sec^{6}(x)sec^{2}(tan(tan(x)))sec^{4}(tan(x)) + 8tan(tan(tan(x)))sec^{2}(tan(x))(sec^{2}(x)(1))sec^{6}(x)sec^{2}(tan(tan(x)))sec^{4}(tan(x)) + 8tan(tan(tan(x)))tan(tan(x))*6sec^{6}(x)tan(x)sec^{2}(tan(tan(x)))sec^{4}(tan(x)) + 8tan(tan(tan(x)))tan(tan(x))sec^{6}(x)*2sec^{2}(tan(tan(x)))tan(tan(tan(x)))sec^{2}(tan(x))(sec^{2}(x)(1))sec^{4}(tan(x)) + 8tan(tan(tan(x)))tan(tan(x))sec^{6}(x)sec^{2}(tan(tan(x)))*4sec^{4}(tan(x))tan(tan(x))sec^{2}(x)(1) + 2*4sec^{4}(x)tan(x)sec^{2}(tan(tan(x)))sec^{2}(tan(x)) + 2sec^{4}(x)*2sec^{2}(tan(tan(x)))tan(tan(tan(x)))sec^{2}(tan(x))(sec^{2}(x)(1))sec^{2}(tan(x)) + 2sec^{4}(x)sec^{2}(tan(tan(x)))*2sec^{2}(tan(x))tan(tan(x))sec^{2}(x)(1) + 4sec^{2}(tan(tan(x)))(sec^{2}(tan(x))(sec^{2}(x)(1)))tan(x)sec^{2}(tan(tan(x)))sec^{4}(x)sec^{4}(tan(x)) + 4tan(tan(tan(x)))sec^{2}(x)(1)sec^{2}(tan(tan(x)))sec^{4}(x)sec^{4}(tan(x)) + 4tan(tan(tan(x)))tan(x)*2sec^{2}(tan(tan(x)))tan(tan(tan(x)))sec^{2}(tan(x))(sec^{2}(x)(1))sec^{4}(x)sec^{4}(tan(x)) + 4tan(tan(tan(x)))tan(x)sec^{2}(tan(tan(x)))*4sec^{4}(x)tan(x)sec^{4}(tan(x)) + 4tan(tan(tan(x)))tan(x)sec^{2}(tan(tan(x)))sec^{4}(x)*4sec^{4}(tan(x))tan(tan(x))sec^{2}(x)(1) + 4*2tan(x)sec^{2}(x)(1)sec^{2}(tan(tan(x)))sec^{2}(x)sec^{2}(tan(x)) + 4tan^{2}(x)*2sec^{2}(tan(tan(x)))tan(tan(tan(x)))sec^{2}(tan(x))(sec^{2}(x)(1))sec^{2}(x)sec^{2}(tan(x)) + 4tan^{2}(x)sec^{2}(tan(tan(x)))*2sec^{2}(x)tan(x)sec^{2}(tan(x)) + 4tan^{2}(x)sec^{2}(tan(tan(x)))sec^{2}(x)*2sec^{2}(tan(x))tan(tan(x))sec^{2}(x)(1) + 12sec^{2}(x)(1)tan(tan(x))sec^{4}(x)sec^{2}(tan(tan(x)))sec^{2}(tan(x)) + 12tan(x)sec^{2}(tan(x))(sec^{2}(x)(1))sec^{4}(x)sec^{2}(tan(tan(x)))sec^{2}(tan(x)) + 12tan(x)tan(tan(x))*4sec^{4}(x)tan(x)sec^{2}(tan(tan(x)))sec^{2}(tan(x)) + 12tan(x)tan(tan(x))sec^{4}(x)*2sec^{2}(tan(tan(x)))tan(tan(tan(x)))sec^{2}(tan(x))(sec^{2}(x)(1))sec^{2}(tan(x)) + 12tan(x)tan(tan(x))sec^{4}(x)sec^{2}(tan(tan(x)))*2sec^{2}(tan(x))tan(tan(x))sec^{2}(x)(1) + 2*6sec^{6}(x)tan(x)sec^{4}(tan(x))sec^{2}(tan(tan(x))) + 