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 2 questions in this calculation: for each question, the 4 derivative of x is calculated.
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/2]Find\ the\ 4th\ derivative\ of\ function\ tan(x + y)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( tan(x + y)\right)}{dx}\\=&sec^{2}(x + y)(1 + 0)\\=&sec^{2}(x + y)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( sec^{2}(x + y)\right)}{dx}\\=&2sec^{2}(x + y)tan(x + y)(1 + 0)\\=&2tan(x + y)sec^{2}(x + y)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 2tan(x + y)sec^{2}(x + y)\right)}{dx}\\=&2sec^{2}(x + y)(1 + 0)sec^{2}(x + y) + 2tan(x + y)*2sec^{2}(x + y)tan(x + y)(1 + 0)\\=&2sec^{4}(x + y) + 4tan^{2}(x + y)sec^{2}(x + y)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 2sec^{4}(x + y) + 4tan^{2}(x + y)sec^{2}(x + y)\right)}{dx}\\=&2*4sec^{4}(x + y)tan(x + y)(1 + 0) + 4*2tan(x + y)sec^{2}(x + y)(1 + 0)sec^{2}(x + y) + 4tan^{2}(x + y)*2sec^{2}(x + y)tan(x + y)(1 + 0)\\=&16tan(x + y)sec^{4}(x + y) + 8tan^{3}(x + y)sec^{2}(x + y)\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/2]Find\ the\ 4th\ derivative\ of\ function\ \frac{(tan(x) + tan(y))}{(1 - tan(x)tan(y))}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{tan(x)}{(-tan(x)tan(y) + 1)} + \frac{tan(y)}{(-tan(x)tan(y) + 1)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{tan(x)}{(-tan(x)tan(y) + 1)} + \frac{tan(y)}{(-tan(x)tan(y) + 1)}\right)}{dx}\\=&(\frac{-(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{2}})tan(x) + \frac{sec^{2}(x)(1)}{(-tan(x)tan(y) + 1)} + (\frac{-(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{2}})tan(y) + \frac{sec^{2}(y)(0)}{(-tan(x)tan(y) + 1)}\\=&\frac{tan(y)tan(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{sec^{2}(x)}{(-tan(x)tan(y) + 1)} + \frac{tan^{2}(y)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{tan(y)tan(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{sec^{2}(x)}{(-tan(x)tan(y) + 1)} + \frac{tan^{2}(y)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}}\right)}{dx}\\=&(\frac{-2(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{3}})tan(y)tan(x)sec^{2}(x) + \frac{sec^{2}(y)(0)tan(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{tan(y)sec^{2}(x)(1)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{tan(y)tan(x)*2sec^{2}(x)tan(x)}{(-tan(x)tan(y) + 1)^{2}} + (\frac{-(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{2}})sec^{2}(x) + \frac{2sec^{2}(x)tan(x)}{(-tan(x)tan(y) + 1)} + (\frac{-2(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{3}})tan^{2}(y)sec^{2}(x) + \frac{2tan(y)sec^{2}(y)(0)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{tan^{2}(y)*2sec^{2}(x)tan(x)}{(-tan(x)tan(y) + 1)^{2}}\\=&\frac{2tan^{2}(y)tan(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan(y)tan^{2}(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan^{3}(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)} + \frac{2tan(x)tan^{2}(y)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{2tan^{2}(y)tan(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan(y)tan^{2}(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan^{3}(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)} + \frac{2tan(x)tan^{2}(y)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}}\right)}{dx}\\=&2(\frac{-3(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{4}})tan^{2}(y)tan(x)sec^{4}(x) + \frac{2*2tan(y)sec^{2}(y)(0)tan(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan^{2}(y)sec^{2}(x)(1)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan^{2}(y)tan(x)*4sec^{4}(x)tan(x)}{(-tan(x)tan(y) + 1)^{3}} + 2(\frac{-2(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{3}})tan(y)sec^{4}(x) + \frac{2sec^{2}(y)(0)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan(y)*4sec^{4}(x)tan(x)}{(-tan(x)tan(y) + 1)^{2}} + 