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.
\[ \begin{equation}\begin{split}[1/2]Find\ the\ 4th\ derivative\ of\ function\ \frac{ln(ln(ln(x)))}{cos(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{ln(ln(ln(x)))}{cos(x)}\right)}{dx}\\=&\frac{1}{(ln(ln(x)))(ln(x))(x)cos(x)} + \frac{ln(ln(ln(x)))sin(x)}{cos^{2}(x)}\\=&\frac{1}{xln(ln(x))ln(x)cos(x)} + \frac{ln(ln(ln(x)))sin(x)}{cos^{2}(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{1}{xln(ln(x))ln(x)cos(x)} + \frac{ln(ln(ln(x)))sin(x)}{cos^{2}(x)}\right)}{dx}\\=&\frac{-1}{x^{2}ln(ln(x))ln(x)cos(x)} + \frac{-1}{xln^{2}(ln(x))(ln(x))(x)ln(x)cos(x)} + \frac{-1}{xln(ln(x))ln^{2}(x)(x)cos(x)} + \frac{sin(x)}{xln(ln(x))ln(x)cos^{2}(x)} + \frac{sin(x)}{(ln(ln(x)))(ln(x))(x)cos^{2}(x)} + \frac{ln(ln(ln(x)))cos(x)}{cos^{2}(x)} + \frac{ln(ln(ln(x)))sin(x)*2sin(x)}{cos^{3}(x)}\\=&\frac{-1}{x^{2}ln(ln(x))ln(x)cos(x)} - \frac{1}{x^{2}ln^{2}(ln(x))ln^{2}(x)cos(x)} - \frac{1}{x^{2}ln^{2}(x)ln(ln(x))cos(x)} + \frac{2sin(x)}{xln(ln(x))ln(x)cos^{2}(x)} + \frac{ln(ln(ln(x)))}{cos(x)} + \frac{2ln(ln(ln(x)))sin^{2}(x)}{cos^{3}(x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-1}{x^{2}ln(ln(x))ln(x)cos(x)} - \frac{1}{x^{2}ln^{2}(ln(x))ln^{2}(x)cos(x)} - \frac{1}{x^{2}ln^{2}(x)ln(ln(x))cos(x)} + \frac{2sin(x)}{xln(ln(x))ln(x)cos^{2}(x)} + \frac{ln(ln(ln(x)))}{cos(x)} + \frac{2ln(ln(ln(x)))sin^{2}(x)}{cos^{3}(x)}\right)}{dx}\\=&\frac{--2}{x^{3}ln(ln(x))ln(x)cos(x)} - \frac{-1}{x^{2}ln^{2}(ln(x))(ln(x))(x)ln(x)cos(x)} - \frac{-1}{x^{2}ln(ln(x))ln^{2}(x)(x)cos(x)} - \frac{sin(x)}{x^{2}ln(ln(x))ln(x)cos^{2}(x)} - \frac{-2}{x^{3}ln^{2}(ln(x))ln^{2}(x)cos(x)} - \frac{-2}{x^{2}ln^{3}(ln(x))(ln(x))(x)ln^{2}(x)cos(x)} - \frac{-2}{x^{2}ln^{2}(ln(x))ln^{3}(x)(x)cos(x)} - \frac{sin(x)}{x^{2}ln^{2}(ln(x))ln^{2}(x)cos^{2}(x)} - \frac{-2}{x^{3}ln^{2}(x)ln(ln(x))cos(x)} - \frac{-2}{x^{2}ln^{3}(x)(x)ln(ln(x))cos(x)} - \frac{-1}{x^{2}ln^{2}(x)ln^{2}(ln(x))(ln(x))(x)cos(x)} - \frac{sin(x)}{x^{2}ln^{2}(x)ln(ln(x))cos^{2}(x)} + \frac{2*-sin(x)}{x^{2}ln(ln(x))ln(x)cos^{2}(x)} + \frac{2*-sin(x)}{xln^{2}(ln(x))(ln(x))(x)ln(x)cos^{2}(x)} + \frac{2*-sin(x)}{xln(ln(x))ln^{2}(x)(x)cos^{2}(x)} + \frac{2cos(x)}{xln(ln(x))ln(x)cos^{2}(x)} + \frac{2sin(x)*2sin(x)}{xln(ln(x))ln(x)cos^{3}(x)} + \frac{1}{(ln(ln(x)))(ln(x))(x)cos(x)} + \frac{ln(ln(ln(x)))sin(x)}{cos^{2}(x)} + \frac{2sin^{2}(x)}{(ln(ln(x)))(ln(x))(x)cos^{3}(x)} + \frac{2ln(ln(ln(x)))*2sin(x)cos(x)}{cos^{3}(x)} + \frac{2ln(ln(ln(x)))sin^{2}(x)*3sin(x)}{cos^{4}(x)}\\=&\frac{2}{xln(x)ln(ln(x))cos(x)} + \frac{3}{x^{3}ln^{2}(ln(x))ln^{2}(x)cos(x)} + \frac{3}{x^{3}ln^{2}(x)ln(ln(x))cos(x)} - \frac{3sin(x)}{x^{2}ln(ln(x))ln(x)cos^{2}(x)} + \frac{2}{x^{3}ln^{3}(ln(x))ln^{3}(x)cos(x)} + \frac{3}{x^{3}ln^{3}(x)ln^{2}(ln(x))cos(x)} - \frac{3sin(x)}{x^{2}ln^{2}(ln(x))ln^{2}(x)cos^{2}(x)} + \frac{2}{x^{3}ln^{3}(x)ln(ln(x))cos(x)} - \frac{3sin(x)}{x^{2}ln^{2}(x)ln(ln(x))cos^{2}(x)} + \frac{2}{x^{3}ln(ln(x))ln(x)cos(x)} + \frac{6sin^{2}(x)}{xln(ln(x))ln(x)cos^{3}(x)} + \frac{1}{xln(ln(x))ln(x)cos(x)} + \frac{5ln(ln(ln(x)))sin(x)}{cos^{2}(x)} + \frac{6ln(ln(ln(x)))sin^{3}(x)}{cos^{4}(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{2}{xln(x)ln(ln(x))cos(x)} + \frac{3}{x^{3}ln^{2}(ln(x))ln^{2}(x)cos(x)} + \frac{3}{x^{3}ln^{2}(x)ln(ln(x))cos(x)} - \frac{3sin(x)}{x^{2}ln(ln(x))ln(x)cos^{2}(x)} + \frac{2}{x^{3}ln^{3}(ln(x))ln^{3}(x)cos(x)} + \frac{3}{x^{3}ln^{3}(x)ln^{2}(ln(x))cos(x)} - \frac{3sin(x)}{x^{2}ln^{2}(ln(x))ln^{2}(x)cos^{2}(x)} + \frac{2}{x^{3}ln^{3}(x)ln(ln(x))cos(x)} - \frac{3sin(x)}{x^{2}ln^{2}(x)ln(ln(x))cos^{2}(x)} + \frac{2}{x^{3}ln(ln(x))ln(x)cos(x)} + \frac{6sin^{2}(x)}{xln(ln(x))ln(x)cos^{3}(x)} + \frac{1}{xln(ln(x))ln(x)cos(x)} + \frac{5ln(ln(ln(x)))sin(x)}{cos^{2}(x)} + \frac{6ln(ln(ln(x)))sin^{3}(x)}{cos^{4}(x)}\right)}{dx}\\=&\frac{2*-1}{x^{2}ln(x)ln(ln(x))cos(x)} + \frac{2*-1}{xln^{2}(x)(x)ln(ln(x))cos(x)} + \frac{2*-1}{xln(x)ln^{2}(ln(x))(ln(x))(x)cos(x)} + \frac{2sin(x)}{xln(x)ln(ln(x))cos^{2}(x)} + \frac{3*-3}{x^{4}ln^{2}(ln(x))ln^{2}(x)cos(x)} + \frac{3*-2}{x^{3}ln^{3}(ln(x))(ln(x))(x)ln^{2}(x)cos(x)} + \frac{3*-2}{x^{3}ln^{2}(ln(x))ln^{3}(x)(x)cos(x)} + \frac{3sin(x)}{x^{3}ln^{2}(ln(x))ln^{2}(x)cos^{2}(x)} + \frac{3*-3}{x^{4}ln^{2}(x)ln(ln(x))cos(x)} + \frac{3*-2}{x^{3}ln^{3}(x)(x)ln(ln(x))cos(x)} + \frac{3*-1}{x^{3}ln^{2}(x)ln^{2}(