2sec^{6}(x)*4sec^{4}(tan(x))tan(tan(x))sec^{2}(x)(1)sec^{2}(tan(tan(x))) + 2sec^{6}(x)sec^{4}(tan(x))*2sec^{2}(tan(tan(x)))tan(tan(tan(x)))sec^{2}(tan(x))(sec^{2}(x)(1)) + 4sec^{2}(tan(tan(x)))(sec^{2}(tan(x))(sec^{2}(x)(1)))tan(tan(x))sec^{4}(tan(x))sec^{6}(x)sec^{2}(tan(tan(x))) + 4tan(tan(tan(x)))sec^{2}(tan(x))(sec^{2}(x)(1))sec^{4}(tan(x))sec^{6}(x)sec^{2}(tan(tan(x))) + 4tan(tan(tan(x)))tan(tan(x))*4sec^{4}(tan(x))tan(tan(x))sec^{2}(x)(1)sec^{6}(x)sec^{2}(tan(tan(x))) + 4tan(tan(tan(x)))tan(tan(x))sec^{4}(tan(x))*6sec^{6}(x)tan(x)sec^{2}(tan(tan(x))) + 4tan(tan(tan(x)))tan(tan(x))sec^{4}(tan(x))sec^{6}(x)*2sec^{2}(tan(tan(x)))tan(tan(tan(x)))sec^{2}(tan(x))(sec^{2}(x)(1)) + 4*2tan(tan(x))sec^{2}(tan(x))(sec^{2}(x)(1))sec^{2}(tan(tan(x)))sec^{6}(x)sec^{2}(tan(x)) + 4tan^{2}(tan(x))*2sec^{2}(tan(tan(x)))tan(tan(tan(x)))sec^{2}(tan(x))(sec^{2}(x)(1))sec^{6}(x)sec^{2}(tan(x)) + 4tan^{2}(tan(x))sec^{2}(tan(tan(x)))*6sec^{6}(x)tan(x)sec^{2}(tan(x)) + 4tan^{2}(tan(x))sec^{2}(tan(tan(x)))sec^{6}(x)*2sec^{2}(tan(x))tan(tan(x))sec^{2}(x)(1)\\=&16tan(tan(tan(x)))sec^{4}(tan(tan(x)))sec^{8}(x)sec^{8}(tan(x)) + 16tan(tan(tan(x)))sec^{8}(x)sec^{6}(tan(x))sec^{2}(tan(tan(x))) + 20tan(x)sec^{4}(tan(tan(x)))sec^{6}(x)sec^{6}(tan(x)) + 8tan^{3}(tan(tan(x)))sec^{2}(tan(tan(x)))sec^{8}(x)sec^{8}(tan(x)) + 32tan(x)tan^{2}(tan(tan(x)))sec^{2}(tan(tan(x)))sec^{6}(x)sec^{6}(tan(x)) + 24tan^{2}(tan(tan(x)))tan(tan(x))sec^{8}(x)sec^{2}(tan(tan(x)))sec^{6}(tan(x)) + 12tan(tan(tan(x)))sec^{6}(x)sec^{2}(tan(tan(x)))sec^{4}(tan(x)) + 12tan(tan(x))sec^{8}(x)sec^{6}(tan(x))sec^{4}(tan(tan(x))) + 16tan^{2}(tan(tan(x)))tan(x)sec^{2}(tan(tan(x)))sec^{6}(x)sec^{6}(tan(x)) + 32tan^{2}(x)tan(tan(tan(x)))sec^{2}(tan(tan(x)))sec^{4}(x)sec^{4}(tan(x)) + 32tan(tan(x))tan(tan(tan(x)))tan(x)sec^{6}(x)sec^{2}(tan(tan(x)))sec^{4}(tan(x)) + 4tan(x)sec^{6}(x)sec^{6}(tan(x))sec^{4}(tan(tan(x))) + 24tan(x)sec^{6}(x)sec^{4}(tan(x))sec^{2}(tan(tan(x))) + 