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.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ tan(\frac{ln(x)}{x})\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = tan(\frac{ln(x)}{x})\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( tan(\frac{ln(x)}{x})\right)}{dx}\\=&sec^{2}(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})\\=&\frac{-ln(x)sec^{2}(\frac{ln(x)}{x})}{x^{2}} + \frac{sec^{2}(\frac{ln(x)}{x})}{x^{2}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-ln(x)sec^{2}(\frac{ln(x)}{x})}{x^{2}} + \frac{sec^{2}(\frac{ln(x)}{x})}{x^{2}}\right)}{dx}\\=&\frac{--2ln(x)sec^{2}(\frac{ln(x)}{x})}{x^{3}} - \frac{sec^{2}(\frac{ln(x)}{x})}{x^{2}(x)} - \frac{ln(x)*2sec^{2}(\frac{ln(x)}{x})tan(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})}{x^{2}} + \frac{-2sec^{2}(\frac{ln(x)}{x})}{x^{3}} + \frac{2sec^{2}(\frac{ln(x)}{x})tan(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})}{x^{2}}\\=&\frac{2ln(x)sec^{2}(\frac{ln(x)}{x})}{x^{3}} - \frac{3sec^{2}(\frac{ln(x)}{x})}{x^{3}} + \frac{2ln^{2}(x)tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{4}} - \frac{4ln(x)tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{4}} + \frac{2tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{4}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{2ln(x)sec^{2}(\frac{ln(x)}{x})}{x^{3}} - \frac{3sec^{2}(\frac{ln(x)}{x})}{x^{3}} + \frac{2ln^{2}(x)tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{4}} - \frac{4ln(x)tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{4}} + \frac{2tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{4}}\right)}{dx}\\=&\frac{2*-3ln(x)sec^{2}(\frac{ln(x)}{x})}{x^{4}} + \frac{2sec^{2}(\frac{ln(x)}{x})}{x^{3}(x)} + \frac{2ln(x)*2sec^{2}(\frac{ln(x)}{x})tan(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})}{x^{3}} - \frac{3*-3sec^{2}(\frac{ln(x)}{x})}{x^{4}} - \frac{3*2sec^{2}(\frac{ln(x)}{x})tan(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})}{x^{3}} + \frac{2*-4ln^{2}(x)tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{5}} + \frac{2*2ln(x)tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{4}(x)} + \frac{2ln^{2}(x)sec^{2}(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})sec^{2}(\frac{ln(x)}{x})}{x^{4}} + \frac{2ln^{2}(x)tan(\frac{ln(x)}{x})*2sec^{2}(\frac{ln(x)}{x})tan(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})}{x^{4}} - \frac{4*-4ln(x)tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{5}} - \frac{4tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{4}(x)} - \frac{4ln(x)sec^{2}(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})sec^{2}(\frac{ln(x)}{x})}{x^{4}} - \frac{4ln(x)tan(\frac{ln(x)}{x})*2sec^{2}(\frac{ln(x)}{x})tan(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})}{x^{4}} + \frac{2*-4tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{5}} + \frac{2sec^{2}(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})sec^{2}(\frac{ln(x)}{x})}{x^{4}} + \frac{2tan(\frac{ln(x)}{x})*2sec^{2}(\frac{ln(x)}{x})tan(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})}{x^{4}}\\=&\frac{-6ln(x)sec^{2}(\frac{ln(x)}{x})}{x^{4}} + \frac{11sec^{2}(\frac{ln(x)}{x})}{x^{4}} - \frac{12ln^{2}(x)tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{5}} + \frac{30ln(x)tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{5}} - \frac{18tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{5}} - \frac{2