ln(x))(ln(x))(x)cos(x)} + \frac{3sin(x)}{x^{3}ln^{2}(x)ln(ln(x))cos^{2}(x)} - \frac{3*-2sin(x)}{x^{3}ln(ln(x))ln(x)cos^{2}(x)} - \frac{3*-sin(x)}{x^{2}ln^{2}(ln(x))(ln(x))(x)ln(x)cos^{2}(x)} - \frac{3*-sin(x)}{x^{2}ln(ln(x))ln^{2}(x)(x)cos^{2}(x)} - \frac{3cos(x)}{x^{2}ln(ln(x))ln(x)cos^{2}(x)} - \frac{3sin(x)*2sin(x)}{x^{2}ln(ln(x))ln(x)cos^{3}(x)} + \frac{2*-3}{x^{4}ln^{3}(ln(x))ln^{3}(x)cos(x)} + \frac{2*-3}{x^{3}ln^{4}(ln(x))(ln(x))(x)ln^{3}(x)cos(x)} + \frac{2*-3}{x^{3}ln^{3}(ln(x))ln^{4}(x)(x)cos(x)} + \frac{2sin(x)}{x^{3}ln^{3}(ln(x))ln^{3}(x)cos^{2}(x)} + \frac{3*-3}{x^{4}ln^{3}(x)ln^{2}(ln(x))cos(x)} + \frac{3*-3}{x^{3}ln^{4}(x)(x)ln^{2}(ln(x))cos(x)} + \frac{3*-2}{x^{3}ln^{3}(x)ln^{3}(ln(x))(ln(x))(x)cos(x)} + \frac{3sin(x)}{x^{3}ln^{3}(x)ln^{2}(ln(x))cos^{2}(x)} - \frac{3*-2sin(x)}{x^{3}ln^{2}(ln(x))ln^{2}(x)cos^{2}(x)} - \frac{3*-2sin(x)}{x^{2}ln^{3}(ln(x))(ln(x))(x)ln^{2}(x)cos^{2}(x)} - \frac{3*-2sin(x)}{x^{2}ln^{2}(ln(x))ln^{3}(x)(x)cos^{2}(x)} - \frac{3cos(x)}{x^{2}ln^{2}(ln(x))ln^{2}(x)cos^{2}(x)} - \frac{3sin(x)*2sin(x)}{x^{2}ln^{2}(ln(x))ln^{2}(x)cos^{3}(x)} + \frac{2*-3}{x^{4}ln^{3}(x)ln(ln(x))cos(x)} + \frac{2*-3}{x^{3}ln^{4}(x)(x)ln(ln(x))cos(x)} + \frac{2*-1}{x^{3}ln^{3}(x)ln^{2}(ln(x))(ln(x))(x)cos(x)} + \frac{2sin(x)}{x^{3}ln^{3}(x)ln(ln(x))cos^{2}(x)} - \frac{3*-2sin(x)}{x^{3}ln^{2}(x)ln(ln(x))cos^{2}(x)} - \frac{3*-2sin(x)}{x^{2}ln^{3}(x)(x)ln(ln(x))cos^{2}(x)} - \frac{3*-sin(x)}{x^{2}ln^{2}(x)ln^{2}(ln(x))(ln(x))(x)cos^{2}(x)} - \frac{3cos(x)}{x^{2}ln^{2}(x)ln(ln(x))cos^{2}(x)} - \frac{3sin(x)*2sin(x)}{x^{2}ln^{2}(x)ln(ln(x))cos^{3}(x)} + \frac{2*-3}{x^{4}ln(ln(x))ln(x)cos(x)} + \frac{2*-1}{x^{3}ln^{2}(ln(x))(ln(x))(x)ln(x)cos(x)} + \frac{2*-1}{x^{3}ln(ln(x))ln^{2}(x)(x)cos(x)} + \frac{2sin(x)}{x^{3}ln(ln(x))ln(x)cos^{2}(x)} + \frac{6*-sin^{2}(x)}{x^{2}ln(ln(x))ln(x)cos^{3}(x)} + \frac{6*-sin^{2}(x)}{xln^{2}(ln(x))(ln(x))(x)ln(x)cos^{3}(x)} + \frac{6*-sin^{2}(x)}{xln(ln(x))ln^{2}(x)(x)cos^{3}(x)} + \frac{6*2sin(x)cos(x)}{xln(ln(x))ln(x)cos^{3}(x)} + \frac{6sin^{2}(x)*3sin(x)}{xln(ln(x))ln(x)cos^{4}(x)} + \frac{-1}{x^{2}ln(ln(x))ln(x)cos(x)} + \frac{-1}{xln^{2}(ln(x))(ln(x))(x)ln(x)cos(x)} + \frac{-1}{xln(ln(x))ln^{2}(x)(x)cos(x)} + \frac{sin(x)}{xln(ln(x))ln(x)cos^{2}(x)} + \frac{5sin(x)}{(ln(ln(x)))(ln(x))(x)cos^{2}(x)} + \frac{5ln(ln(ln(x)))cos(x)}{cos^{2}(x)} + \frac{5ln(ln(ln(x)))sin(x)*2sin(x)}{cos^{3}(x)} + \frac{6sin^{3}(x)}{(ln(ln(x)))(ln(x))(x)cos^{4}(x)} + \frac{6ln(ln(ln(x)))*3sin^{2}(x)cos(x)}{cos^{4}(x)} + \frac{6ln(ln(ln(x)))sin^{3}(x)*4sin(x)}{cos^{5}(x)}\\=& - \frac{5}{x^{2}ln(x)ln(ln(x))cos(x)} - \frac{3}{x^{2}ln^{2}(x)ln(ln(x))cos(x)} - \frac{5}{x^{2}ln^{2}(x)ln^{2}(ln(x))cos(x)} + \frac{18sin(x)}{xln(ln(x))ln(x)cos^{2}(x)} - \frac{12}{x^{4}ln^{3}(ln(x))ln^{3}(x)cos(x)} - \frac{18}{x^{4}ln^{3}(x)ln^{2}(ln(x))cos(x)} + \frac{12sin(x)}{x^{3}ln^{2}(ln(x))ln^{2}(x)cos^{2}(x)} - \frac{3}{x^{2}ln(ln(x))ln^{2}(x)cos(x)} - \frac{12}{x^{4}ln^{3}(x)ln(ln(x))cos(x)} + \frac{12sin(x)}{x^{3}ln^{2}(x)ln(ln(x))cos^{2}(x)} + \frac{8sin(x)}{x^{3}ln(ln(x))ln(x)cos^{2}(x)} - \frac{12sin^{2}(x)}{x^{2}ln(ln(x))ln(x)cos^{3}(x)} - \frac{6}{x^{4}ln^{4}(ln(x))ln^{4}(x)cos(x)} - \frac{12}{x^{4}ln^{4}(x)ln^{3}(ln(x))cos(x)} + \frac{8sin(x)}{x^{3}ln^{3}(ln(x))ln^{3}(x)cos^{2}(x)} - \frac{11}{x^{4}ln^{4}(x)ln^{2}(ln(x))cos(x)} + \frac{12sin(x)}{x^{3}ln^{3}(x)ln^{2}(ln(x))cos^{2}(x)} - \frac{11}{x^{4}ln^{2}(ln(x))ln^{2}(x)cos(x)} - \frac{12sin^{2}(x)}{x^{2}ln^{2}(ln(x))ln^{2}(x)cos^{3}(x)} - \frac{6}{x^{4}ln^{4}(x)ln(ln(x))cos(x)} + \frac{8sin(x)}{x^{3}ln^{3}(x)ln(ln(x))cos^{2}(x)} - \frac{11}{x^{4}ln^{2}(x)ln(ln(x))cos(x)} - \frac{12sin^{2}(x)}{x^{2}ln^{2}(x)ln(ln(x))cos^{3}(x)} - \frac{6}{x^{4}ln(ln(x))ln(x)cos(x)} + \frac{2sin(x)}{xln(x)ln(ln(x))cos^{2}(x)} + \frac{24sin^{3}(x)}{xln(ln(x))ln(x)cos^{4}(x)} - \frac{1}{x^{2}ln(ln(x))ln(x)cos(x)} - \frac{1}{x^{2}ln^{2}(ln(x))ln^{2}(x)cos(x)} + \frac{5ln(ln(ln(x)))}{cos(x)} + \frac{28ln(ln(ln(x)))sin^{2}(x)}{cos^{3}(x)} + \frac{24ln(ln(ln(x)))sin^{4}(x)}{cos^{5}(x)}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/2]Find\ the\ 4th\ derivative\ of\ function\ frac(ln(ln(ln(x))))(cos(x))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = fracln(ln(ln(x)))cos(x)\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( fracln(ln(ln(x)))cos(x)\right)}{dx}\\=&\frac{fraccos(x)}{(ln(ln(x)))(ln(x))(x)} + fracln(ln(ln(x)))*-sin(x)\\=&\frac{fraccos(x)}{xln(ln(x))ln(x)} - fracln(ln(ln(x)))sin(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{fraccos(x)}{xln(ln(x))ln(x)} - fracln(ln(ln(x)))sin(x)\right)}{dx}\\=&\frac{frac*-cos(x)}{x^{2}ln(ln(x))ln(x)} + \frac{frac*-cos(x)}{xln^{2}(ln(x))(ln(x))(x)ln(x)} + \frac{frac*-cos(x)}{xln(ln(x))ln^{2}(x)(x)} + \frac{frac*-sin(x)}{xln(ln(x))ln(x)} - \frac{fracsin(x)}{(ln(ln(x)))(ln(x))(x)} - fracln(ln(ln(x)))cos(x)\\=&\frac{-fraccos(x)}{x^{2}ln(ln(x))ln(x)} - \frac{fraccos(x)}{x^{2}ln^{2}(ln(x))ln^{2}(x)} - \frac{fraccos(x)}{x^{2}ln^{2}(x)ln(ln(x))} - \frac{2fracsin(x)}{xln(ln(x))ln(x)} - fracln(ln(ln(x)))cos(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-fraccos(x)}{x^{2}ln(ln(x))ln(x)} - \frac{fraccos(x)}{x^{2}ln^{2}(ln(x))ln^{2}(x)} - \frac{fraccos(x)}{x^{2}ln^{2}(x)ln(ln(x))} - \frac{2fracsin(x)}{xln(ln(x))ln(x)} - fracln(ln(ln(x)))cos(x)\right)}{dx}\\=&\frac{-frac*-2cos(x)}{x^{3}ln(ln(x))ln(x)} - \frac{frac*-cos(x)}{x^{2}ln^{2}(ln(x))(ln(x))(x)ln(x)} - \frac{frac*-cos(x)}{x^{2}ln(ln(x))ln^{2}(x)(x)} - \frac{frac*-sin(x)}{x^{2}ln(ln(x))ln(x)} - \frac{frac*-2cos(x)}{x^{3}ln^{2}(ln(x))ln^{2}(x)} - \frac{frac*-2cos(x)}{x^{2}ln^{3}(ln(x))(ln(x))(x)ln^{2}(x)} - \frac{frac*-2cos(x)}{x^{2}ln^{2}(ln(x))ln^{3}(x)(x)} - \frac{frac*-sin(x)}{x^{2}ln^{2}(ln(x))ln^{2}(x)} - \frac{frac*-2cos(x)}{x^{3}ln^{2}(x)ln(ln(x))} - \frac{frac*-2cos(x)}{x^{2}ln^{3}(x)(x)ln(ln(x))} - \frac{frac*-cos(x)}{x^{2}ln^{2}(x)ln^{2}(ln(x))(ln(x))(x)} - \frac{frac*-sin(x)}{x^{2}ln^{2}(x)ln(ln(x))} - \frac{2frac*-sin(x)}{x^{2}ln(ln(x))ln(x)} - \frac{2frac*-sin(x)}{xln^{2}(ln(x))(ln(x))(x)ln(x)} - \frac{2frac*-sin(x)}{xln(ln(x))ln^{2}(x)(x)} - \frac{2fraccos(x)}{xln(ln(x))ln(x)} - \frac{fraccos(x)}{(ln(ln(x)))(ln(x))(x)} - fracln(ln(ln(x)))*-sin(x)\\=&\frac{2fraccos(x)}{x^{3}ln(ln(x))ln(x)} + \frac{3fraccos(x)}{x^{3}ln^{2}(ln(x))ln^{2}(x)} + \frac{3fraccos(x)}{x^{3}ln^{2}(x)ln(ln(x))} + \frac{3fracsin(x)}{x^{2}ln(ln(x))ln(x)} + \frac{2fraccos(x)}{x^{3}ln^{3}(ln(x))ln^{3}(x)} + \frac{3fraccos(x)}{x^{3}ln^{3}(x)ln^{2}(ln(x))} + \frac{3fracsin(x)}{x^{2}ln^{2}(ln(x))ln^{2}(x)