48tan(tan(tan(x)))tan(x)tan(tan(x))sec^{6}(x)sec^{2}(tan(tan(x)))sec^{4}(tan(x)) + 16tan(tan(x))tan^{2}(tan(tan(x)))sec^{6}(tan(x))sec^{8}(x)sec^{2}(tan(tan(x))) + 32tan^{2}(tan(x))tan(tan(tan(x)))sec^{2}(tan(tan(x)))sec^{8}(x)sec^{4}(tan(x)) + 4tan(tan(tan(x)))sec^{4}(tan(x))sec^{6}(x)sec^{2}(tan(tan(x))) + 8tan(tan(x))sec^{8}(x)sec^{4}(tan(x))sec^{2}(tan(tan(x))) + 24tan(tan(tan(x)))tan^{2}(x)sec^{2}(tan(tan(x)))sec^{4}(x)sec^{4}(tan(x)) + 16tan(tan(x))tan(x)tan(tan(tan(x)))sec^{6}(x)sec^{2}(tan(tan(x)))sec^{4}(tan(x)) + 16tan(x)sec^{4}(x)sec^{2}(tan(tan(x)))sec^{2}(tan(x)) + 8tan^{3}(x)sec^{2}(tan(tan(x)))sec^{2}(x)sec^{2}(tan(x)) + 56tan^{2}(x)tan(tan(x))sec^{4}(x)sec^{2}(tan(tan(x)))sec^{2}(tan(x)) + 12tan(tan(x))sec^{6}(x)sec^{2}(tan(tan(x)))sec^{2}(tan(x)) + 24tan(tan(x))tan(x)tan(tan(tan(x)))sec^{4}(tan(x))sec^{6}(x)sec^{2}(tan(tan(x))) + 24tan^{2}(tan(x))tan(x)sec^{2}(tan(tan(x)))sec^{6}(x)sec^{2}(tan(x)) + 8tan(tan(x))sec^{4}(tan(x))sec^{8}(x)sec^{2}(tan(tan(x))) + 4tan(tan(x))sec^{2}(tan(tan(x)))sec^{6}(x)sec^{2}(tan(x)) + 16tan^{2}(tan(x))tan(tan(tan(x)))sec^{8}(x)sec^{4}(tan(x))sec^{2}(tan(tan(x))) + 24tan(tan(tan(x)))tan(x)tan(tan(x))sec^{4}(tan(x))sec^{6}(x)sec^{2}(tan(tan(x))) + 8tan(tan(x))tan^{2}(tan(tan(x)))sec^{6}(tan(x))sec^{2}(tan(tan(x)))sec^{8}(x) + 12tan(tan(x))sec^{8}(x)sec^{4}(tan(tan(x)))sec^{6}(tan(x)) + 8tan(tan(tan(x)))tan^{2}(tan(x))sec^{2}(tan(tan(x)))sec^{8}(x)sec^{4}(tan(x)) + 24tan(x)tan^{2}(tan(x))sec^{2}(tan(tan(x)))sec^{6}(x)sec^{2}(tan(x)) + 8tan^{3}(tan(x))sec^{8}(x)sec^{2}(tan(tan(x)))sec^{2}(tan(x))\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ tan(tan(tan(x)))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( tan(tan(tan(x)))\right)}{dx}\\=&sec^{2}(tan(tan(x)))(sec^{2}(tan(x))(sec^{2}(x)(1)))\\=&sec^{2}(tan(tan(x)))sec^{2}(x)sec^{2}(tan(x))\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( sec^{2}(tan(tan(x)))sec^{2}(x)sec^{2}(tan(x))\right)}{dx}\\=&2sec^{2}(tan(tan(x)))tan(tan(tan(x)))sec^{2}(tan(x))(sec^{2}(x)(1))sec^{2}(x)sec^{2}(tan(x)) + sec^{2}(tan(tan(x)))*2sec^{2}(x)tan(x)sec^{2}(tan(x)) + sec^{2}(tan(tan(x)))sec^{2}(x)*2sec^{2}(tan(x))tan(tan(x))sec^{2}(x)(1)\\=&2tan(tan(tan(x)))sec^{2}(tan(tan(x)))sec^{4}(x)sec^{4}(tan(x)) + 