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.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ lg(lg(lg(lg(lg(x)))))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( lg(lg(lg(lg(lg(x)))))\right)}{dx}\\=&\frac{1}{ln{10}(lg(lg(lg(lg(x)))))ln{10}(lg(lg(lg(x))))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)}\\=&\frac{1}{xln^{5}{10}lg(lg(lg(lg(x))))lg(lg(x))lg(x)lg(lg(lg(x)))}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{1}{xln^{5}{10}lg(lg(lg(lg(x))))lg(lg(x))lg(x)lg(lg(lg(x)))}\right)}{dx}\\=&\frac{-1}{x^{2}ln^{5}{10}lg(lg(lg(lg(x))))lg(lg(x))lg(x)lg(lg(lg(x)))} + \frac{-5*0}{xln^{6}{10}lg(lg(lg(lg(x))))lg(lg(x))lg(x)lg(lg(lg(x)))} + \frac{-1}{xln^{5}{10}lg^{2}(lg(lg(lg(x))))ln{10}(lg(lg(lg(x))))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)lg(lg(x))lg(x)lg(lg(lg(x)))} + \frac{-1}{xln^{5}{10}lg(lg(lg(lg(x))))lg^{2}(lg(x))ln{10}(lg(x))ln{10}(x)lg(x)lg(lg(lg(x)))} + \frac{-1}{xln^{5}{10}lg(lg(lg(lg(x))))lg(lg(x))lg^{2}(x)ln{10}(x)lg(lg(lg(x)))} + \frac{-1}{xln^{5}{10}lg(lg(lg(lg(x))))lg(lg(x))lg(x)lg^{2}(lg(lg(x)))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)}\\=&\frac{-1}{x^{2}ln^{5}{10}lg(lg(lg(lg(x))))lg(lg(x))lg(x)lg(lg(lg(x)))} - \frac{1}{x^{2}ln^{9}{10}lg^{2}(lg(lg(lg(x))))lg^{2}(lg(x))lg^{2}(x)lg^{2}(lg(lg(x)))} - \frac{1}{x^{2}ln^{7}{10}lg^{2}(x)lg(lg(lg(lg(x))))lg^{2}(lg(x))lg(lg(lg(x)))} - \frac{1}{x^{2}ln^{6}{10}lg(lg(lg(lg(x))))lg^{2}(x)lg(lg(x))lg(lg(lg(x)))} - \frac{1}{x^{2}ln^{8}{10}lg^{2}(x)lg^{2}(lg(x))lg(lg(lg(lg(x))))lg^{2}(lg(lg(x)))}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-1}{x^{2}ln^{5}{10}lg(lg(lg(lg(x))))lg(lg(x))lg(x)lg(lg(lg(x)))} - \frac{1}{x^{2}ln^{9}{10}lg^{2}(lg(lg(lg(x))))lg^{2}(lg(x))lg^{2}(x)lg^{2}(lg(lg(x)))} - \frac{1}{x^{2}ln^{7}{10}lg^{2}(x)lg(lg(lg(lg(x))))lg^{2}(lg(x))lg(lg(lg(x)))} - \frac{1}{x^{2}ln^{6}{10}lg(lg(lg(lg(x))))lg^{2}(x)lg(lg(x))lg(lg(lg(x)))} - \frac{1}{x^{2}ln^{8}{10}lg^{2}(x)lg^{2}(lg(x))lg(lg(lg(lg(x))))lg^{2}(lg(lg(x)))}\right)}{dx}\\=&\frac{--2}{x^{3}ln^{5}{10}lg(lg(lg(lg(x))))lg(lg(x))lg(x)lg(lg(lg(x)))} - \frac{-5*0}{x^{2}ln^{6}{10}lg(lg(lg(lg(x))))lg(lg(x))lg(x)lg(lg(lg(x)))} - \frac{-1}{x^{2}ln^{5}{10}lg^{2}(lg(lg(lg(x))))ln{10}(lg(lg(lg(x))))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)lg(lg(x))lg(x)lg(lg(lg(x)))} - \frac{-1}{x^{2}ln^{5}{10}lg(lg(lg(lg(x))))lg^{2}(lg(x))ln{10}(lg(x))ln{10}(x)lg(x)lg(lg(lg(x)))} - \frac{-1}{x^{2}ln^{5}{10}lg(lg(lg(lg(x))))lg(lg(x))lg^{2}(x)ln{10}(x)lg(lg(lg(x)))} - \frac{-1}{x^{2}ln^{5}{10}lg(lg(lg(lg(x))))lg(lg(x))lg(x)lg^{2}(lg(lg(x)))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)} - \frac{-2}{x^{3}ln^{9}{10}lg^{2}(lg(lg(lg(x))))lg^{2}(lg(x))lg^{2}(x)lg^{2}(lg(lg(x)))} - \frac{-9*0}{x^{2}ln^{10}{10}lg^{2}(lg(lg(lg(x))))lg^{2}(lg(x))lg^{2}(x)lg^{2}(lg(lg(x)))} - \frac{-2}{x^{2}ln^{9}{10}lg^{3}(lg(lg(lg(x))))ln{10}(lg(lg(lg(x))))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)lg^{2}(lg(x))lg^{2}(x)lg^{2}(lg(lg(x)))} - \frac{-2}{x^{2}ln^{9}{10}lg^{2}(lg(lg(lg(x))))lg^{3}(lg(x))ln{10}(lg(x))ln{10}(x)lg^{2}(x)lg^{2}(lg(lg(x)))} - \frac{-2}{x^{2}ln^{9}{10}lg^{2}(lg(lg(lg(x))))lg^{2}(lg(x))lg^{3}(x)ln{10}(x)lg^{2}(lg(lg(x)))} - \frac{-2}{x^{2}ln^{9}{10}lg^{2}(lg(lg(lg(x))))lg^{2}(lg(x))lg^{2}(x)lg^{3}(lg(lg(x)))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)} - \frac{-2}{x^{3}ln