2(\frac{-2(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{3}})tan(y)tan^{2}(x)sec^{2}(x) + \frac{2sec^{2}(y)(0)tan^{2}(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan(y)*2tan(x)sec^{2}(x)(1)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan(y)tan^{2}(x)*2sec^{2}(x)tan(x)}{(-tan(x)tan(y) + 1)^{2}} + 2(\frac{-3(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{4}})tan^{3}(y)sec^{4}(x) + \frac{2*3tan^{2}(y)sec^{2}(y)(0)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan^{3}(y)*4sec^{4}(x)tan(x)}{(-tan(x)tan(y) + 1)^{3}} + 2(\frac{-(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{2}})tan(x)sec^{2}(x) + \frac{2sec^{2}(x)(1)sec^{2}(x)}{(-tan(x)tan(y) + 1)} + \frac{2tan(x)*2sec^{2}(x)tan(x)}{(-tan(x)tan(y) + 1)} + 2(\frac{-2(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{3}})tan(x)tan^{2}(y)sec^{2}(x) + \frac{2sec^{2}(x)(1)tan^{2}(y)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan(x)*2tan(y)sec^{2}(y)(0)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan(x)tan^{2}(y)*2sec^{2}(x)tan(x)}{(-tan(x)tan(y) + 1)^{2}}\\=&\frac{6tan^{3}(y)tan(x)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{6tan^{2}(y)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{12tan^{2}(y)tan^{2}(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{12tan(x)tan(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{4tan(y)tan^{3}(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{6tan^{4}(y)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{8tan(x)tan^{3}(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan(y)tan(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2sec^{4}(x)}{(-tan(x)tan(y) + 1)} + \frac{4tan^{2}(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)} + \frac{4tan^{3}(y)tan(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan^{2}(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{4tan^{2}(x)tan^{2}(y)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{6tan^{3}(y)tan(x)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{6tan^{2}(y)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{12tan^{2}(y)tan^{2}(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{12tan(x)tan(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{4tan(y)tan^{3}(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{6tan^{4}(y)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{8tan(x)tan^{3}(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan(y)tan(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2sec^{4}(x)}{(-tan(x)tan(y) + 1)} + \frac{4tan^{2}(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)} + \frac{4tan^{3}(y)tan(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan^{2}(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{4tan^{2}(x)tan^{2}(y)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}}\right)}{dx}\\=&6(\frac{-4(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{5}})tan^{3}(y)tan(x)sec^{6}(x) + \frac{6*3tan^{2}(y)sec^{2}(y)(0)tan(x)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{6tan^{3}(y)sec^{2}(x)(1)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{6tan^{3}(y)tan(x)*6sec^{6}(x)tan(x)}{(-tan(x)tan(y) + 1)^{4}} + 6(\frac{-3(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{4}})tan^{2}(y)sec^{6}(x) + \frac{6*2tan(y)sec^{2}(y)(0)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{6tan^{2}(y)*6sec^{6}(x)tan(x)}{(-tan(x)tan(y) + 1)^{3}} + 12(\frac{-3(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{4}})tan^{2}(y)tan^{2}(x)sec^{4}(x) + \frac{12*2tan(y)sec^{2}(y)(0)tan^{2}(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{12tan^{2}(y)*2tan(x)sec^{2}(x)(1)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{12tan^{2}(y)tan^{2}(x)*4sec^{4}(x)tan(x)}{(-tan(x)tan(y) + 1)^{3}} + 12(\frac{-2(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{3}})tan(x)tan(y)sec^{4}(x) + \frac{12sec^{2}(x)(1)tan(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{12tan(x)sec^{2}(y)(0)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{12tan(x)tan(y)*4sec^{4}(x)tan(x)}{(-tan(x)tan(y) + 