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.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ log_{ln(x)}^{lg(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( log_{ln(x)}^{lg(x)}\right)}{dx}\\=&(\frac{(\frac{(\frac{1}{ln{10}(x)})}{(lg(x))} - \frac{(\frac{1}{(x)})log_{ln(x)}^{lg(x)}}{(ln(x))})}{(ln(ln(x)))})\\=&\frac{1}{xln{10}ln(ln(x))lg(x)} - \frac{log_{ln(x)}^{lg(x)}}{xln(x)ln(ln(x))}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{1}{xln{10}ln(ln(x))lg(x)} - \frac{log_{ln(x)}^{lg(x)}}{xln(x)ln(ln(x))}\right)}{dx}\\=&\frac{-1}{x^{2}ln{10}ln(ln(x))lg(x)} + \frac{-0}{xln^{2}{10}ln(ln(x))lg(x)} + \frac{-1}{xln{10}ln^{2}(ln(x))(ln(x))(x)lg(x)} + \frac{-1}{xln{10}ln(ln(x))lg^{2}(x)ln{10}(x)} - \frac{-log_{ln(x)}^{lg(x)}}{x^{2}ln(x)ln(ln(x))} - \frac{(\frac{(\frac{(\frac{1}{ln{10}(x)})}{(lg(x))} - \frac{(\frac{1}{(x)})log_{ln(x)}^{lg(x)}}{(ln(x))})}{(ln(ln(x)))})}{xln(x)ln(ln(x))} - \frac{log_{ln(x)}^{lg(x)}*-1}{xln^{2}(x)(x)ln(ln(x))} - \frac{log_{ln(x)}^{lg(x)}*-1}{xln(x)ln^{2}(ln(x))(ln(x))(x)}\\=&\frac{-1}{x^{2}ln{10}ln(ln(x))lg(x)} - \frac{1}{x^{2}ln{10}ln(x)ln^{2}(ln(x))lg(x)} - \frac{1}{x^{2}ln(ln(x))ln^{2}{10}lg^{2}(x)} + \frac{log_{ln(x)}^{lg(x)}}{x^{2}ln(x)ln(ln(x))} - \frac{1}{x^{2}ln{10}ln^{2}(ln(x))ln(x)lg(x)} + \frac{2log_{ln(x)}^{lg(x)}}{x^{2}ln^{2}(x)ln^{2}(ln(x))} + \frac{log_{ln(x)}^{lg(x)}}{x^{2}ln^{2}(x)ln(ln(x))}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-1}{x^{2}ln{10}ln(ln(x))lg(x)} - \frac{1}{x^{2}ln{10}ln(x)ln^{2}(ln(x))lg(x)} - \frac{1}{x^{2}ln(ln(x))ln^{2}{10}lg^{2}(x)} + \frac{log_{ln(x)}^{lg(x)}}{x^{2}ln(x)ln(ln(x))} - \frac{1}{x^{2}ln{10}ln^{2}(ln(x))ln(x)lg(x)} + \frac{2log_{ln(x)}^{lg(x)}}{x^{2}ln^{2}(x)ln^{2}(ln(x))} + \frac{log_{ln(x)}^{lg(x)}}{x^{2}ln^{2}(x)ln(ln(x))}\right)}{dx}\\=&\frac{--2}{x^{3}ln{10}ln(ln(x))lg(x)} - \frac{-0}{x^{2}ln^{2}{10}ln(ln(x))lg(x)} - \frac{-1}{x^{2}ln{10}ln^{2}(ln(x))(ln(x))(x)lg(x)} - \frac{-1}{x^{2}ln{10}ln(ln(x))lg^{2}(x)ln{10}(x)} - \frac{-2}{x^{3}ln{10}ln(x)ln^{2}(ln(x))lg(x)} - \frac{-0}{x^{2}ln^{2}{10}ln(x)ln^{2}(ln(x))lg(x)} - \frac{-1}{x^{2}ln{10}ln^{2}(x)(x)ln^{2}(ln(x))lg(x)} - \frac{-2}{x