2tan(x)sec^{2}(tan(tan(x)))sec^{2}(x)sec^{2}(tan(x)) + 2tan(tan(x))sec^{4}(x)sec^{2}(tan(tan(x)))sec^{2}(tan(x))\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 2tan(tan(tan(x)))sec^{2}(tan(tan(x)))sec^{4}(x)sec^{4}(tan(x)) + 2tan(x)sec^{2}(tan(tan(x)))sec^{2}(x)sec^{2}(tan(x)) + 2tan(tan(x))sec^{4}(x)sec^{2}(tan(tan(x)))sec^{2}(tan(x))\right)}{dx}\\=&2sec^{2}(tan(tan(x)))(sec^{2}(tan(x))(sec^{2}(x)(1)))sec^{2}(tan(tan(x)))sec^{4}(x)sec^{4}(tan(x)) + 2tan(tan(tan(x)))*2sec^{2}(tan(tan(x)))tan(tan(tan(x)))sec^{2}(tan(x))(sec^{2}(x)(1))sec^{4}(x)sec^{4}(tan(x)) + 2tan(tan(tan(x)))sec^{2}(tan(tan(x)))*4sec^{4}(x)tan(x)sec^{4}(tan(x)) + 2tan(tan(tan(x)))sec^{2}(tan(tan(x)))sec^{4}(x)*4sec^{4}(tan(x))tan(tan(x))sec^{2}(x)(1) + 2sec^{2}(x)(1)sec^{2}(tan(tan(x)))sec^{2}(x)sec^{2}(tan(x)) + 2tan(x)*2sec^{2}(tan(tan(x)))tan(tan(tan(x)))sec^{2}(tan(x))(sec^{2}(x)(1))sec^{2}(x)sec^{2}(tan(x)) + 2tan(x)sec^{2}(tan(tan(x)))*2sec^{2}(x)tan(x)sec^{2}(tan(x)) + 2tan(x)sec^{2}(tan(tan(x)))sec^{2}(x)*2sec^{2}(tan(x))tan(tan(x))sec^{2}(x)(1) + 2sec^{2}(tan(x))(sec^{2}(x)(1))sec^{4}(x)sec^{2}(tan(tan(x)))sec^{2}(tan(x)) + 2tan(tan(x))*4sec^{4}(x)tan(x)sec^{2}(tan(tan(x)))sec^{2}(tan(x)) + 2tan(tan(x))sec^{4}(x)*2sec^{2}(tan(tan(x)))tan(tan(tan(x)))sec^{2}(tan(x))(sec^{2}(x)(1))sec^{2}(tan(x)) + 2tan(tan(x))sec^{4}(x)sec^{2}(tan(tan(x)))*2sec^{2}(tan(x))tan(tan(x))sec^{2}(x)(1)\\=&2sec^{4}(tan(tan(x)))sec^{6}(x)sec^{6}(tan(x)) + 4tan^{2}(tan(tan(x)))sec^{2}(tan(tan(x)))sec^{6}(x)sec^{6}(tan(x)) + 8tan(x)tan(tan(tan(x)))sec^{2}(tan(tan(x)))sec^{4}(x)sec^{4}(tan(x)) + 8tan(tan(tan(x)))tan(tan(x))sec^{6}(x)sec^{2}(tan(tan(x)))sec^{4}(tan(x)) + 2sec^{4}(x)sec^{2}(tan(tan(x)))sec^{2}(tan(x)) + 