^{7}{10}lg^{2}(x)lg(lg(lg(lg(x))))lg^{2}(lg(x))lg(lg(lg(x)))} - \frac{-7*0}{x^{2}ln^{8}{10}lg^{2}(x)lg(lg(lg(lg(x))))lg^{2}(lg(x))lg(lg(lg(x)))} - \frac{-2}{x^{2}ln^{7}{10}lg^{3}(x)ln{10}(x)lg(lg(lg(lg(x))))lg^{2}(lg(x))lg(lg(lg(x)))} - \frac{-1}{x^{2}ln^{7}{10}lg^{2}(x)lg^{2}(lg(lg(lg(x))))ln{10}(lg(lg(lg(x))))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)lg^{2}(lg(x))lg(lg(lg(x)))} - \frac{-2}{x^{2}ln^{7}{10}lg^{2}(x)lg(lg(lg(lg(x))))lg^{3}(lg(x))ln{10}(lg(x))ln{10}(x)lg(lg(lg(x)))} - \frac{-1}{x^{2}ln^{7}{10}lg^{2}(x)lg(lg(lg(lg(x))))lg^{2}(lg(x))lg^{2}(lg(lg(x)))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)} - \frac{-2}{x^{3}ln^{6}{10}lg(lg(lg(lg(x))))lg^{2}(x)lg(lg(x))lg(lg(lg(x)))} - \frac{-6*0}{x^{2}ln^{7}{10}lg(lg(lg(lg(x))))lg^{2}(x)lg(lg(x))lg(lg(lg(x)))} - \frac{-1}{x^{2}ln^{6}{10}lg^{2}(lg(lg(lg(x))))ln{10}(lg(lg(lg(x))))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)lg^{2}(x)lg(lg(x))lg(lg(lg(x)))} - \frac{-2}{x^{2}ln^{6}{10}lg(lg(lg(lg(x))))lg^{3}(x)ln{10}(x)lg(lg(x))lg(lg(lg(x)))} - \frac{-1}{x^{2}ln^{6}{10}lg(lg(lg(lg(x))))lg^{2}(x)lg^{2}(lg(x))ln{10}(lg(x))ln{10}(x)lg(lg(lg(x)))} - \frac{-1}{x^{2}ln^{6}{10}lg(lg(lg(lg(x))))lg^{2}(x)lg(lg(x))lg^{2}(lg(lg(x)))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)} - \frac{-2}{x^{3}ln^{8}{10}lg^{2}(x)lg^{2}(lg(x))lg(lg(lg(lg(x))))lg^{2}(lg(lg(x)))} - \frac{-8*0}{x^{2}ln^{9}{10}lg^{2}(x)lg^{2}(lg(x))lg(lg(lg(lg(x))))lg^{2}(lg(lg(x)))} - \frac{-2}{x^{2}ln^{8}{10}lg^{3}(x)ln{10}(x)lg^{2}(lg(x))lg(lg(lg(lg(x))))lg^{2}(lg(lg(x)))} - \frac{-2}{x^{2}ln^{8}{10}lg^{2}(x)lg^{3}(lg(x))ln{10}(lg(x))ln{10}(x)lg(lg(lg(lg(x))))lg^{2}(lg(lg(x)))} - \frac{-1}{x^{2}ln^{8}{10}lg^{2}(x)lg^{2}(lg(x))lg^{2}(lg(lg(lg(x))))ln{10}(lg(lg(lg(x))))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)lg^{2}(lg(lg(x)))} - \frac{-2}{x^{2}ln^{8}{10}lg^{2}(x)lg^{2}(lg(x))lg(lg(lg(lg(x))))lg^{3}(lg(lg(x)))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)}\\=&\frac{2}{x^{3}ln^{5}{10}lg(lg(lg(lg(x))))lg(lg(x))lg(x)lg(lg(lg(x)))} + \frac{3}{x^{3}ln^{9}{10}lg^{2}(lg(lg(lg(x))))lg^{2}(lg(x))lg^{2}(x)lg^{2}(lg(lg(x)))} + \frac{3}{x^{3}ln^{7}{10}lg^{2}(x)lg(lg(lg(lg(x))))lg^{2}(lg(x))lg(lg(lg(x)))} + \frac{3}{x^{3}ln^{6}{10}lg(lg(lg(lg(x))))lg^{2}(x)lg(lg(x))lg(lg(lg(x)))} + \frac{3}{x^{3}ln^{8}{10}lg^{2}(x)lg^{2}(lg(x))lg(lg(lg(lg(x))))lg^{2}(lg(lg(x)))} + \frac{2}{x^{3}ln^{13}{10}lg^{3}(lg(lg(lg(x))))lg^{3}(lg(x))lg^{3}(x)lg^{3}(lg(lg(x)))} + \frac{2}{x^{3}ln^{11}{10}lg^{3}(x)lg^{2}(lg(lg(lg(x))))lg^{3}(lg(x))lg^{2}(lg(lg(x)))} + \frac{2}{x^{3}ln^{10}{10}lg^{2}(lg(lg(lg(x))))lg^{3}(x)lg^{2}(lg(x))lg^{2}(lg(lg(x)))} + \frac{2}{x^{3}ln^{12}{10}lg^{3}(x)lg^{3}(lg(x))lg^{2}(lg(lg(lg(x))))lg^{3}(lg(lg(x)))} + \frac{3}{x^{3}ln^{8}{10}lg^{3}(x)lg(lg(lg(lg(x))))lg^{2}(lg(x))lg(lg(lg(x)))} + \frac{1}{x^{3}ln^{11}{10}lg^{3}(lg(x))lg^{3}(x)lg^{2}(lg(lg(lg(x))))lg^{2}(lg(lg(x)))} + \frac{2}{x^{3}ln^{9}{10}lg(lg(lg(lg(x))))lg^{3}(x)lg^{3}(lg(x))lg(lg(lg(x)))} + \frac{1}{x^{3}ln^{10}{10}lg^{3}(x)lg^{3}(lg(x))lg^{2}(lg(lg(x)))lg(lg(lg(lg(x))))} + \frac{1}{x^{3}ln^{10}{10}lg^{2}(lg(lg(lg(x))))lg^{2}(lg(x))lg^{3}(x)lg^{2}(lg(lg(x)))} + \frac{2}{x^{3}ln^{7}{10}lg(lg(lg(lg(x))))lg^{3}(x)lg(lg(x))lg(lg(lg(x)))} + \frac{3}{x^{3}ln^{9}{10}lg^{3}(x)lg^{2}(lg(x))lg(lg(lg(lg(x))))lg^{2}(lg(lg(x)))} + \frac{2}{x^{3}ln^{10}{10}lg^{3}(x)lg^{3}(lg(x))lg(lg(lg(lg(x))))lg^{2}(lg(lg(x)))} + \frac{1}{x^{3}ln^{12}{10}lg^{3}(x)lg^{3}(lg(x))lg^{3}(lg(lg(x)))lg^{2}(lg(lg(lg(x))))} + \frac{2}{x^{3}ln^{11}{10}lg^{3}(x)lg(lg(lg(lg(x))))lg^{3}(lg(x))lg^{3}(lg(lg(x)))}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{2}{x^{3}ln^{5}{10}lg(lg(lg(lg(x))))lg(lg(x))lg(x)lg(lg(lg(x)))} + \frac{3}{x^{3}ln^{9}{10}lg^{2}(lg(lg(lg(x))))lg^{2}(lg(x))lg^{2}(x)lg^{2}(lg(lg(x)))} + \frac{3}{x^{3}ln^{7}{10}lg^{2}(x)lg(lg(lg(lg(x))))lg^{2}(lg(x))lg(lg(lg(x)))} + \frac{3}{x^{3}ln^{6}{10}lg(lg(lg(lg(x))))lg^{2}(x)lg(lg(x))lg(lg(lg(x)))} + \frac{3}{x^{3}ln^{8}{10}lg^{2}(x)lg^{2}(lg(x))lg(lg(lg(lg(x))))lg^{2}(lg(lg(x)))} + \frac{2}{x^{3}ln^{13}{10}lg^{3}(lg(lg(lg(x))))lg^{3}(lg(x))lg^{3}(x)lg^{3}(lg(lg(x)))} + \frac{2}{x^{3}ln^{11}{10}lg^{3}(x)lg^{2}(lg(lg(lg(x))))lg^{3}(lg(x))lg^{2}(lg(lg(x)))} + \frac{2}{x^{3}ln^{10}{10}lg^{2}(lg(lg(lg(x))))lg^{3}(x)lg^{2}(lg(x))lg^{2}(lg(lg(x)))} + \frac{2}{x^{3}ln^{12}{10}lg^{3}(x)lg^{3}(lg(x))lg^{2}(lg(lg(lg(x))))lg^{3}(lg(lg(x)))} + \frac{3}{x^{3}ln^{8}{10}lg^{3}(x)lg(lg(lg(lg(x))))lg^{2}(lg(x))lg(lg(lg(x)))} + \frac{1}{x^{3}ln^{11}{10}lg^{3}(lg(x))lg^{3}(x)lg^{2}(lg(lg(lg(x))))lg^{2}(lg(lg(x)))} + \frac{2}{x^{3}ln^{9}{10}lg(lg(lg(lg(x))))lg^{3}(x)lg^{3}(lg(x))lg(lg(lg(x)))} + \frac{1}{x^{3}ln^{10}{10}lg^{3}(x)lg^{3}(lg(x))lg^{2}(lg(lg(x)))lg(lg(lg(lg(x))))} + \frac{1}{x^{3}ln^{10}{10}lg^{2}(lg(lg(lg(x))))lg^{2}(lg(x))lg^{3}(x)lg^{2}(lg(lg(x)))} + \frac{2}{x^{3}ln^{7}{10}lg(lg(lg(lg(x))))lg^{3}(x)lg(lg(x))lg(lg(lg(x)))} + \frac{3}{x^{3}ln^{9}{10}lg^{3}(x)lg^{2}(lg(x))lg(lg(lg(lg(x))))lg^{2}(lg(lg(x)))} + \frac{2}{x^{3}ln^{10}{10}lg^{3}(x)lg^{3}(lg(x))lg(lg(lg(lg(x))))lg^{2}(lg(lg(x)))} + \frac{1}{x^{3}ln^{12}{10}lg^{3}(x)lg^{3}(lg(x))lg^{3}(lg(lg(x)))lg^{2}(lg(lg(lg(x))))} + \frac{2}{x^{3}ln^{11}{10}lg^{3}(x)lg(lg(lg(lg(x))))lg^{3}(lg(x))lg^{3}(lg(lg(x)))}\right)}{dx}\\=&\frac{2*-3}{x^{4}ln^{5}{10}lg(lg(lg(lg(x))))lg(lg(x))lg(x)lg(lg(lg(x)))} + \frac{2*-5*0}{x^{3}ln^{6}{10}lg(lg(lg(lg(x))))lg(lg(x))lg(x)lg(lg(lg(x)))} + \frac{2*-1}{x^{3}ln^{5}{10}lg^{2}(lg(lg(lg(x))))ln{10}(lg(lg(lg(x))))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)lg(lg(x))lg(x)lg(lg(lg(x)))} + \frac{2*-1}{x^{3}ln^{5}{10}lg(lg(lg(lg(x))))lg^{2}(lg(x))ln{10}(lg(x))ln{10}(x)lg(x)lg(lg(lg(x)))} + \frac{2*-1}{x^{3}ln^{5}{10}lg(lg(lg(lg(x))))lg(lg(x))lg^{2}(x)ln{10}(x)lg(lg(lg(x)))} + \frac{2*-1}{x^{3}ln^{5}{10}lg(lg(lg(lg(x))))lg(lg(x))lg(x)lg^{2}(lg(lg(x)))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)} + \frac{3*-3}{x^{4}ln^{9}{10}lg^{2}(lg(lg(lg(x))))lg^{2}(lg(x))lg^{2}(x)lg^{2}(lg(lg(x)))} + \frac{3*-9*0}{x^{3}ln^{10}{10}lg^{2}(lg(lg(lg(x))))lg^{2}(lg(x))lg^{2}(x)lg^{2}(lg(lg(x)))} + \frac{3*-2}{x^{3}ln^{9}{10}lg^{3}(lg(lg(lg(x))))ln{10}(lg(lg(lg(x))))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)lg^{2}(lg(x))lg^{2}(x)lg^{2}(lg(lg(x)))} + \frac{3*-2}{x^{3}ln^{9}{10}lg^{2}(lg(lg(lg(x))))lg^{3}(lg(x))ln{10}(lg(x))ln{10}(x)lg^{2}(x)lg^{2}(lg(lg(x)))} + \frac{3*-2}{x^{3}ln^{9}{10}lg^{2}(lg(lg(lg(x))))lg^{2}(lg(x))lg^{3}(x)ln{10}(x)lg^{2}(lg(lg(x)))} + \frac{3*-2}{x^{3}ln^{9}{10}lg^{2}(lg(lg(lg(x))))lg^{2}(lg(x))lg^{2}(x)lg^{3}(lg(lg(x)))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)} + \frac{3*-3}{x^{4}ln^{7}{10}lg^{2}(x)lg(lg(lg(lg(x))))lg^{2}(lg(x))lg(lg(lg(x)))} + \frac{3*