ln^{3}(x)sec^{4}(\frac{ln(x)}{x})}{x^{6}} + \frac{6ln^{2}(x)sec^{4}(\frac{ln(x)}{x})}{x^{6}} - \frac{4ln^{3}(x)tan^{2}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{6}} + \frac{12ln^{2}(x)tan^{2}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{6}} - \frac{6ln(x)sec^{4}(\frac{ln(x)}{x})}{x^{6}} - \frac{12ln(x)tan^{2}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{6}} + \frac{2sec^{4}(\frac{ln(x)}{x})}{x^{6}} + \frac{4tan^{2}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{6}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-6ln(x)sec^{2}(\frac{ln(x)}{x})}{x^{4}} + \frac{11sec^{2}(\frac{ln(x)}{x})}{x^{4}} - \frac{12ln^{2}(x)tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{5}} + \frac{30ln(x)tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{5}} - \frac{18tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{5}} - \frac{2ln^{3}(x)sec^{4}(\frac{ln(x)}{x})}{x^{6}} + \frac{6ln^{2}(x)sec^{4}(\frac{ln(x)}{x})}{x^{6}} - \frac{4ln^{3}(x)tan^{2}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{6}} + \frac{12ln^{2}(x)tan^{2}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{6}} - \frac{6ln(x)sec^{4}(\frac{ln(x)}{x})}{x^{6}} - \frac{12ln(x)tan^{2}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{6}} + \frac{2sec^{4}(\frac{ln(x)}{x})}{x^{6}} + \frac{4tan^{2}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{6}}\right)}{dx}\\=&\frac{-6*-4ln(x)sec^{2}(\frac{ln(x)}{x})}{x^{5}} - \frac{6sec^{2}(\frac{ln(x)}{x})}{x^{4}(x)} - \frac{6ln(x)*2sec^{2}(\frac{ln(x)}{x})tan(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})}{x^{4}} + \frac{11*-4sec^{2}(\frac{ln(x)}{x})}{x^{5}} + \frac{11*2sec^{2}(\frac{ln(x)}{x})tan(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})}{x^{4}} - \frac{12*-5ln^{2}(x)tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{6}} - \frac{12*2ln(x)tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{5}(x)} - \frac{12ln^{2}(x)sec^{2}(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})sec^{2}(\frac{ln(x)}{x})}{x^{5}} - \frac{12ln^{2}(x)tan(\frac{ln(x)}{x})*2sec^{2}(\frac{ln(x)}{x})tan(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})}{x^{5}} + \frac{30*-5ln(x)tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{6}} + \frac{30tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{5}(x)} + \frac{30ln(x)sec^{2}(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})sec^{2}(\frac{ln(x)}{x})}{x^{5}} + \frac{30ln(x)tan(\frac{ln(x)}{x})*2sec^{2}(\frac{ln(x)}{x})tan(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})}{x^{5}} - \frac{18*-5tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{6}} - \frac{18sec^{2}(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})sec^{2}(\frac{ln(x)}{x})}{x^{5}} - \frac{18tan(\frac{ln(x)}{x})*2sec^{2}(\frac{ln(x)}{x})tan(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})}{x^{5}} - \frac{2*-6ln^{3}(x)sec^{4}(\frac{ln(x)}{x})}{x^{7}} - \frac{2*3ln^{2}(x)sec^{4}(\frac{ln(x)}{x})}{x^{6}(x)} - \frac{2ln^{3}(x)*4sec^{4}(\frac{ln(x)}{x})tan(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})}{x^{6}} + \frac{6*-6ln^{2}(x)sec^{4}(\frac{ln(x)}{x})}{x^{7}} + \frac{6*2ln(x)sec^{4}(\frac{ln(x)}{x})}{x^{6}(x)} + \frac{6ln^{2}(x)*4sec^{4}(\frac{ln(x)}{x})tan(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})}{x^{6}} - \frac{4*-6ln^{3}(x)tan^{2}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{7}} - \frac{4