^{2}ln{10}ln(x)ln^{3}(ln(x))(ln(x))(x)lg(x)} - \frac{-1}{x^{2}ln{10}ln(x)ln^{2}(ln(x))lg^{2}(x)ln{10}(x)} - \frac{-2}{x^{3}ln(ln(x))ln^{2}{10}lg^{2}(x)} - \frac{-1}{x^{2}ln^{2}(ln(x))(ln(x))(x)ln^{2}{10}lg^{2}(x)} - \frac{-2*0}{x^{2}ln(ln(x))ln^{3}{10}lg^{2}(x)} - \frac{-2}{x^{2}ln(ln(x))ln^{2}{10}lg^{3}(x)ln{10}(x)} + \frac{-2log_{ln(x)}^{lg(x)}}{x^{3}ln(x)ln(ln(x))} + \frac{(\frac{(\frac{(\frac{1}{ln{10}(x)})}{(lg(x))} - \frac{(\frac{1}{(x)})log_{ln(x)}^{lg(x)}}{(ln(x))})}{(ln(ln(x)))})}{x^{2}ln(x)ln(ln(x))} + \frac{log_{ln(x)}^{lg(x)}*-1}{x^{2}ln^{2}(x)(x)ln(ln(x))} + \frac{log_{ln(x)}^{lg(x)}*-1}{x^{2}ln(x)ln^{2}(ln(x))(ln(x))(x)} - \frac{-2}{x^{3}ln{10}ln^{2}(ln(x))ln(x)lg(x)} - \frac{-0}{x^{2}ln^{2}{10}ln^{2}(ln(x))ln(x)lg(x)} - \frac{-2}{x^{2}ln{10}ln^{3}(ln(x))(ln(x))(x)ln(x)lg(x)} - \frac{-1}{x^{2}ln{10}ln^{2}(ln(x))ln^{2}(x)(x)lg(x)} - \frac{-1}{x^{2}ln{10}ln^{2}(ln(x))ln(x)lg^{2}(x)ln{10}(x)} + \frac{2*-2log_{ln(x)}^{lg(x)}}{x^{3}ln^{2}(x)ln^{2}(ln(x))} + \frac{2(\frac{(\frac{(\frac{1}{ln{10}(x)})}{(lg(x))} - \frac{(\frac{1}{(x)})log_{ln(x)}^{lg(x)}}{(ln(x))})}{(ln(ln(x)))})}{x^{2}ln^{2}(x)ln^{2}(ln(x))} + \frac{2log_{ln(x)}^{lg(x)}*-2}{x^{2}ln^{3}(x)(x)ln^{2}(ln(x))} + \frac{2log_{ln(x)}^{lg(x)}*-2}{x^{2}ln^{2}(x)ln^{3}(ln(x))(ln(x))(x)} + \frac{-2log_{ln(x)}^{lg(x)}}{x^{3}ln^{2}(x)ln(ln(x))} + \frac{(\frac{(\frac{(\frac{1}{ln{10}(x)})}{(lg(x))} - \frac{(\frac{1}{(x)})log_{ln(x)}^{lg(x)}}{(ln(x))})}{(ln(ln(x)))})}{x^{2}ln^{2}(x)ln(ln(x))} + \frac{log_{ln(x)}^{lg(x)}*-2}{x^{2}ln^{3}(x)(x)ln(ln(x))} + \frac{log_{ln(x)}^{lg(x)}*-1}{x^{2}ln^{2}(x)ln^{2}(ln(x))(ln(x))(x)}\\=&\frac{2}{x^{3}ln{10}ln(ln(x))lg(x)} + \frac{3}{x^{3}ln{10}ln(x)ln^{2}(ln(x))lg(x)} + \frac{3}{x^{3}ln(ln(x))ln^{2}{10}lg^{2}(x)} + \frac{1}{x^{3}ln^{2}(x)ln{10}ln^{2}(ln(x))lg(x)} + \frac{4}{x^{3}ln{10}ln^{3}(ln(x))ln^{2}(x)lg(x)} + \frac{1}{x^{3}ln^{2}{10}ln(x)ln^{2}(ln(x))lg^{2}(x)} + \frac{1}{x^{3}ln^{2}(ln(x))ln(x)ln^{2}{10}lg^{2}(x)} + \frac{2}{x^{3}ln^{3}{10}ln(ln(x))lg^{3}(x)} - \frac{2log_{ln(x)}^{lg(x)}}{x^{3}ln(x)ln(ln(x))} + \frac{3}{x^{3}ln{10}ln^{2}(ln(x))ln(x)lg(x)} - \frac{6log_{ln(x)}^{lg(x)}}{x^{3}ln^