-7*0}{x^{3}ln^{8}{10}lg^{2}(x)lg(lg(lg(lg(x))))lg^{2}(lg(x))lg(lg(lg(x)))} + \frac{3*-2}{x^{3}ln^{7}{10}lg^{3}(x)ln{10}(x)lg(lg(lg(lg(x))))lg^{2}(lg(x))lg(lg(lg(x)))} + \frac{3*-1}{x^{3}ln^{7}{10}lg^{2}(x)lg^{2}(lg(lg(lg(x))))ln{10}(lg(lg(lg(x))))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)lg^{2}(lg(x))lg(lg(lg(x)))} + \frac{3*-2}{x^{3}ln^{7}{10}lg^{2}(x)lg(lg(lg(lg(x))))lg^{3}(lg(x))ln{10}(lg(x))ln{10}(x)lg(lg(lg(x)))} + \frac{3*-1}{x^{3}ln^{7}{10}lg^{2}(x)lg(lg(lg(lg(x))))lg^{2}(lg(x))lg^{2}(lg(lg(x)))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)} + \frac{3*-3}{x^{4}ln^{6}{10}lg(lg(lg(lg(x))))lg^{2}(x)lg(lg(x))lg(lg(lg(x)))} + \frac{3*-6*0}{x^{3}ln^{7}{10}lg(lg(lg(lg(x))))lg^{2}(x)lg(lg(x))lg(lg(lg(x)))} + \frac{3*-1}{x^{3}ln^{6}{10}lg^{2}(lg(lg(lg(x))))ln{10}(lg(lg(lg(x))))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)lg^{2}(x)lg(lg(x))lg(lg(lg(x)))} + \frac{3*-2}{x^{3}ln^{6}{10}lg(lg(lg(lg(x))))lg^{3}(x)ln{10}(x)lg(lg(x))lg(lg(lg(x)))} + \frac{3*-1}{x^{3}ln^{6}{10}lg(lg(lg(lg(x))))lg^{2}(x)lg^{2}(lg(x))ln{10}(lg(x))ln{10}(x)lg(lg(lg(x)))} + \frac{3*-1}{x^{3}ln^{6}{10}lg(lg(lg(lg(x))))lg^{2}(x)lg(lg(x))lg^{2}(lg(lg(x)))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)} + \frac{3*-3}{x^{4}ln^{8}{10}lg^{2}(x)lg^{2}(lg(x))lg(lg(lg(lg(x))))lg^{2}(lg(lg(x)))} + \frac{3*-8*0}{x^{3}ln^{9}{10}lg^{2}(x)lg^{2}(lg(x))lg(lg(lg(lg(x))))lg^{2}(lg(lg(x)))} + \frac{3*-2}{x^{3}ln^{8}{10}lg^{3}(x)ln{10}(x)lg^{2}(lg(x))lg(lg(lg(lg(x))))lg^{2}(lg(lg(x)))} + \frac{3*-2}{x^{3}ln^{8}{10}lg^{2}(x)lg^{3}(lg(x))ln{10}(lg(x))ln{10}(x)lg(lg(lg(lg(x))))lg^{2}(lg(lg(x)))} + \frac{3*-1}{x^{3}ln^{8}{10}lg^{2}(x)lg^{2}(lg(x))lg^{2}(lg(lg(lg(x))))ln{10}(lg(lg(lg(x))))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)lg^{2}(lg(lg(x)))} + \frac{3*-2}{x^{3}ln^{8}{10}lg^{2}(x)lg^{2}(lg(x))lg(lg(lg(lg(x))))lg^{3}(lg(lg(x)))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)} + \frac{2*-3}{x^{4}ln^{13}{10}lg^{3}(lg(lg(lg(x))))lg^{3}(lg(x))lg^{3}(x)lg^{3}(lg(lg(x)))} + \frac{2*-13*0}{x^{3}ln^{14}{10}lg^{3}(lg(lg(lg(x))))lg^{3}(lg(x))lg^{3}(x)lg^{3}(lg(lg(x)))} + \frac{2*-3}{x^{3}ln^{13}{10}lg^{4}(lg(lg(lg(x))))ln{10}(lg(lg(lg(x))))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)lg^{3}(lg(x))lg^{3}(x)lg^{3}(lg(lg(x)))} + \frac{2*-3}{x^{3}ln^{13}{10}lg^{3}(lg(lg(lg(x))))lg^{4}(lg(x))ln{10}(lg(x))ln{10}(x)lg^{3}(x)lg^{3}(lg(lg(x)))} + \frac{2*-3}{x^{3}ln^{13}{10}lg^{3}(lg(lg(lg(x))))lg^{3}(lg(x))lg^{4}(x)ln{10}(x)lg^{3}(lg(lg(x)))} + \frac{2*-3}{x^{3}ln^{13}{10}lg^{3}(lg(lg(lg(x))))lg^{3}(lg(x))lg^{3}(x)lg^{4}(lg(lg(x)))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)} + \frac{2*-3}{x^{4}ln^{11}{10}lg^{3}(x)lg^{2}(lg(lg(lg(x))))lg^{3}(lg(x))lg^{2}(lg(lg(x)))} + \frac{2*-11*0}{x^{3}ln^{12}{10}lg^{3}(x)lg^{2}(lg(lg(lg(x))))lg^{3}(lg(x))lg^{2}(lg(lg(x)))} + \frac{2*-3}{x^{3}ln^{11}{10}lg^{4}(x)ln{10}(x)lg^{2}(lg(lg(lg(x))))lg^{3}(lg(x))lg^{2}(lg(lg(x)))} + \frac{2*-2}{x^{3}ln^{11}{10}lg^{3}(x)lg^{3}(lg(lg(lg(x))))ln{10}(lg(lg(lg(x))))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)lg^{3}(lg(x))lg^{2}(lg(lg(x)))} + \frac{2*-3}{x^{3}ln^{11}{10}lg^{3}(x)lg^{2}(lg(lg(lg(x))))lg^{4}(lg(x))ln{10}(lg(x))ln{10}(x)lg^{2}(lg(lg(x)))} + \frac{2*-2}{x^{3}ln^{11}{10}lg^{3}(x)lg^{2}(lg(lg(lg(x))))lg^{3}(lg(x))lg^{3}(lg(lg(x)))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