1)^{2}} + 4(\frac{-2(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{3}})tan(y)tan^{3}(x)sec^{2}(x) + \frac{4sec^{2}(y)(0)tan^{3}(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{4tan(y)*3tan^{2}(x)sec^{2}(x)(1)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{4tan(y)tan^{3}(x)*2sec^{2}(x)tan(x)}{(-tan(x)tan(y) + 1)^{2}} + 6(\frac{-4(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{5}})tan^{4}(y)sec^{6}(x) + \frac{6*4tan^{3}(y)sec^{2}(y)(0)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{6tan^{4}(y)*6sec^{6}(x)tan(x)}{(-tan(x)tan(y) + 1)^{4}} + 8(\frac{-3(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{4}})tan(x)tan^{3}(y)sec^{4}(x) + \frac{8sec^{2}(x)(1)tan^{3}(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{8tan(x)*3tan^{2}(y)sec^{2}(y)(0)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{8tan(x)tan^{3}(y)*4sec^{4}(x)tan(x)}{(-tan(x)tan(y) + 1)^{3}} + 2(\frac{-2(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{3}})tan(y)tan(x)sec^{4}(x) + \frac{2sec^{2}(y)(0)tan(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan(y)sec^{2}(x)(1)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan(y)tan(x)*4sec^{4}(x)tan(x)}{(-tan(x)tan(y) + 1)^{2}} + 2(\frac{-(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{2}})sec^{4}(x) + \frac{2*4sec^{4}(x)tan(x)}{(-tan(x)tan(y) + 1)} + 4(\frac{-(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{2}})tan^{2}(x)sec^{2}(x) + \frac{4*2tan(x)sec^{2}(x)(1)sec^{2}(x)}{(-tan(x)tan(y) + 1)} + \frac{4tan^{2}(x)*2sec^{2}(x)tan(x)}{(-tan(x)tan(y) + 1)} + 4(\frac{-3(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{4}})tan^{3}(y)tan(x)sec^{4}(x) + \frac{4*3tan^{2}(y)sec^{2}(y)(0)tan(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{4tan^{3}(y)sec^{2}(x)(1)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{4tan^{3}(y)tan(x)*4sec^{4}(x)tan(x)}{(-tan(x)tan(y) + 1)^{3}} + 2(\frac{-2(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{3}})tan^{2}(y)sec^{4}(x) + \frac{2*2tan(y)sec^{2}(y)(0)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan^{2}(y)*4sec^{4}(x)tan(x)}{(-tan(x)tan(y) + 1)^{2}} + 4(\frac{-2(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{3}})tan^{2}(x)tan^{2}(y)sec^{2}(x) + \frac{4*2tan(x)sec^{2}(x)(1)tan^{2}(y)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{4tan^{2}(x)*2tan(y)sec^{2}(y)(0)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{4tan^{2}(x)tan^{2}(y)*2sec^{2}(x)tan(x)}{(-tan(x)tan(y) + 1)^{2}}\\=&\frac{24tan^{4}(y)tan(x)sec^{8}(x)}{(-tan(x)tan(y) + 1)^{5}} + \frac{24tan^{3}(y)sec^{8}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{72tan^{3}(y)tan^{2}(x)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{60tan(x)tan^{2}(y)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{60tan^{2}(x)tan(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{56tan^{2}(y)tan^{3}(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{28tan^{2}(y)tan(x)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{16tan(y)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{8tan(y)tan^{4}(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{24tan^{5}(y)sec^{8}(x)}{(-tan(x)tan(y) + 1)^{5}} + \frac{36tan^{4}(y)tan(x)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{16tan^{3}(y)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{16tan(x)tan^{2}(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{32tan^{2}(x)tan^{3}(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{12tan(y)tan^{2}(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{16tan(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)} + \frac{36tan(x)tan^{4}(y)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{24tan^{3}(y)tan^{2}(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{8tan^{3}(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)} + \frac{8tan^{3}(x)tan^{2}(y)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/2]Find\ the\ 4th\ derivative\ of\ function\ tan(x + y)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( tan(x + y)\right)}{dx}\\=&sec^{2}(x + y)(1 + 0)\\=&sec^{2}(x + y)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( sec^{2}(x + y)\right)}{dx}\\=&2sec^{2}(x + y)tan(x + y)(1 + 0)\\=&2tan(x + y)sec^{2}(x + y)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 2tan(x + y)sec^{2}(x + y)\right)}{dx}\\=&2sec^{2}(x + y)(1 + 0)sec^{2}(x + y) + 2tan(x + y)*2sec^{2}(x + y)tan(x + y)(1 + 0)\\=&2sec^{4}(x + y) + 4tan^{2}(x + y)sec^{2}(x + y)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 2sec^{4}(x + y) + 4tan^{2}(x + y)sec^{2}(x + y)\right)}{dx}\\=&2*4sec^{4}(x + y)tan(x + y)(1 + 0) + 4*2tan(x + y)sec^{2}(x + y)(1 + 0)sec^{2}(x + y) + 4tan^{2}(x + y)*2sec^{2}(x + y)tan(x + y)(1 + 0)\\=&16tan(x + y)sec^{4}(x + y) + 8tan^{3}(x + y)sec^{2}(x + y)\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/2]Find\ the\ 4th\ derivative\ of\ function\ \frac{(tan(x) + tan(y))}{(1 - tan(x)tan(y))}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{tan(x)}{(-tan(x)tan(y) + 1)} + \frac{tan(y)}{(-tan(x)tan(y) + 1)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{tan(x)}{(-tan(x)tan(y) + 1)} + \frac{tan(y)}{(-tan(x)tan(y) + 1)}\right)}{dx}\\=&(\frac{-(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{2}})tan(x) + \frac{sec^{2}(x)(1)}{(-tan(x)tan(y) + 1)} + (\frac{-(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{2}})tan(y) + \frac{sec^{2}(y)(0)}{(-tan(x)tan(y) + 1)}\\=&\frac{tan(y)tan(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{sec^{2}(x)}{(-tan(x)tan(y) + 1)} + \frac{tan^{2}(y)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{tan(y)tan(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{sec^{2}(x)}{(-tan(x)tan(y) + 1)} + \frac{tan^{2}(y)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}}\right)}{dx}\\=&(\frac{-2(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{3}})tan(y)tan(x)sec^{2}(x) + \frac{sec^{2}(y)(0)tan(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{tan(y)sec^{2}(x)(1)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{tan(y)tan(x)*2sec^{2}(x)tan(x)}{(-tan(x)tan(y) + 1)^{2}} + (\frac{-(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{2}})sec^{2}(x) + \frac{2sec^{2}(x)tan(x)}{(-tan(x)tan(y) + 1)} + (\frac{-2(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{3}})tan^{2}(y)sec^{2}(x) + \frac{2tan(y)sec^{2}(y)(0)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{tan^{2}(y)*2sec^{2}(x)tan(x)}{(-tan(x)tan(y) + 1)^{2}}\\=&\frac{2tan^{2}(y)tan(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan(y)tan^{2}(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan^{3}(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)} + \frac{2tan(x)tan^{2}(y)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{2tan^{2}(y)tan(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan(y)tan^{2}(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan^{3}(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)} + \frac{2tan(x)tan^{2}(y)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}}\right)}{dx}\\=&2(\frac{-3(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{4}})tan^{2}(y)tan(x)sec^{4}(x) + \frac{2*2tan(y)sec^{2}(y)(0)tan(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan^{2}(y)sec^{2}(x)(1)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan^{2}(y)tan(x)*4sec^{4}(x)tan(x)}{(-tan(x)tan(y) + 1)^{3}} + 2(\frac{-2(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{3}})tan(y)sec^{4}(x) + \frac{2sec^{2}(y)(0)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan(y)*4sec^{4}(x)tan(x)}{(-tan(x)tan(y) + 1)^{2}} + 2(\frac{-2(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{3}})tan(y)tan^{2}(x)sec^{2}(x) + \frac{2sec^{2}(y)(0)tan^{2}(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan(y)*2tan(x)sec^{2}(x)(1)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan(y)tan^{2}(x)*2sec^{2}(x)tan(x)}{(-tan(x)tan(y) + 