*3ln^{2}(x)tan^{2}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{6}(x)} - \frac{4ln^{3}(x)*2tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})sec^{2}(\frac{ln(x)}{x})}{x^{6}} - \frac{4ln^{3}(x)tan^{2}(\frac{ln(x)}{x})*2sec^{2}(\frac{ln(x)}{x})tan(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})}{x^{6}} + \frac{12*-6ln^{2}(x)tan^{2}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{7}} + \frac{12*2ln(x)tan^{2}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{6}(x)} + \frac{12ln^{2}(x)*2tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})sec^{2}(\frac{ln(x)}{x})}{x^{6}} + \frac{12ln^{2}(x)tan^{2}(\frac{ln(x)}{x})*2sec^{2}(\frac{ln(x)}{x})tan(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})}{x^{6}} - \frac{6*-6ln(x)sec^{4}(\frac{ln(x)}{x})}{x^{7}} - \frac{6sec^{4}(\frac{ln(x)}{x})}{x^{6}(x)} - \frac{6ln(x)*4sec^{4}(\frac{ln(x)}{x})tan(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})}{x^{6}} - \frac{12*-6ln(x)tan^{2}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{7}} - \frac{12tan^{2}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{6}(x)} - \frac{12ln(x)*2tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})sec^{2}(\frac{ln(x)}{x})}{x^{6}} - \frac{12ln(x)tan^{2}(\frac{ln(x)}{x})*2sec^{2}(\frac{ln(x)}{x})tan(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})}{x^{6}} + \frac{2*-6sec^{4}(\frac{ln(x)}{x})}{x^{7}} + \frac{2*4sec^{4}(\frac{ln(x)}{x})tan(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})}{x^{6}} + \frac{4*-6tan^{2}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{7}} + \frac{4*2tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})sec^{2}(\frac{ln(x)}{x})}{x^{6}} + \frac{4tan^{2}(\frac{ln(x)}{x})*2sec^{2}(\frac{ln(x)}{x})tan(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})}{x^{6}}\\=&\frac{24ln(x)sec^{2}(\frac{ln(x)}{x})}{x^{5}} - \frac{50sec^{2}(\frac{ln(x)}{x})}{x^{5}} + \frac{16ln^{4}(x)tan(\frac{ln(x)}{x})sec^{4}(\frac{ln(x)}{x})}{x^{8}} - \frac{64ln^{3}(x)tan(\frac{ln(x)}{x})sec^{4}(\frac{ln(x)}{x})}{x^{8}} - \frac{64ln(x)tan(\frac{ln(x)}{x})sec^{4}(\frac{ln(x)}{x})}{x^{8}} + \frac{16tan(\frac{ln(x)}{x})sec^{4}(\frac{ln(x)}{x})}{x^{8}} + \frac{72ln^{2}(x)tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{6}} - \frac{208ln(x)tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{6}} + \frac{24ln^{3}(x)sec^{4}(\frac{ln(x)}{x})}{x^{7}} - \frac{84ln^{2}(x)sec^{4}(\frac{ln(x)}{x})}{x^{7}} + \frac{48ln^{3}(x)tan^{2}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{7}} - \frac{168ln^{2}(x)tan^{2}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{7}} + \frac{142tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{6}} + \frac{96ln(x)sec^{4}(\frac{ln(x)}{x})}{x^{7}} + \frac{192ln(x)tan^{2}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{7}} - \frac{36sec^{4}(\frac{ln(x)}{x})}{x^{7}} - \frac{72tan^{2}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{7}} + \frac{96ln^{2}(x)tan(\frac{ln(x)}{x})sec^{4}(\frac{ln(x)}{x})}{x^{8}} + \frac{8ln^{4}(x)tan^{3}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{8}} - \frac{32ln^{3}(x)tan^{3}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{8}} + \frac{48ln^{2}(x)tan^{3}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{8}} - \frac{32ln(x)tan^{3}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{8}} + \frac{8tan^{3}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{8}}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ tan(\frac{ln(x)}{x})\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = tan(\frac{ln(x)}{x})\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( tan(\frac{ln(x)}{x})\right)}{dx}\\=&sec^{2}(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})\\=&\frac{-ln(x)sec^{2}(\frac{ln(x)}{x})}{x^{2}} + \frac{sec^{2}(\frac{ln(x)}{x})}{x^{2}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-ln(x)sec^{2}(\frac{ln(x)}{x})}{x^{2}} + \frac{sec^{2}(\frac{ln(x)}{x})}{x^{2}}\right)}{dx}\\=&\frac{--2ln(x)sec^{2}(\frac{ln(x)}{x})}{x^{3}} - \frac{sec^{2}(\frac{ln(x)}{x})}{x^{2}(x)} - \frac{ln(x)*2sec^{2}(\frac{ln(x)}{x})tan(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})}{x^{2}} + \frac{-2sec^{2}(\frac{ln(x)}{x})}{x^{3}} + \frac{2sec^{2}(\frac{ln(x)}{x})tan(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})}{x^{2}}\\=&\frac{2ln(x)sec^{2}(\frac{ln(x)}{x})}{x^{3}} - \frac{3sec^{2}(\frac{ln(x)}{x})}{x^{3}} + \frac{2ln^{2}(x)tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{4}} - \frac{4ln(x)tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{4}} + \frac{2tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{4}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{2ln(x)sec^{2}(\frac{ln(x)}{x})}{x^{3}} - \frac{3sec^{2}(\frac{ln(x)}{x})}{x^{3}} + \frac{2ln^{2}(x)tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{4}} - \frac{4ln(x)tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{4}} + \frac{2tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{4}}\right)}{dx}\\=&\frac{2*-3ln(x)sec^{2}(\frac{ln(x)}{x})}{x^{4}} + \frac{2sec^{2}(\frac{ln(x)}{x})}{x^{3}(x)} + \frac{2ln(x)*2sec^{2}(\frac{ln(x)}{x})tan(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})}{x^{3}} - \frac{3*-3sec^{2}(\frac{ln(x)}{x})}{x^{4}} - \frac{3*2sec^{2}(\frac{ln(x)}{x})tan(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})}{x^{3}} + \frac{2*-4ln^{2}(x)tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{5}} + \frac{2*2ln(x)tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{4}(x)} + \frac{2ln^{2}(x)sec^{2}(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})sec^{2}(\frac{ln(x)}{x})}{x^{4}} + \frac{2ln^{2}(x)tan(\frac{ln(x)}{x})*2sec^{2}(\frac{ln(x)}{x})tan(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})}{x^{4}} - \frac{4*-4ln(x)tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{5}} - \frac{4tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{4}(x)} - \frac{4ln(x)sec^{2}(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})sec^{2}(\frac{ln(x)}{x})}{x^{4}} - \frac{4ln(x)tan(\frac{ln(x)}{x})*2sec^{2}(\frac{ln(x)}{x})tan(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})}{x^{4}} + \frac{2*-4tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{5}} + \frac{2sec^{2}(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})sec^{2}(\frac{ln(x)}{x})}{x^{4}} + \frac{2tan(\frac{ln(x)}{x})*2sec^{2}(\frac{ln(x)}{x})tan(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})}{x^{4}}\\=&\frac{-6ln(x)sec^{2}(\frac{ln(x)}{x})}{x^{4}} + \frac{11sec^{2}(\frac{ln(x)}{x})}{x^{4}} - \frac{12ln^{2}(x)tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{5}} + \frac{30ln(x)tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{5}} - \frac{18tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{5}} - \frac{2ln^{3}(x)sec^{4}(\frac{ln(x)}{x})}{x^{6}