)} + \frac{2*-3}{x^{4}ln^{10}{10}lg^{2}(lg(lg(lg(x))))lg^{3}(x)lg^{2}(lg(x))lg^{2}(lg(lg(x)))} + \frac{2*-10*0}{x^{3}ln^{11}{10}lg^{2}(lg(lg(lg(x))))lg^{3}(x)lg^{2}(lg(x))lg^{2}(lg(lg(x)))} + \frac{2*-2}{x^{3}ln^{10}{10}lg^{3}(lg(lg(lg(x))))ln{10}(lg(lg(lg(x))))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)lg^{3}(x)lg^{2}(lg(x))lg^{2}(lg(lg(x)))} + \frac{2*-3}{x^{3}ln^{10}{10}lg^{2}(lg(lg(lg(x))))lg^{4}(x)ln{10}(x)lg^{2}(lg(x))lg^{2}(lg(lg(x)))} + \frac{2*-2}{x^{3}ln^{10}{10}lg^{2}(lg(lg(lg(x))))lg^{3}(x)lg^{3}(lg(x))ln{10}(lg(x))ln{10}(x)lg^{2}(lg(lg(x)))} + \frac{2*-2}{x^{3}ln^{10}{10}lg^{2}(lg(lg(lg(x))))lg^{3}(x)lg^{2}(lg(x))lg^{3}(lg(lg(x)))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)} + \frac{2*-3}{x^{4}ln^{12}{10}lg^{3}(x)lg^{3}(lg(x))lg^{2}(lg(lg(lg(x))))lg^{3}(lg(lg(x)))} + \frac{2*-12*0}{x^{3}ln^{13}{10}lg^{3}(x)lg^{3}(lg(x))lg^{2}(lg(lg(lg(x))))lg^{3}(lg(lg(x)))} + \frac{2*-3}{x^{3}ln^{12}{10}lg^{4}(x)ln{10}(x)lg^{3}(lg(x))lg^{2}(lg(lg(lg(x))))lg^{3}(lg(lg(x)))} + \frac{2*-3}{x^{3}ln^{12}{10}lg^{3}(x)lg^{4}(lg(x))ln{10}(lg(x))ln{10}(x)lg^{2}(lg(lg(lg(x))))lg^{3}(lg(lg(x)))} + \frac{2*-2}{x^{3}ln^{12}{10}lg^{3}(x)lg^{3}(lg(x))lg^{3}(lg(lg(lg(x))))ln{10}(lg(lg(lg(x))))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)lg^{3}(lg(lg(x)))} + \frac{2*-3}{x^{3}ln^{12}{10}lg^{3}(x)lg^{3}(lg(x))lg^{2}(lg(lg(lg(x))))lg^{4}(lg(lg(x)))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)} + \frac{3*-3}{x^{4}ln^{8}{10}lg^{3}(x)lg(lg(lg(lg(x))))lg^{2}(lg(x))lg(lg(lg(x)))} + \frac{3*-8*0}{x^{3}ln^{9}{10}lg^{3}(x)lg(lg(lg(lg(x))))lg^{2}(lg(x))lg(lg(lg(x)))} + \frac{3*-3}{x^{3}ln^{8}{10}lg^{4}(x)ln{10}(x)lg(lg(lg(lg(x))))lg^{2}(lg(x))lg(lg(lg(x)))} + \frac{3*-1}{x^{3}ln^{8}{10}lg^{3}(x)lg^{2}(lg(lg(lg(x))))ln{10}(lg(lg(lg(x))))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)lg^{2}(lg(x))lg(lg(lg(x)))} + \frac{3*-2}{x^{3}ln^{8}{10}lg^{3}(x)lg(lg(lg(lg(x))))lg^{3}(lg(x))ln{10}(lg(x))ln{10}(x)lg(lg(lg(x)))} + \frac{3*-1}{x^{3}ln^{8}{10}lg^{3}(x)lg(lg(lg(lg(x))))lg^{2}(lg(x))lg^{2}(lg(lg(x)))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)} + \frac{-3}{x^{4}ln^{11}{10}lg^{3}(lg(x))lg^{3}(x)lg^{2}(lg(lg(lg(x))))lg^{2}(lg(lg(x)))} + \frac{-11*0}{x^{3}ln^{12}{10}lg^{3}(lg(x))lg^{3}(x)lg^{2}(lg(lg(lg(x))))lg^{2}(lg(lg(x)))} + \frac{-3}{x^{3}ln^{11}{10}lg^{4}(lg(x))ln{10}(lg(x))ln{10}(x)lg^{3}(x)lg^{2}(lg(lg(lg(x))))lg^{2}(lg(lg(x)))} + \frac{-3}{x^{3}ln^{11}{10}lg^{3}(lg(x))lg^{4}(x)ln{10}(x)lg^{2}(lg(lg(lg(x))))lg^{2}(lg(lg(x)))} + \frac{-2}{x^{3}ln^{11}{10}lg^{3}(lg(x))lg^{3}(x)lg^{3}(lg(lg(lg(x))))ln{10}(lg(lg(lg(x))))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)lg^{2}(lg(lg(x)))} + \frac{-2}{x^{3}ln^{11}{10}lg^{3}(lg(x))lg^{3}(x)lg^{2}(lg(lg(lg(x))))lg^{3}(lg(lg(x)))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)} + \frac{2*-3}{x^{4}ln^{9}{10}lg(lg(lg(lg(x))))lg^{3}(x)lg^{3}(lg(x))lg(lg(lg(x)))} + \frac{2*-9*0}{x^{3}ln^{10}{10}lg(lg(lg(lg(x))))lg^{3}(x)lg^{3}(lg(x))lg(lg(lg(x)))} + \frac{2*-1}{x^{3}ln^{9}{10}lg^{2}(lg(lg(lg(x))))ln{10}(lg(lg(lg(x))))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)lg^{3}(x)lg^{3}(lg(x))lg(lg(lg(x)))} + \frac{2*-3}{x^{3}ln^{9}{10}lg(lg(lg(lg(x))))lg^{4}(x)ln{10}(x)lg^{3}(lg(x))lg(lg(lg(x)))} + \frac{2*-3}{x^{3}ln^{9}{10}lg(lg(lg(lg(x))))lg^{3}(x)lg^{4}(lg(x))