1)^{2}} + 2(\frac{-3(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{4}})tan^{3}(y)sec^{4}(x) + \frac{2*3tan^{2}(y)sec^{2}(y)(0)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan^{3}(y)*4sec^{4}(x)tan(x)}{(-tan(x)tan(y) + 1)^{3}} + 2(\frac{-(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{2}})tan(x)sec^{2}(x) + \frac{2sec^{2}(x)(1)sec^{2}(x)}{(-tan(x)tan(y) + 1)} + \frac{2tan(x)*2sec^{2}(x)tan(x)}{(-tan(x)tan(y) + 1)} + 2(\frac{-2(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{3}})tan(x)tan^{2}(y)sec^{2}(x) + \frac{2sec^{2}(x)(1)tan^{2}(y)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan(x)*2tan(y)sec^{2}(y)(0)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan(x)tan^{2}(y)*2sec^{2}(x)tan(x)}{(-tan(x)tan(y) + 1)^{2}}\\=&\frac{6tan^{3}(y)tan(x)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{6tan^{2}(y)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{12tan^{2}(y)tan^{2}(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{12tan(x)tan(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{4tan(y)tan^{3}(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{6tan^{4}(y)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{8tan(x)tan^{3}(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan(y)tan(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2sec^{4}(x)}{(-tan(x)tan(y) + 1)} + \frac{4tan^{2}(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)} + \frac{4tan^{3}(y)tan(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan^{2}(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{4tan^{2}(x)tan^{2}(y)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{6tan^{3}(y)tan(x)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{6tan^{2}(y)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{12tan^{2}(y)tan^{2}(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{12tan(x)tan(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{4tan(y)tan^{3}(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{6tan^{4}(y)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{8tan(x)tan^{3}(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan(y)tan(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2sec^{4}(x)}{(-tan(x)tan(y) + 1)} + \frac{4tan^{2}(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)} + \frac{4tan^{3}(y)tan(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan^{2}(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{4tan^{2}(x)tan^{2}(y)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}}\right)}{dx}\\=&6(\frac{-4(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{5}})tan^{3}(y)tan(x)sec^{6}(x) + \frac{6*3tan^{2}(y)sec^{2}(y)(0)tan(x)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{6tan^{3}(y)sec^{2}(x)(1)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{6tan^{3}(y)tan(x)*6sec^{6}(x)tan(x)}{(-tan(x)tan(y) + 1)^{4}} + 6(\frac{-3(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{4}})tan^{2}(y)sec^{6}(x) + \frac{6*2tan(y)sec^{2}(y)(0)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{6tan^{2}(y)*6sec^{6}(x)tan(x)}{(-tan(x)tan(y) + 1)^{3}} + 12(\frac{-3(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{4}})tan^{2}(y)tan^{2}(x)sec^{4}(x) + \frac{12*2tan(y)sec^{2}(y)(0)tan^{2}(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{12tan^{2}(y)*2tan(x)sec^{2}(x)(1)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{12tan^{2}(y)tan^{2}(x)*4sec^{4}(x)tan(x)}{(-tan(x)tan(y) + 1)^{3}} + 12(\frac{-2(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{3}})tan(x)tan(y)sec^{4}(x) + \frac{12sec^{2}(x)(1)tan(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{12tan(x)sec^{2}(y)(0)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{12tan(x)tan(y)*4sec^{4}(x)tan(x)}{(-tan(x)tan(y) + 1)^{2}} + 