4tan(tan(tan(x)))tan(x)sec^{2}(tan(tan(x)))sec^{4}(x)sec^{4}(tan(x)) + 4tan^{2}(x)sec^{2}(tan(tan(x)))sec^{2}(x)sec^{2}(tan(x)) + 12tan(x)tan(tan(x))sec^{4}(x)sec^{2}(tan(tan(x)))sec^{2}(tan(x)) + 2sec^{6}(x)sec^{4}(tan(x))sec^{2}(tan(tan(x))) + 4tan(tan(tan(x)))tan(tan(x))sec^{4}(tan(x))sec^{6}(x)sec^{2}(tan(tan(x))) + 4tan^{2}(tan(x))sec^{2}(tan(tan(x)))sec^{6}(x)sec^{2}(tan(x))\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 2sec^{4}(tan(tan(x)))sec^{6}(x)sec^{6}(tan(x)) + 4tan^{2}(tan(tan(x)))sec^{2}(tan(tan(x)))sec^{6}(x)sec^{6}(tan(x)) + 8tan(x)tan(tan(tan(x)))sec^{2}(tan(tan(x)))sec^{4}(x)sec^{4}(tan(x)) + 8tan(tan(tan(x)))tan(tan(x))sec^{6}(x)sec^{2}(tan(tan(x)))sec^{4}(tan(x)) + 2sec^{4}(x)sec^{2}(tan(tan(x)))sec^{2}(tan(x)) + 4tan(tan(tan(x)))tan(x)sec^{2}(tan(tan(x)))sec^{4}(x)sec^{4}(tan(x)) + 4tan^{2}(x)sec^{2}(tan(tan(x)))sec^{2}(x)sec^{2}(tan(x)) + 12tan(x)tan(tan(x))sec^{4}(x)sec^{2}(tan(tan(x)))sec^{2}(tan(x)) + 2sec^{6}(x)sec^{4}(tan(x))sec^{2}(tan(tan(x))) + 4tan(tan(tan(x)))tan(tan(x))sec^{4}(tan(x))sec^{6}(x)sec^{2}(tan(tan(x))) + 4tan^{2}(tan(x))sec^{2}(tan(tan(x)))sec^{6}(x)sec^{2}(tan(x))\right)}{dx}\\=&2*4sec^{4}(tan(tan(x)))tan(tan(tan(x)))sec^{2}(tan(x))(sec^{2}(x)(1))sec^{6}(x)sec^{6}(tan(x)) + 2sec^{4}(tan(tan(x)))*6sec^{6}(x)tan(x)sec^{6}(tan(x)) + 2sec^{4}(tan(tan(x)))sec^{6}(x)*6sec^{6}(tan(x))tan(tan(x))sec^{2}(x)(1) + 4*2tan(tan(tan(x)))sec^{2}(tan(tan(x)))(sec^{2}(tan(x))(sec^{2}(x)(1)))sec^{2}(tan(tan(x)))sec^{6}(x)sec^{6}(tan(x)) + 4tan^{2}(tan(tan(x)))*2sec^{2}(tan(tan(x)))tan(tan(tan(x)))sec^{2}(tan(x))(sec^{2}(x)(1))sec^{6}(x)sec^{6}(tan(x)) + 4tan^{2}(tan(tan(x)))sec^{2}(tan(tan(x)))*6sec^{6}(x)tan(x)sec^{6}(tan(x)) + 4tan^{2}(tan(tan(x)))sec^{2}(tan(tan(x)))sec^{6}(x)*6sec^{6}(tan(x))tan(tan(x))sec^{2}(x)(1) + 8sec^{2}(x)(1)tan(tan(tan(x)))sec^{2}(tan(tan(x)))sec^{4}(x)sec^{4}(tan(x)) + 8tan(x)sec^{2}(tan(tan(x)))(sec^{2}(tan(x))(sec^{2}(x)(1)))sec^{2}(tan(tan(x)))sec^{4}(x)sec^{4}(tan(x)) + 