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