} + \frac{6ln^{2}(x)sec^{4}(\frac{ln(x)}{x})}{x^{6}} - \frac{4ln^{3}(x)tan^{2}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{6}} + \frac{12ln^{2}(x)tan^{2}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{6}} - \frac{6ln(x)sec^{4}(\frac{ln(x)}{x})}{x^{6}} - \frac{12ln(x)tan^{2}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{6}} + \frac{2sec^{4}(\frac{ln(x)}{x})}{x^{6}} + \frac{4tan^{2}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{6}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-6ln(x)sec^{2}(\frac{ln(x)}{x})}{x^{4}} + \frac{11sec^{2}(\frac{ln(x)}{x})}{x^{4}} - \frac{12ln^{2}(x)tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{5}} + \frac{30ln(x)tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{5}} - \frac{18tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{5}} - \frac{2ln^{3}(x)sec^{4}(\frac{ln(x)}{x})}{x^{6}} + \frac{6ln^{2}(x)sec^{4}(\frac{ln(x)}{x})}{x^{6}} - \frac{4ln^{3}(x)tan^{2}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{6}} + \frac{12ln^{2}(x)tan^{2}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{6}} - \frac{6ln(x)sec^{4}(\frac{ln(x)}{x})}{x^{6}} - \frac{12ln(x)tan^{2}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{6}} + \frac{2sec^{4}(\frac{ln(x)}{x})}{x^{6}} + \frac{4tan^{2}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{6}}\right)}{dx}\\=&\frac{-6*-4ln(x)sec^{2}(\frac{ln(x)}{x})}{x^{5}} - \frac{6sec^{2}(\frac{ln(x)}{x})}{x^{4}(x)} - \frac{6ln(x)*2sec^{2}(\frac{ln(x)}{x})tan(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})}{x^{4}} + \frac{11*-4sec^{2}(\frac{ln(x)}{x})}{x^{5}} + \frac{11*2sec^{2}(\frac{ln(x)}{x})tan(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})}{x^{4}} - \frac{12*-5ln^{2}(x)tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{6}} - \frac{12*2ln(x)tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{5}(x)} - \frac{12ln^{2}(x)sec^{2}(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})sec^{2}(\frac{ln(x)}{x})}{x^{5}} - \frac{12ln^{2}(x)tan(\frac{ln(x)}{x})*2sec^{2}(\frac{ln(x)}{x})tan(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})}{x^{5}} + \frac{30*-5ln(x)tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{6}} + \frac{30tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{5}(x)} + \frac{30ln(x)sec^{2}(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})sec^{2}(\frac{ln(x)}{x})}{x^{5}} + \frac{30ln(x)tan(\frac{ln(x)}{x})*2sec^{2}(\frac{ln(x)}{x})tan(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})}{x^{5}} - \frac{18*-5tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{6}} - \frac{18sec^{2}(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})sec^{2}(\frac{ln(x)}{x})}{x^{5}} - \frac{18tan(\frac{ln(x)}{x})*2sec^{2}(\frac{ln(x)}{x})tan(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})}{x^{5}} - \frac{2*-6ln^{3}(x)sec^{4}(\frac{ln(x)}{x})}{x^{7}} - \frac{2*3ln^{2}(x)sec^{4}(\frac{ln(x)}{x})}{x^{6}(x)} - \frac{2ln^{3}(x)*4sec^{4}(\frac{ln(x)}{x})tan(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})}{x^{6}} + \frac{6*-6ln^{2}(x)sec^{4}(\frac{ln(x)}{x})}{x^{7}} + \frac{6*2ln(x)sec^{4}(\frac{ln(x)}{x})}{x^{6}(x)} + \frac{6ln^{2}(x)*4sec^{4}(\frac{ln(x)}{x})tan(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})}{x^{6}} - \frac{4*-6ln^{3}(x)tan^{2}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{7}} - \frac{4*3ln^{2}(x)tan^{2}