8tan(x)tan(tan(tan(x)))*2sec^{2}(tan(tan(x)))tan(tan(tan(x)))sec^{2}(tan(x))(sec^{2}(x)(1))sec^{4}(x)sec^{4}(tan(x)) + 8tan(x)tan(tan(tan(x)))sec^{2}(tan(tan(x)))*4sec^{4}(x)tan(x)sec^{4}(tan(x)) + 8tan(x)tan(tan(tan(x)))sec^{2}(tan(tan(x)))sec^{4}(x)*4sec^{4}(tan(x))tan(tan(x))sec^{2}(x)(1) + 8sec^{2}(tan(tan(x)))(sec^{2}(tan(x))(sec^{2}(x)(1)))tan(tan(x))sec^{6}(x)sec^{2}(tan(tan(x)))sec^{4}(tan(x)) + 8tan(tan(tan(x)))sec^{2}(tan(x))(sec^{2}(x)(1))sec^{6}(x)sec^{2}(tan(tan(x)))sec^{4}(tan(x)) + 8tan(tan(tan(x)))tan(tan(x))*6sec^{6}(x)tan(x)sec^{2}(tan(tan(x)))sec^{4}(tan(x)) + 8tan(tan(tan(x)))tan(tan(x))sec^{6}(x)*2sec^{2}(tan(tan(x)))tan(tan(tan(x)))sec^{2}(tan(x))(sec^{2}(x)(1))sec^{4}(tan(x)) + 8tan(tan(tan(x)))tan(tan(x))sec^{6}(x)sec^{2}(tan(tan(x)))*4sec^{4}(tan(x))tan(tan(x))sec^{2}(x)(1) + 2*4sec^{4}(x)tan(x)sec^{2}(tan(tan(x)))sec^{2}(tan(x)) + 2sec^{4}(x)*2sec^{2}(tan(tan(x)))tan(tan(tan(x)))sec^{2}(tan(x))(sec^{2}(x)(1))sec^{2}(tan(x)) + 2sec^{4}(x)sec^{2}(tan(tan(x)))*2sec^{2}(tan(x))tan(tan(x))sec^{2}(x)(1) + 4sec^{2}(tan(tan(x)))(sec^{2}(tan(x))(sec^{2}(x)(1)))tan(x)sec^{2}(tan(tan(x)))sec^{4}(x)sec^{4}(tan(x)) + 4tan(tan(tan(x)))sec^{2}(x)(1)sec^{2}(tan(tan(x)))sec^{4}(x)sec^{4}(tan(x)) + 4tan(tan(tan(x)))tan(x)*2sec^{2}(tan(tan(x)))tan(tan(tan(x)))sec^{2}(tan(x))(sec^{2}(x)(1))sec^{4}(x)sec^{4}(tan(x)) + 4tan(tan(tan(x)))tan(x)sec^{2}(tan(tan(x)))*4sec^{4}(x)tan(x)sec^{4}(tan(x)) + 4tan(tan(tan(x)))tan(x)sec^{2}(tan(tan(x)))sec^{4}(x)*4sec^{4}(tan(x))tan(tan(x))sec^{2}(x)(1) + 4*2tan(x)sec^{2}(x)(1)sec^{2}(tan(tan(x)))sec^{2}(x)sec^{2}(tan(x)) + 4tan^{2}(x)*2sec^{2}(tan(tan(x)))tan(tan(tan(x)))sec^{2}(tan(x))(sec^{2}(x)(1))sec^{2}(x)sec^{2}(tan(x)) + 4tan^{2}(x)sec^{2}(tan(tan(x)))*2sec^{2}(x)tan(x)sec^{2}(tan(x)) + 4tan^{2}(x)sec^{2}(tan(tan(x)))sec^{2}(x)*2sec^{2}(tan(x))tan(tan(x))sec^{2}(x)(1) + 12sec^{2}(x)(1)tan(tan(x))sec^{4}(x)sec^{2}(tan(tan(x)))sec^{2}(tan(x)) + 