} + \frac{2fraccos(x)}{x^{3}ln^{3}(x)ln(ln(x))} + \frac{3fracsin(x)}{x^{2}ln^{2}(x)ln(ln(x))} - \frac{2fraccos(x)}{xln(x)ln(ln(x))} - \frac{fraccos(x)}{xln(ln(x))ln(x)} + fracln(ln(ln(x)))sin(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{2fraccos(x)}{x^{3}ln(ln(x))ln(x)} + \frac{3fraccos(x)}{x^{3}ln^{2}(ln(x))ln^{2}(x)} + \frac{3fraccos(x)}{x^{3}ln^{2}(x)ln(ln(x))} + \frac{3fracsin(x)}{x^{2}ln(ln(x))ln(x)} + \frac{2fraccos(x)}{x^{3}ln^{3}(ln(x))ln^{3}(x)} + \frac{3fraccos(x)}{x^{3}ln^{3}(x)ln^{2}(ln(x))} + \frac{3fracsin(x)}{x^{2}ln^{2}(ln(x))ln^{2}(x)} + \frac{2fraccos(x)}{x^{3}ln^{3}(x)ln(ln(x))} + \frac{3fracsin(x)}{x^{2}ln^{2}(x)ln(ln(x))} - \frac{2fraccos(x)}{xln(x)ln(ln(x))} - \frac{fraccos(x)}{xln(ln(x))ln(x)} + fracln(ln(ln(x)))sin(x)\right)}{dx}\\=&\frac{2frac*-3cos(x)}{x^{4}ln(ln(x))ln(x)} + \frac{2frac*-cos(x)}{x^{3}ln^{2}(ln(x))(ln(x))(x)ln(x)} + \frac{2frac*-cos(x)}{x^{3}ln(ln(x))ln^{2}(x)(x)} + \frac{2frac*-sin(x)}{x^{3}ln(ln(x))ln(x)} + \frac{3frac*-3cos(x)}{x^{4}ln^{2}(ln(x))ln^{2}(x)} + \frac{3frac*-2cos(x)}{x^{3}ln^{3}(ln(x))(ln(x))(x)ln^{2}(x)} + \frac{3frac*-2cos(x)}{x^{3}ln^{2}(ln(x))ln^{3}(x)(x)} + \frac{3frac*-sin(x)}{x^{3}ln^{2}(ln(x))ln^{2}(x)} + \frac{3frac*-3cos(x)}{x^{4}ln^{2}(x)ln(ln(x))} + \frac{3frac*-2cos(x)}{x^{3}ln^{3}(x)(x)ln(ln(x))} + \frac{3frac*-cos(x)}{x^{3}ln^{2}(x)ln^{2}(ln(x))(ln(x))(x)} + \frac{3frac*-sin(x)}{x^{3}ln^{2}(x)ln(ln(x))} + \frac{3frac*-2sin(x)}{x^{3}ln(ln(x))ln(x)} + \frac{3frac*-sin(x)}{x^{2}ln^{2}(ln(x))(ln(x))(x)ln(x)} + \frac{3frac*-sin(x)}{x^{2}ln(ln(x))ln^{2}(x)(x)} + \frac{3fraccos(x)}{x^{2}ln(ln(x))ln(x)} + \frac{2frac*-3cos(x)}{x^{4}ln^{3}(ln(x))ln^{3}(x)} + \frac{2frac*-3cos(x)}{x^{3}ln^{4}(ln(x))(ln(x))(x)ln^{3}(x)} + \frac{2frac*-3cos(x)}{x^{3}ln^{3}(ln(x))ln^{4}(x)(x)} + \frac{2frac*-sin(x)}{x^{3}ln^{3}(ln(x))ln^{3}(x)} + \frac{3frac*-3cos(x)}{x^{4}ln^{3}(x)ln^{2}(ln(x))} + \frac{3frac*-3cos(x)}{x^{3}ln^{4}(x)(x)ln^{2}(ln(x))} + \frac{3frac*-2cos(x)}{x^{3}ln^{3}(x)ln^{3}(ln(x))(ln(x))(x)} + \frac{3frac*-sin(x)}{x^{3}ln^{3}(x)ln^{2}(ln(x))} + \frac{3frac*-2sin(x)}{x^{3}ln^{2}(ln(x))ln^{2}(x)} + \frac{3frac*-2sin(x)}{x^{2}ln^{3}(ln(x))(ln(x))(x