ln{10}(lg(x))ln{10}(x)lg(lg(lg(x)))} + \frac{2*-1}{x^{3}ln^{9}{10}lg(lg(lg(lg(x))))lg^{3}(x)lg^{3}(lg(x))lg^{2}(lg(lg(x)))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)} + \frac{-3}{x^{4}ln^{10}{10}lg^{3}(x)lg^{3}(lg(x))lg^{2}(lg(lg(x)))lg(lg(lg(lg(x))))} + \frac{-10*0}{x^{3}ln^{11}{10}lg^{3}(x)lg^{3}(lg(x))lg^{2}(lg(lg(x)))lg(lg(lg(lg(x))))} + \frac{-3}{x^{3}ln^{10}{10}lg^{4}(x)ln{10}(x)lg^{3}(lg(x))lg^{2}(lg(lg(x)))lg(lg(lg(lg(x))))} + \frac{-3}{x^{3}ln^{10}{10}lg^{3}(x)lg^{4}(lg(x))ln{10}(lg(x))ln{10}(x)lg^{2}(lg(lg(x)))lg(lg(lg(lg(x))))} + \frac{-2}{x^{3}ln^{10}{10}lg^{3}(x)lg^{3}(lg(x))lg^{3}(lg(lg(x)))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)lg(lg(lg(lg(x))))} + \frac{-1}{x^{3}ln^{10}{10}lg^{3}(x)lg^{3}(lg(x))lg^{2}(lg(lg(x)))lg^{2}(lg(lg(lg(x))))ln{10}(lg(lg(lg(x))))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)} + \frac{-3}{x^{4}ln^{10}{10}lg^{2}(lg(lg(lg(x))))lg^{2}(lg(x))lg^{3}(x)lg^{2}(lg(lg(x)))} + \frac{-10*0}{x^{3}ln^{11}{10}lg^{2}(lg(lg(lg(x))))lg^{2}(lg(x))lg^{3}(x)lg^{2}(lg(lg(x)))} + \frac{-2}{x^{3}ln^{10}{10}lg^{3}(lg(lg(lg(x))))ln{10}(lg(lg(lg(x))))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)lg^{2}(lg(x))lg^{3}(x)lg^{2}(lg(lg(x)))} + \frac{-2}{x^{3}ln^{10}{10}lg^{2}(lg(lg(lg(x))))lg^{3}(lg(x))ln{10}(lg(x))ln{10}(x)lg^{3}(x)lg^{2}(lg(lg(x)))} + \frac{-3}{x^{3}ln^{10}{10}lg^{2}(lg(lg(lg(x))))lg^{2}(lg(x))lg^{4}(x)ln{10}(x)lg^{2}(lg(lg(x)))} + \frac{-2}{x^{3}ln^{10}{10}lg^{2}(lg(lg(lg(x))))lg^{2}(lg(x))lg^{3}(x)lg^{3}(lg(lg(x)))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)} + \frac{2*-3}{x^{4}ln^{7}{10}lg(lg(lg(lg(x))))lg^{3}(x)lg(lg(x))lg(lg(lg(x)))} + \frac{2*-7*0}{x^{3}ln^{8}{10}lg(lg(lg(lg(x))))lg^{3}(x)lg(lg(x))lg(lg(lg(x)))} + \frac{2*-1}{x^{3}ln^{7}{10}lg^{2}(lg(lg(lg(x))))ln{10}(lg(lg(lg(x))))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)lg^{3}(x)lg(lg(x))lg(lg(lg(x)))} + \frac{2*-3}{x^{3}ln^{7}{10}lg(lg(lg(lg(x))))lg^{4}(x)ln{10}(x)lg(lg(x))lg(lg(lg(x)))} + \frac{2*-1}{x^{3}ln^{7}{10}lg(lg(lg(lg(x))))lg^{3}(x)lg^{2}(lg(x))ln{10}(lg(x))ln{10}(x)lg(lg(lg(x)))} + \frac{2*-1}{x^{3}ln^{7}{10}lg(lg(lg(lg(x))))lg^{3}(x)lg(lg(x))lg^{2}(lg(lg(x)))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)} + \frac{3*-3}{x^{4}ln^{9}{10}lg^{3}(x)lg^{2}(lg(x))lg(lg(lg(lg(x))))lg^{2}(lg(lg(x)))} + \frac{3*-9*0}{x^{3}ln^{10}{10}lg^{3}(x)lg^{2}(lg(x))lg(lg(lg(lg(x))))lg^{2}(lg(lg(x)))} + \frac{3*-3}{x^{3}ln^{9}{10}lg^{4}(x)ln{10}(x)lg^{2}(lg(x))lg(lg(lg(lg(x))))lg^{2}(lg(lg(x)))} + \frac{3*-2}{x^{3}ln^{9}{10}lg^{3}(x)lg^{3}(lg(x))ln{10}(lg(x))ln{10}(x)lg(lg(lg(lg(x))))lg^{2}(lg(lg(x)))} + \frac{3*-1}{x^{3}ln^{9}{10}lg^{3}(x)lg^{2}(lg(x))lg^{2}(lg(lg(lg(x))))ln{10}(lg(lg(lg(x))))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)lg^{2}(lg(lg(x)))} + \frac{3*-2}{x^{3}ln^{9}{10}lg^{3}(x)lg^{2}(lg(x))lg(lg(lg(lg(x))))lg^{3}(lg(lg(x)))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)} + \frac{2*-3}{x^{4}ln^{10}{10}lg^{3}(x)lg^{3}(lg(x))lg(lg(lg(lg(x))))lg^{2}(lg(lg(x)))} + \frac{2*-10*0}{x^{3}ln^{11}{10}lg^{3}(x)lg^{3}(lg(x))lg(lg(lg(lg(x))))lg^{2}(lg(lg(x)))} + \frac{2*-3}{x^{3}ln^{10}{10}lg^{4}(x)ln{10}(x)lg^{3}(lg(x))lg(lg(lg(lg(x))))lg^{2}(lg(lg(x)))} + \frac{2*-3}{x^{3}ln^{10}{10}lg^{3}(x)lg^{4}(lg(x))ln{10}(lg(x))ln{10}(x)lg(lg(lg(lg(x))))lg^{2}(lg(lg(x)))} + \frac{2*-1}{x^{3}ln^{10}{10