12tan(x)sec^{2}(tan(x))(sec^{2}(x)(1))sec^{4}(x)sec^{2}(tan(tan(x)))sec^{2}(tan(x)) + 12tan(x)tan(tan(x))*4sec^{4}(x)tan(x)sec^{2}(tan(tan(x)))sec^{2}(tan(x)) + 12tan(x)tan(tan(x))sec^{4}(x)*2sec^{2}(tan(tan(x)))tan(tan(tan(x)))sec^{2}(tan(x))(sec^{2}(x)(1))sec^{2}(tan(x)) + 12tan(x)tan(tan(x))sec^{4}(x)sec^{2}(tan(tan(x)))*2sec^{2}(tan(x))tan(tan(x))sec^{2}(x)(1) + 2*6sec^{6}(x)tan(x)sec^{4}(tan(x))sec^{2}(tan(tan(x))) + 2sec^{6}(x)*4sec^{4}(tan(x))tan(tan(x))sec^{2}(x)(1)sec^{2}(tan(tan(x))) + 2sec^{6}(x)sec^{4}(tan(x))*2sec^{2}(tan(tan(x)))tan(tan(tan(x)))sec^{2}(tan(x))(sec^{2}(x)(1)) + 4sec^{2}(tan(tan(x)))(sec^{2}(tan(x))(sec^{2}(x)(1)))tan(tan(x))sec^{4}(tan(x))sec^{6}(x)sec^{2}(tan(tan(x))) + 4tan(tan(tan(x)))sec^{2}(tan(x))(sec^{2}(x)(1))sec^{4}(tan(x))sec^{6}(x)sec^{2}(tan(tan(x))) + 4tan(tan(tan(x)))tan(tan(x))*4sec^{4}(tan(x))tan(tan(x))sec^{2}(x)(1)sec^{6}(x)sec^{2}(tan(tan(x))) + 4tan(tan(tan(x)))tan(tan(x))sec^{4}(tan(x))*6sec^{6}(x)tan(x)sec^{2}(tan(tan(x))) + 4tan(tan(tan(x)))tan(tan(x))sec^{4}(tan(x))sec^{6}(x)*2sec^{2}(tan(tan(x)))tan(tan(tan(x)))sec^{2}(tan(x))(sec^{2}(x)(1)) + 4*2tan(tan(x))sec^{2}(tan(x))(sec^{2}(x)(1))sec^{2}(tan(tan(x)))sec^{6}(x)sec^{2}(tan(x)) + 4tan^{2}(tan(x))*2sec^{2}(tan(tan(x)))tan(tan(tan(x)))sec^{2}(tan(x))(sec^{2}(x)(1))sec^{6}(x)sec^{2}(tan(x)) + 4tan^{2}(tan(x))sec^{2}(tan(tan(x)))*6sec^{6}(x)tan(x)sec^{2}(tan(x)) + 4tan^{2}(tan(x))sec^{2}(tan(tan(x)))sec^{6}(x)*2sec^{2}(tan(x))tan(tan(x))sec^{2}(x)(1)\\=&16tan(tan(tan(x)))sec^{4}(tan(tan(x)))sec^{8}(x)sec^{8}(tan(x)) + 16tan(tan(tan(x)))sec^{8}(x)sec^{6}(tan(x))sec^{2}(tan(tan(x))) + 20tan(x)sec^{4}(tan(tan(x)))sec^{6}(x)sec^{6}(tan(x)) + 8tan^{3}(tan(tan(x)))sec^{2}(tan(tan(x)))sec^{8}(x)sec^{8}(tan(x)) + 32tan(x)tan^{2}(tan(tan(x)))sec^{2}(tan(tan(x)))sec^{6}(x)sec^{6}(tan(x)) + 24tan^{2}(tan(tan(x)))tan(tan(x))sec^{8}(x)sec^{2}(tan(tan(x)))sec^{6}(tan(x)) + 12tan(tan(tan(x)))sec^{6}(x)sec^{2}(tan(tan(x)))sec^{4}(tan(x)) + 12tan(tan(x))sec^{8}(x)sec^{6}(tan(x))sec^{4}(tan(tan(x))) + 