{2}(x)ln^{2}(ln(x))} - \frac{3log_{ln(x)}^{lg(x)}}{x^{3}ln^{2}(x)ln(ln(x))} + \frac{2}{x^{3}ln{10}ln^{2}(x)ln^{3}(ln(x))lg(x)} + \frac{1}{x^{3}ln^{2}(ln(x))ln{10}ln^{2}(x)lg(x)} + \frac{1}{x^{3}ln^{2}{10}ln^{2}(ln(x))ln(x)lg^{2}(x)} - \frac{6log_{ln(x)}^{lg(x)}}{x^{3}ln^{3}(x)ln^{3}(ln(x))} - \frac{6log_{ln(x)}^{lg(x)}}{x^{3}ln^{3}(x)ln^{2}(ln(x))} + \frac{1}{x^{3}ln{10}ln^{2}(ln(x))ln^{2}(x)lg(x)} - \frac{2log_{ln(x)}^{lg(x)}}{x^{3}ln^{3}(x)ln(ln(x))}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{2}{x^{3}ln{10}ln(ln(x))lg(x)} + \frac{3}{x^{3}ln{10}ln(x)ln^{2}(ln(x))lg(x)} + \frac{3}{x^{3}ln(ln(x))ln^{2}{10}lg^{2}(x)} + \frac{1}{x^{3}ln^{2}(x)ln{10}ln^{2}(ln(x))lg(x)} + \frac{4}{x^{3}ln{10}ln^{3}(ln(x))ln^{2}(x)lg(x)} + \frac{1}{x^{3}ln^{2}{10}ln(x)ln^{2}(ln(x))lg^{2}(x)} + \frac{1}{x^{3}ln^{2}(ln(x))ln(x)ln^{2}{10}lg^{2}(x)} + \frac{2}{x^{3}ln^{3}{10}ln(ln(x))lg^{3}(x)} - \frac{2log_{ln(x)}^{lg(x)}}{x^{3}ln(x)ln(ln(x))} + \frac{3}{x^{3}ln{10}ln^{2}(ln(x))ln(x)lg(x)} - \frac{6log_{ln(x)}^{lg(x)}}{x^{3}ln^{2}(x)ln^{2}(ln(x))} - \frac{3log_{ln(x)}^{lg(x)}}{x^{3}ln^{2}(x)ln(ln(x))} + \frac{2}{x^{3}ln{10}ln^{2}(x)ln^{3}(ln(x))lg(x)} + \frac{1}{x^{3}ln^{2}(ln(x))ln{10}ln^{2}(x)lg(x)} + \frac{1}{x^{3}ln^{2}{10}ln^{2}(ln(x))ln(x)lg^{2}(x)} - \frac{6log_{ln(x)}^{lg(x)}}{x^{3}ln^{3}(x)ln^{3}(ln(x))} - \frac{6log_{ln(x)}^{lg(x)}}{x^{3}ln^{3}(x)ln^{2}(ln(x))} + \frac{1}{x^{3}ln{10}ln^{2}(ln(x))ln^{2}(x)lg(x)} - \frac{2log_{ln(x)}^{lg(x)}}{x^{3}ln^{3}(x)ln(ln(x))}\right)}{dx}\\=&\frac{2*-3}{x^{4}ln{10}ln(ln(x))lg(x)} + \frac{2*-0}{x^{3}ln^{2}{10}ln(ln(x))lg(x)} + \frac{2*-1}{x^{3}ln{10}ln^{2}(ln(x))(ln(x))(x)lg(x)} + \frac{2*-1}{x^{3}ln{10}ln(ln(x))lg^{2}(x)ln{10}(x)} + \frac{3*-3}{x^{4}ln{10}ln(x)ln^{2}(ln(x))lg(x)} + \frac{3*-0}{x^{3}ln^{2}{10}ln(x)ln^{2}(ln(x))lg(x)} + \frac{3*-1}{x^{3}ln{10}ln^{2}(x)(x)ln^{2}(ln(x))lg(x)} + \frac{3*-2}{x^{3}ln{10}ln(x)ln^{3}(ln(x))(ln(x))(x)lg(x)} + \frac{3*-1}{x^{3}ln{10}ln(x)ln^{2}(ln(x))lg^{2}(x)ln{10}(x)} + \frac{3*-3}{x^{4}ln(ln(x))ln^{2}{10}lg^{2}(x)} + \frac{3*-1}{x^{3}ln^{2}(ln(x))(ln(x))(x)ln^{2}{10}lg^{2}(x)} + \frac{3*-2*0}{x^{3}ln(ln(x))ln^{3}{10}lg^{2