}lg^{3}(x)lg^{3}(lg(x))lg^{2}(lg(lg(lg(x))))ln{10}(lg(lg(lg(x))))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)lg^{2}(lg(lg(x)))} + \frac{2*-2}{x^{3}ln^{10}{10}lg^{3}(x)lg^{3}(lg(x))lg(lg(lg(lg(x))))lg^{3}(lg(lg(x)))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)} + \frac{-3}{x^{4}ln^{12}{10}lg^{3}(x)lg^{3}(lg(x))lg^{3}(lg(lg(x)))lg^{2}(lg(lg(lg(x))))} + \frac{-12*0}{x^{3}ln^{13}{10}lg^{3}(x)lg^{3}(lg(x))lg^{3}(lg(lg(x)))lg^{2}(lg(lg(lg(x))))} + \frac{-3}{x^{3}ln^{12}{10}lg^{4}(x)ln{10}(x)lg^{3}(lg(x))lg^{3}(lg(lg(x)))lg^{2}(lg(lg(lg(x))))} + \frac{-3}{x^{3}ln^{12}{10}lg^{3}(x)lg^{4}(lg(x))ln{10}(lg(x))ln{10}(x)lg^{3}(lg(lg(x)))lg^{2}(lg(lg(lg(x))))} + \frac{-3}{x^{3}ln^{12}{10}lg^{3}(x)lg^{3}(lg(x))lg^{4}(lg(lg(x)))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)lg^{2}(lg(lg(lg(x))))} + \frac{-2}{x^{3}ln^{12}{10}lg^{3}(x)lg^{3}(lg(x))lg^{3}(lg(lg(x)))lg^{3}(lg(lg(lg(x))))ln{10}(lg(lg(lg(x))))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)} + \frac{2*-3}{x^{4}ln^{11}{10}lg^{3}(x)lg(lg(lg(lg(x))))lg^{3}(lg(x))lg^{3}(lg(lg(x)))} + \frac{2*-11*0}{x^{3}ln^{12}{10}lg^{3}(x)lg(lg(lg(lg(x))))lg^{3}(lg(x))lg^{3}(lg(lg(x)))} + \frac{2*-3}{x^{3}ln^{11}{10}lg^{4}(x)ln{10}(x)lg(lg(lg(lg(x))))lg^{3}(lg(x))lg^{3}(lg(lg(x)))} + \frac{2*-1}{x^{3}ln^{11}{10}lg^{3}(x)lg^{2}(lg(lg(lg(x))))ln{10}(lg(lg(lg(x))))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)lg^{3}(lg(x))lg^{3}(lg(lg(x)))} + \frac{2*-3}{x^{3}ln^{11}{10}lg^{3}(x)lg(lg(lg(lg(x))))lg^{4}(lg(x))ln{10}(lg(x))ln{10}(x)lg^{3}(lg(lg(x)))} + \frac{2*-3}{x^{3}ln^{11}{10}lg^{3}(x)lg(lg(lg(lg(x))))lg^{3}(lg(x))lg^{4}(lg(lg(x)))ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)}\\=&\frac{-6}{x^{4}ln^{5}{10}lg(lg(lg(lg(x))))lg(lg(x))lg(x)lg(lg(lg(x)))} - \frac{11}{x^{4}ln^{9}{10}lg^{2}(lg(lg(lg(x))))lg^{2}(lg(x))lg^{2}(x)lg^{2}(lg(lg(x)))} - \frac{11}{x^{4}ln^{7}{10}lg^{2}(x)lg(lg(lg(lg(x))))lg^{2}(lg(x))lg(lg(lg(x)))} - \frac{11}{x^{4}ln^{6}{10}lg(lg(lg(lg(x))))lg^{2}(x)lg(lg(x))lg(lg(lg(x)))} - \frac{11}{x^{4}ln^{8}{10}lg^{2}(x)lg^{2}(lg(x))lg(lg(lg(lg(x))))lg^{2}(lg(lg(x)))} - \frac{12}{x^{4}ln^{13}{10}lg^{3}(lg(lg(lg(x))))lg^{3}(lg(x))lg^{3}(x)lg^{3}(lg(lg(x)))} - \frac{12}{x^{4}ln^{11}{10}lg^{3}(x)lg^{2}(lg(lg(lg(x))))lg^{3}(lg(x))lg^{2}(lg(lg(x)))} - \frac{12}{x^{4}ln^{10}{10}lg^{2}(lg(lg(lg(x))))lg^{3}(x)lg^{2}(lg(x))lg^{2}(lg(lg(x)))} - \frac{12}{x^{4}ln^{12}{10}lg^{3}(x)lg^{3}(lg(x))lg^{2}(lg(lg(lg(x))))lg^{3}(lg(lg(x)))} - \frac{18}{x^{4}ln^{8}{10}lg^{3}(x)lg(lg(lg(lg(x))))lg^{2}(lg(x))lg(lg(lg(x)))} - \frac{6}{x^{4}ln^{11}{10}lg^{3}(lg(x))lg^{3}(x)lg^{2}(lg(lg(lg(x))))lg^{2}(lg(lg(x)))} - \frac{12}{x^{4}ln^{9}{10}lg(lg(lg(lg(x))))lg^{3}(x)lg^{3}(lg(x))lg(lg(lg(x)))} - \frac{6}{x^{4}ln^{10}{10}lg^{3}(x)lg^{3}(lg(x))lg^{2}(lg(lg(x)))lg(lg(lg(lg(x))))} - \frac{6}{x^{4}ln^{10}{10}lg^{2}(lg(lg(lg(x))))lg^{2}(lg(x))lg^{3}(x)lg^{2}(lg(lg(x)))} - \frac{12}{x^{4}ln^{7}{10}lg(lg(lg(lg(x))))lg^{3}(x)lg(lg(x))lg(lg(lg(x)))} - \frac{18}{x^{4}ln^{9}{10}lg^{3}(x)lg^{2}(lg(x))lg(lg(lg(lg(x))))lg^{2}(lg(lg(x)))} - \frac{12}{x^{4}ln^{10}{10}lg^{3}(x)lg^{3}(lg(x))lg(lg(lg(lg(x))))lg^{2}(lg(lg(x)))} - \frac{6}{x^{4}ln^{12}{10}lg^{3}(x)lg^{3}(lg(x))lg^{3}(lg(lg(x)))lg^{2}(lg(lg(lg(x))))} - \frac{12}{x^{4}ln^{11}{10}lg^{3}(x)lg(lg(lg(lg(x))))lg^{3}(lg(x))lg^{3}(lg(lg(x)))} - \frac{6}{x^{4}ln^{17}{10