16tan^{2}(tan(tan(x)))tan(x)sec^{2}(tan(tan(x)))sec^{6}(x)sec^{6}(tan(x)) + 32tan^{2}(x)tan(tan(tan(x)))sec^{2}(tan(tan(x)))sec^{4}(x)sec^{4}(tan(x)) + 32tan(tan(x))tan(tan(tan(x)))tan(x)sec^{6}(x)sec^{2}(tan(tan(x)))sec^{4}(tan(x)) + 4tan(x)sec^{6}(x)sec^{6}(tan(x))sec^{4}(tan(tan(x))) + 24tan(x)sec^{6}(x)sec^{4}(tan(x))sec^{2}(tan(tan(x))) + 48tan(tan(tan(x)))tan(x)tan(tan(x))sec^{6}(x)sec^{2}(tan(tan(x)))sec^{4}(tan(x)) + 16tan(tan(x))tan^{2}(tan(tan(x)))sec^{6}(tan(x))sec^{8}(x)sec^{2}(tan(tan(x))) + 32tan^{2}(tan(x))tan(tan(tan(x)))sec^{2}(tan(tan(x)))sec^{8}(x)sec^{4}(tan(x)) + 4tan(tan(tan(x)))sec^{4}(tan(x))sec^{6}(x)sec^{2}(tan(tan(x))) + 8tan(tan(x))sec^{8}(x)sec^{4}(tan(x))sec^{2}(tan(tan(x))) + 24tan(tan(tan(x)))tan^{2}(x)sec^{2}(tan(tan(x)))sec^{4}(x)sec^{4}(tan(x)) + 16tan(tan(x))tan(x)tan(tan(tan(x)))sec^{6}(x)sec^{2}(tan(tan(x)))sec^{4}(tan(x)) + 16tan(x)sec^{4}(x)sec^{2}(tan(tan(x)))sec^{2}(tan(x)) + 8tan^{3}(x)sec^{2}(tan(tan(x)))sec^{2}(x)sec^{2}(tan(x)) + 56tan^{2}(x)tan(tan(x))sec^{4}(x)sec^{2}(tan(tan(x)))sec^{2}(tan(x)) + 12tan(tan(x))sec^{6}(x)sec^{2}(tan(tan(x)))sec^{2}(tan(x)) + 24tan(tan(x))tan(x)tan(tan(tan(x)))sec^{4}(tan(x))sec^{6}(x)sec^{2}(tan(tan(x))) + 24tan^{2}(tan(x))tan(x)sec^{2}(tan(tan(x)))sec^{6}(x)sec^{2}(tan(x)) + 8tan(tan(x))sec^{4}(tan(x))sec^{8}(x)sec^{2}(tan(tan(x))) + 4tan(tan(x))sec^{2}(tan(tan(x)))sec^{6}(x)sec^{2}(tan(x)) + 16tan^{2}(tan(x))tan(tan(tan(x)))sec^{8}(x)sec^{4}(tan(x))sec^{2}(tan(tan(x))) + 24tan(tan(tan(x)))tan(x)tan(tan(x))sec^{4}(tan(x))sec^{6}(x)sec^{2}(tan(tan(x))) + 8tan(tan(x))tan^{2}(tan(tan(x)))sec^{6}(tan(x))sec^{2}(tan(tan(x)))sec^{8}(x) + 12tan(tan(x))sec^{8}(x)sec^{4}(tan(tan(x)))sec^{6}(tan(x)) + 8tan(tan(tan(x)))tan^{2}(tan(x))sec^{2}(tan(tan(x)))sec^{8}(x)sec^{4}(tan(x)) + 24tan(x)tan^{2}(tan(x))sec^{2}(tan(tan(x)))sec^{6}(x)sec^{2}(tan(x)) + 8tan^{3}(tan(x))sec^{8}(x)sec^{2}(tan(tan(x)))sec^{2}(tan(x))\\ \end{split}\end{equation} \]