}(x)} + \frac{3*-2}{x^{3}ln(ln(x))ln^{2}{10}lg^{3}(x)ln{10}(x)} + \frac{-3}{x^{4}ln^{2}(x)ln{10}ln^{2}(ln(x))lg(x)} + \frac{-2}{x^{3}ln^{3}(x)(x)ln{10}ln^{2}(ln(x))lg(x)} + \frac{-0}{x^{3}ln^{2}(x)ln^{2}{10}ln^{2}(ln(x))lg(x)} + \frac{-2}{x^{3}ln^{2}(x)ln{10}ln^{3}(ln(x))(ln(x))(x)lg(x)} + \frac{-1}{x^{3}ln^{2}(x)ln{10}ln^{2}(ln(x))lg^{2}(x)ln{10}(x)} + \frac{4*-3}{x^{4}ln{10}ln^{3}(ln(x))ln^{2}(x)lg(x)} + \frac{4*-0}{x^{3}ln^{2}{10}ln^{3}(ln(x))ln^{2}(x)lg(x)} + \frac{4*-3}{x^{3}ln{10}ln^{4}(ln(x))(ln(x))(x)ln^{2}(x)lg(x)} + \frac{4*-2}{x^{3}ln{10}ln^{3}(ln(x))ln^{3}(x)(x)lg(x)} + \frac{4*-1}{x^{3}ln{10}ln^{3}(ln(x))ln^{2}(x)lg^{2}(x)ln{10}(x)} + \frac{-3}{x^{4}ln^{2}{10}ln(x)ln^{2}(ln(x))lg^{2}(x)} + \frac{-2*0}{x^{3}ln^{3}{10}ln(x)ln^{2}(ln(x))lg^{2}(x)} + \frac{-1}{x^{3}ln^{2}{10}ln^{2}(x)(x)ln^{2}(ln(x))lg^{2}(x)} + \frac{-2}{x^{3}ln^{2}{10}ln(x)ln^{3}(ln(x))(ln(x))(x)lg^{2}(x)} + \frac{-2}{x^{3}ln^{2}{10}ln(x)ln^{2}(ln(x))lg^{3}(x)ln{10}(x)} + \frac{-3}{x^{4}ln^{2}(ln(x))ln(x)ln^{2}{10}lg^{2}(x)} + \frac{-2}{x^{3}ln^{3}(ln(x))(ln(x))(x)ln(x)ln^{2}{10}lg^{2}(x)} + \frac{-1}{x^{3}ln^{2}(ln(x))ln^{2}(x)(x)ln^{2}{10}lg^{2}(x)} + \frac{-2*0}{x^{3}ln^{2}(ln(x))ln(x)ln^{3}{10}lg^{2}(x)} + \frac{-2}{x^{3}ln^{2}(ln(x))ln(x)ln^{2}{10}lg^{3}(x)ln{10}(x)} + \frac{2*-3}{x^{4}ln^{3}{10}ln(ln(x))lg^{3}(x)} + \frac{2*-3*0}{x^{3}ln^{4}{10}ln(ln(x))lg^{3}(x)} + \frac{2*-1}{x^{3}ln^{3}{10}ln^{2}(ln(x))(ln(x))(x)lg^{3}(x)} + \frac{2*-3}{x^{3}ln^{3}{10}ln(ln(x))lg^{4}(x)ln{10}(x)} - \frac{2*-3log_{ln(x)}^{lg(x)}}{x^{4}ln(x)ln(ln(x))} - \frac{2(\frac{(\frac{(\frac{1}{ln{10}(x)})}{(lg(x))} - \frac{(\frac{1}{(x)})log_{ln(x)}^{lg(x)}}{(ln(x))})}{(ln(ln(x)))})}{x^{3}ln(x)ln(ln(x))} - \frac{2log_{ln(x)}^{lg(x)}*-1}{x^{3}ln^{2}(x)(x)ln(ln(x))} - \frac{2log_{ln(x)}^{lg(x)}*-1}{x^{3}ln(x)ln^{2}(ln(x))(ln(x))(x)} + \frac{3*-3}{x^{4}ln{10}ln^{2}(ln(x))ln(x)lg(x)} + \frac{3*-0}{x^{3}ln^{2}{10}ln^{2}(ln(x))ln(x)lg(x)} + \frac{3*-2}{x^{3}ln{10}ln^{3}(ln(x))(ln(x))(x)ln(x)lg(x)} + \frac{3*-1}{x^{3}ln{10}ln^{2}(ln(x))ln^{2}(x)(x)lg(x)} + \frac{3*-1}{x^{3}ln{10}ln^{2}(ln(x))ln(x)lg^{2}(x)ln{10}(x)} - \frac{6*-3log_{ln(x)}^{lg(