}lg^{4}(lg(lg(lg(x))))lg^{4}(lg(x))lg^{4}(x)lg^{4}(lg(lg(x)))} - \frac{6}{x^{4}ln^{15}{10}lg^{4}(x)lg^{3}(lg(lg(lg(x))))lg^{4}(lg(x))lg^{3}(lg(lg(x)))} - \frac{6}{x^{4}ln^{14}{10}lg^{3}(lg(lg(lg(x))))lg^{4}(x)lg^{3}(lg(x))lg^{3}(lg(lg(x)))} - \frac{6}{x^{4}ln^{16}{10}lg^{4}(x)lg^{4}(lg(x))lg^{3}(lg(lg(lg(x))))lg^{4}(lg(lg(x)))} - \frac{12}{x^{4}ln^{12}{10}lg^{4}(x)lg^{2}(lg(lg(lg(x))))lg^{3}(lg(x))lg^{2}(lg(lg(x)))} - \frac{4}{x^{4}ln^{15}{10}lg^{4}(lg(x))lg^{4}(x)lg^{3}(lg(lg(lg(x))))lg^{3}(lg(lg(x)))} - \frac{6}{x^{4}ln^{13}{10}lg^{2}(lg(lg(lg(x))))lg^{4}(x)lg^{4}(lg(x))lg^{2}(lg(lg(x)))} - \frac{9}{x^{4}ln^{14}{10}lg^{4}(x)lg^{4}(lg(x))lg^{3}(lg(lg(x)))lg^{2}(lg(lg(lg(x))))} - \frac{6}{x^{4}ln^{14}{10}lg^{3}(lg(lg(lg(x))))lg^{3}(lg(x))lg^{4}(x)lg^{3}(lg(lg(x)))} - \frac{9}{x^{4}ln^{11}{10}lg^{2}(lg(lg(lg(x))))lg^{4}(x)lg^{2}(lg(x))lg^{2}(lg(lg(x)))} - \frac{12}{x^{4}ln^{13}{10}lg^{4}(x)lg^{3}(lg(x))lg^{2}(lg(lg(lg(x))))lg^{3}(lg(lg(x)))} - \frac{6}{x^{4}ln^{14}{10}lg^{4}(x)lg^{4}(lg(x))lg^{2}(lg(lg(lg(x))))lg^{3}(lg(lg(x)))} - \frac{4}{x^{4}ln^{16}{10}lg^{4}(x)lg^{4}(lg(x))lg^{4}(lg(lg(x)))lg^{3}(lg(lg(lg(x))))} - \frac{6}{x^{4}ln^{15}{10}lg^{4}(x)lg^{2}(lg(lg(lg(x))))lg^{4}(lg(x))lg^{4}(lg(lg(x)))} - \frac{11}{x^{4}ln^{9}{10}lg^{4}(x)lg(lg(lg(lg(x))))lg^{2}(lg(x))lg(lg(lg(x)))} - \frac{6}{x^{4}ln^{12}{10}lg^{3}(lg(x))lg^{4}(x)lg^{2}(lg(lg(lg(x))))lg^{2}(lg(lg(x)))} - \frac{12}{x^{4}ln^{10}{10}lg(lg(lg(lg(x))))lg^{4}(x)lg^{3}(lg(x))lg(lg(lg(x)))} - \frac{6}{x^{4}ln^{11}{10}lg^{4}(x)lg^{3}(lg(x))lg^{2}(lg(lg(x)))lg(lg(lg(lg(x))))} - \frac{3}{x^{4}ln^{13}{10}lg^{4}(x)lg^{4}(lg(x))lg^{2}(lg(lg(lg(x))))lg^{2}(lg(lg(x)))} - \frac{2}{x^{4}ln^{15}{10}lg^{4}(x)lg^{4}(lg(x))lg^{3}(lg(lg(x)))lg^{3}(lg(lg(lg(x))))} - \frac{3}{x^{4}ln^{14}{10}lg^{4}(x)lg^{2}(lg(lg(lg(x))))lg^{4}(lg(x))lg^{3}(lg(lg(x)))} - \frac{2}{x^{4}ln^{13}{10}lg^{2}(lg(lg(lg(x))))lg^{4}(lg(x))lg^{4}(x)lg^{2}(lg(lg(x)))} - \frac{6}{x^{4}ln^{11}{10}lg^{4}(x)lg(lg(lg(lg(x))))lg^{4}(lg(x))lg(lg(lg(x)))} - \frac{8}{x^{4}ln^{12}{10}lg^{4}(x)lg^{4}(lg(x))lg(lg(lg(lg(x))))lg^{2}(lg(lg(x)))} - \frac{3}{x^{4}ln^{12}{10}lg^{4}(x)lg^{4}(lg(x))lg^{2}(lg(lg(x)))lg(lg(lg(lg(x))))} - \frac{2}{x^{4}ln^{13}{10}lg^{4}(lg(x))lg^{3}(lg(lg(x)))lg^{4}(x)lg(lg(lg(lg(x))))} - \frac{2}{x^{4}ln^{11}{10}lg^{2}(lg(lg(lg(x))))lg^{2}(lg(x))lg^{4}(x)lg^{2}(lg(lg(x)))} - \frac{6}{x^{4}ln^{8}{10}lg(lg(lg(lg(x))))lg^{4}(x)lg(lg(x))lg(lg(lg(x)))} - \frac{11}{x^{4}ln^{10}{10}lg^{4}(x)lg^{2}(lg(x))lg(lg(lg(lg(x))))lg^{2}(lg(lg(x)))} - \frac{12}{x^{4}ln^{11}{10}lg^{4}(x)lg^{3}(lg(x))lg(lg(lg(lg(x))))lg^{2}(lg(lg(x)))} - \frac{6}{x^{4}ln^{13}{10}lg^{4}(x)lg^{3}(lg(x))lg^{3}(lg(lg(x)))lg^{2}(lg(lg(lg(x))))} - \frac{12}{x^{4}ln^{12}{10}lg^{4}(x)lg(lg(lg(lg(x))))lg^{3}(lg(x))lg^{3}(lg(lg(x)))} - \frac{4}{x^{4}ln^{13}{10}lg^{4}(x)lg(lg(lg(lg(x))))lg^{4}(lg(x))lg^{3}(lg(lg(x)))} - \frac{3}{x^{4}ln^{15}{10}lg^{4}(lg(x))lg^{4}(lg(lg(x)))lg^{4}(x)lg^{2}(lg(lg(lg(x))))} - \frac{2}{x^{4}ln^{16}{10}lg^{4}(x)lg^{3}(lg(lg(lg(x))))lg^{4}(lg(x))lg^{4}(lg(lg(x)))} - \frac{2}{x^{4}ln^{15}{10}lg^{4}(lg(x))lg^{4}(x)lg^{2}(lg(lg(lg(x))))lg^{4}(lg(lg(x)))} - \frac{6}{x^{4}ln^{13}{10}lg(lg(lg(lg(x))))lg^{4}(x)lg^{4}(lg(x))lg^{3}(lg(lg(x)))} - \frac{6}{x^{4}ln^{14}{10}lg^{4}(x)lg^{4}(lg(x))lg^{4}(lg(lg(x)))lg(lg(lg(lg(x))))}\\ \end{split}\end{equation} \]