4(\frac{-2(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{3}})tan(y)tan^{3}(x)sec^{2}(x) + \frac{4sec^{2}(y)(0)tan^{3}(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{4tan(y)*3tan^{2}(x)sec^{2}(x)(1)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{4tan(y)tan^{3}(x)*2sec^{2}(x)tan(x)}{(-tan(x)tan(y) + 1)^{2}} + 6(\frac{-4(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{5}})tan^{4}(y)sec^{6}(x) + \frac{6*4tan^{3}(y)sec^{2}(y)(0)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{6tan^{4}(y)*6sec^{6}(x)tan(x)}{(-tan(x)tan(y) + 1)^{4}} + 8(\frac{-3(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{4}})tan(x)tan^{3}(y)sec^{4}(x) + \frac{8sec^{2}(x)(1)tan^{3}(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{8tan(x)*3tan^{2}(y)sec^{2}(y)(0)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{8tan(x)tan^{3}(y)*4sec^{4}(x)tan(x)}{(-tan(x)tan(y) + 1)^{3}} + 2(\frac{-2(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{3}})tan(y)tan(x)sec^{4}(x) + \frac{2sec^{2}(y)(0)tan(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan(y)sec^{2}(x)(1)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan(y)tan(x)*4sec^{4}(x)tan(x)}{(-tan(x)tan(y) + 1)^{2}} + 2(\frac{-(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{2}})sec^{4}(x) + \frac{2*4sec^{4}(x)tan(x)}{(-tan(x)tan(y) + 1)} + 4(\frac{-(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{2}})tan^{2}(x)sec^{2}(x) + \frac{4*2tan(x)sec^{2}(x)(1)sec^{2}(x)}{(-tan(x)tan(y) + 1)} + \frac{4tan^{2}(x)*2sec^{2}(x)tan(x)}{(-tan(x)tan(y) + 1)} + 4(\frac{-3(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{4}})tan^{3}(y)tan(x)sec^{4}(x) + \frac{4*3tan^{2}(y)sec^{2}(y)(0)tan(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{4tan^{3}(y)sec^{2}(x)(1)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{4tan^{3}(y)tan(x)*4sec^{4}(x)tan(x)}{(-tan(x)tan(y) + 1)^{3}} + 2(\frac{-2(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{3}})tan^{2}(y)sec^{4}(x) + \frac{2*2tan(y)sec^{2}(y)(0)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan^{2}(y)*4sec^{4}(x)tan(x)}{(-tan(x)tan(y) + 1)^{2}} + 4(\frac{-2(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{3}})tan^{2}(x)tan^{2}(y)sec^{2}(x) + \frac{4*2tan(x)sec^{2}(x)(1)tan^{2}(y)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{4tan^{2}(x)*2tan(y)sec^{2}(y)(0)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{4tan^{2}(x)tan^{2}(y)*2sec^{2}(x)tan(x)}{(-tan(x)tan(y) + 1)^{2}}\\=&\frac{24tan^{4}(y)tan(x)sec^{8}(x)}{(-tan(x)tan(y) + 1)^{5}} + \frac{24tan^{3}(y)sec^{8}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{72tan^{3}(y)tan^{2}(x)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{60tan(x)tan^{2}(y)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{60tan^{2}(x)tan(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{56tan^{2}(y)tan^{3}(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{28tan^{2}(y)tan(x)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{16tan(y)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{8tan(y)tan^{4}(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{24tan^{5}(y)sec^{8}(x)}{(-tan(x)tan(y) + 1)^{5}} + \frac{36tan^{4}(y)tan(x)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{16tan^{3}(y)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{16tan(x)tan^{2}(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{32tan^{2}(x)tan^{3}(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{12tan(y)tan^{2}(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{16tan(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)} + \frac{36tan(x)tan^{4}(y)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{24tan^{3}(y)tan^{2}(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{8tan^{3}(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)} + \frac{8tan^{3}(x)tan^{2}(y)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}}\\ \end{split}\end{equation} \]