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{6}(x)} - \frac{4ln^{3}(x)*2tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})sec^{2}(\frac{ln(x)}{x})}{x^{6}} - \frac{4ln^{3}(x)tan^{2}(\frac{ln(x)}{x})*2sec^{2}(\frac{ln(x)}{x})tan(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})}{x^{6}} + \frac{12*-6ln^{2}(x)tan^{2}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{7}} + \frac{12*2ln(x)tan^{2}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{6}(x)} + \frac{12ln^{2}(x)*2tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})sec^{2}(\frac{ln(x)}{x})}{x^{6}} + \frac{12ln^{2}(x)tan^{2}(\frac{ln(x)}{x})*2sec^{2}(\frac{ln(x)}{x})tan(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})}{x^{6}} - \frac{6*-6ln(x)sec^{4}(\frac{ln(x)}{x})}{x^{7}} - \frac{6sec^{4}(\frac{ln(x)}{x})}{x^{6}(x)} - \frac{6ln(x)*4sec^{4}(\frac{ln(x)}{x})tan(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})}{x^{6}} - \frac{12*-6ln(x)tan^{2}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{7}} - \frac{12tan^{2}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{6}(x)} - \frac{12ln(x)*2tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})sec^{2}(\frac{ln(x)}{x})}{x^{6}} - \frac{12ln(x)tan^{2}(\frac{ln(x)}{x})*2sec^{2}(\frac{ln(x)}{x})tan(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})}{x^{6}} + \frac{2*-6sec^{4}(\frac{ln(x)}{x})}{x^{7}} + \frac{2*4sec^{4}(\frac{ln(x)}{x})tan(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})}{x^{6}} + \frac{4*-6tan^{2}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{7}} + \frac{4*2tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})sec^{2}(\frac{ln(x)}{x})}{x^{6}} + \frac{4tan^{2}(\frac{ln(x)}{x})*2sec^{2}(\frac{ln(x)}{x})tan(\frac{ln(x)}{x})(\frac{-ln(x)}{x^{2}} + \frac{1}{x(x)})}{x^{6}}\\=&\frac{24ln(x)sec^{2}(\frac{ln(x)}{x})}{x^{5}} - \frac{50sec^{2}(\frac{ln(x)}{x})}{x^{5}} + \frac{16ln^{4}(x)tan(\frac{ln(x)}{x})sec^{4}(\frac{ln(x)}{x})}{x^{8}} - \frac{64ln^{3}(x)tan(\frac{ln(x)}{x})sec^{4}(\frac{ln(x)}{x})}{x^{8}} - \frac{64ln(x)tan(\frac{ln(x)}{x})sec^{4}(\frac{ln(x)}{x})}{x^{8}} + \frac{16tan(\frac{ln(x)}{x})sec^{4}(\frac{ln(x)}{x})}{x^{8}} + \frac{72ln^{2}(x)tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{6}} - \frac{208ln(x)tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{6}} + \frac{24ln^{3}(x)sec^{4}(\frac{ln(x)}{x})}{x^{7}} - \frac{84ln^{2}(x)sec^{4}(\frac{ln(x)}{x})}{x^{7}} + \frac{48ln^{3}(x)tan^{2}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{7}} - \frac{168ln^{2}(x)tan^{2}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{7}} + \frac{142tan(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{6}} + \frac{96ln(x)sec^{4}(\frac{ln(x)}{x})}{x^{7}} + \frac{192ln(x)tan^{2}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{7}} - \frac{36sec^{4}(\frac{ln(x)}{x})}{x^{7}} - \frac{72tan^{2}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{7}} + \frac{96ln^{2}(x)tan(\frac{ln(x)}{x})sec^{4}(\frac{ln(x)}{x})}{x^{8}} + \frac{8ln^{4}(x)tan^{3}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{8}} - \frac{32ln^{3}(x)tan^{3}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{8}} + \frac{48ln^{2}(x)tan^{3}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{8}} - \frac{32ln(x)tan^{3}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{8}} + \frac{8tan^{3}(\frac{ln(x)}{x})sec^{2}(\frac{ln(x)}{x})}{x^{8}}\\ \end{split}\end{equation} \]