)ln^{2}(x)} + \frac{3frac*-2sin(x)}{x^{2}ln^{2}(ln(x))ln^{3}(x)(x)} + \frac{3fraccos(x)}{x^{2}ln^{2}(ln(x))ln^{2}(x)} + \frac{2frac*-3cos(x)}{x^{4}ln^{3}(x)ln(ln(x))} + \frac{2frac*-3cos(x)}{x^{3}ln^{4}(x)(x)ln(ln(x))} + \frac{2frac*-cos(x)}{x^{3}ln^{3}(x)ln^{2}(ln(x))(ln(x))(x)} + \frac{2frac*-sin(x)}{x^{3}ln^{3}(x)ln(ln(x))} + \frac{3frac*-2sin(x)}{x^{3}ln^{2}(x)ln(ln(x))} + \frac{3frac*-2sin(x)}{x^{2}ln^{3}(x)(x)ln(ln(x))} + \frac{3frac*-sin(x)}{x^{2}ln^{2}(x)ln^{2}(ln(x))(ln(x))(x)} + \frac{3fraccos(x)}{x^{2}ln^{2}(x)ln(ln(x))} - \frac{2frac*-cos(x)}{x^{2}ln(x)ln(ln(x))} - \frac{2frac*-cos(x)}{xln^{2}(x)(x)ln(ln(x))} - \frac{2frac*-cos(x)}{xln(x)ln^{2}(ln(x))(ln(x))(x)} - \frac{2frac*-sin(x)}{xln(x)ln(ln(x))} - \frac{frac*-cos(x)}{x^{2}ln(ln(x))ln(x)} - \frac{frac*-cos(x)}{xln^{2}(ln(x))(ln(x))(x)ln(x)} - \frac{frac*-cos(x)}{xln(ln(x))ln^{2}(x)(x)} - \frac{frac*-sin(x)}{xln(ln(x))ln(x)} + \frac{fracsin(x)}{(ln(ln(x)))(ln(x))(x)} + fracln(ln(ln(x)))cos(x)\\=&\frac{-6fraccos(x)}{x^{4}ln(ln(x))ln(x)} - \frac{11fraccos(x)}{x^{4}ln^{2}(ln(x))ln^{2}(x)} - \frac{11fraccos(x)}{x^{4}ln^{2}(x)ln(ln(x))} - \frac{8fracsin(x)}{x^{3}ln(ln(x))ln(x)} - \frac{12fraccos(x)}{x^{4}ln^{3}(ln(x))ln^{3}(x)} - \frac{18fraccos(x)}{x^{4}ln^{3}(x)ln^{2}(ln(x))} - \frac{12fracsin(x)}{x^{3}ln^{2}(ln(x))ln^{2}(x)} - \frac{12fraccos(x)}{x^{4}ln^{3}(x)ln(ln(x))} - \frac{12fracsin(x)}{x^{3}ln^{2}(x)ln(ln(x))} + \frac{5fraccos(x)}{x^{2}ln(x)ln(ln(x))} - \frac{6fraccos(x)}{x^{4}ln^{4}(ln(x))ln^{4}(x)} - \frac{12fraccos(x)}{x^{4}ln^{4}(x)ln^{3}(ln(x))} - \frac{8fracsin(x)}{x^{3}ln^{3}(ln(x))ln^{3}(x)} - \frac{11fraccos(x)}{x^{4}ln^{4}(x)ln^{2}(ln(x))} - \frac{12fracsin(x)}{x^{3}ln^{3}(x)ln^{2}(ln(x))} + \frac{5fraccos(x)}{x^{2}ln^{2}(x)ln^{2}(ln(x))} - \frac{6fraccos(x)}{x^{4}ln^{4}(x)ln(ln(x))} - \frac{8fracsin(x)}{x^{3}ln^{3}(x)ln(ln(x))} + \frac{3fraccos(x)}{x^{2}ln(ln(x))ln^{2}(x)} + \frac{3fraccos(x)}{x^{2}ln^{2}(x)ln(ln(x))} + \frac{2fracsin(x)}{xln(x)ln(ln(x))} + \frac{fraccos(x)}{x^{2}ln(ln(x))ln(x)} + \frac{fraccos(x)}{x^{2}ln^{2}(ln(x))ln^{2}(x)} + \frac{2fracsin(x)}{xln(ln(x))ln(x)} + fracln(ln(ln(x)))cos(x)\\ \end{split}\end{equation} \]