x)}}{x^{4}ln^{2}(x)ln^{2}(ln(x))} - \frac{6(\frac{(\frac{(\frac{1}{ln{10}(x)})}{(lg(x))} - \frac{(\frac{1}{(x)})log_{ln(x)}^{lg(x)}}{(ln(x))})}{(ln(ln(x)))})}{x^{3}ln^{2}(x)ln^{2}(ln(x))} - \frac{6log_{ln(x)}^{lg(x)}*-2}{x^{3}ln^{3}(x)(x)ln^{2}(ln(x))} - \frac{6log_{ln(x)}^{lg(x)}*-2}{x^{3}ln^{2}(x)ln^{3}(ln(x))(ln(x))(x)} - \frac{3*-3log_{ln(x)}^{lg(x)}}{x^{4}ln^{2}(x)ln(ln(x))} - \frac{3(\frac{(\frac{(\frac{1}{ln{10}(x)})}{(lg(x))} - \frac{(\frac{1}{(x)})log_{ln(x)}^{lg(x)}}{(ln(x))})}{(ln(ln(x)))})}{x^{3}ln^{2}(x)ln(ln(x))} - \frac{3log_{ln(x)}^{lg(x)}*-2}{x^{3}ln^{3}(x)(x)ln(ln(x))} - \frac{3log_{ln(x)}^{lg(x)}*-1}{x^{3}ln^{2}(x)ln^{2}(ln(x))(ln(x))(x)} + \frac{2*-3}{x^{4}ln{10}ln^{2}(x)ln^{3}(ln(x))lg(x)} + \frac{2*-0}{x^{3}ln^{2}{10}ln^{2}(x)ln^{3}(ln(x))lg(x)} + \frac{2*-2}{x^{3}ln{10}ln^{3}(x)(x)ln^{3}(ln(x))lg(x)} + \frac{2*-3}{x^{3}ln{10}ln^{2}(x)ln^{4}(ln(x))(ln(x))(x)lg(x)} + \frac{2*-1}{x^{3}ln{10}ln^{2}(x)ln^{3}(ln(x))lg^{2}(x)ln{10}(x)} + \frac{-3}{x^{4}ln^{2}(ln(x))ln{10}ln^{2}(x)lg(x)} + \frac{-2}{x^{3}ln^{3}(ln(x))(ln(x))(x)ln{10}ln^{2}(x)lg(x)} + \frac{-0}{x^{3}ln^{2}(ln(x))ln^{2}{10}ln^{2}(x)lg(x)} + \frac{-2}{x^{3}ln^{2}(ln(x))ln{10}ln^{3}(x)(x)lg(x)} + \frac{-1}{x^{3}ln^{2}(ln(x))ln{10}ln^{2}(x)lg^{2}(x)ln{10}(x)} + \frac{-3}{x^{4}ln^{2}{10}ln^{2}(ln(x))ln(x)lg^{2}(x)} + \frac{-2*0}{x^{3}ln^{3}{10}ln^{2}(ln(x))ln(x)lg^{2}(x)} + \frac{-2}{x^{3}ln^{2}{10}ln^{3}(ln(x))(ln(x))(x)ln(x)lg^{2}(x)} + \frac{-1}{x^{3}ln^{2}{10}ln^{2}(ln(x))ln^{2}(x)(x)lg^{2}(x)} + \frac{-2}{x^{3}ln^{2}{10}ln^{2}(ln(x))ln(x)lg^{3}(x)ln{10}(x)} - \frac{6*-3log_{ln(x)}^{lg(x)}}{x^{4}ln^{3}(x)ln^{3}(ln(x))} - \frac{6(\frac{(\frac{(\frac{1}{ln{10}(x)})}{(lg(x))} - \frac{(\frac{1}{(x)})log_{ln(x)}^{lg(x)}}{(ln(x))})}{(ln(ln(x)))})}{x^{3}ln^{3}(x)ln^{3}(ln(x))} - \frac{6log_{ln(x)}^{lg(x)}*-3}{x^{3}ln^{4}(x)(x)ln^{3}(ln(x))} - \frac{6log_{ln(x)}^{lg(x)}*-3}{x^{3}ln^{3}(x)ln^{4}(ln(x))(ln(x))(x)} - \frac{6*-3log_{ln(x)}^{lg(x)}}{x^{4}ln^{3}(x)ln^{2}(ln(x))} - \frac{6(\frac{(\frac{(\frac{1}{ln{10}(x)})}{(lg(x))} - \frac{(\frac{1}{(x)})log_{ln(x)}^{lg(x)}}{(ln(x))})}{(ln(ln(x)))})}{x^{3}ln^{3}(x)ln^{2}(ln(x))} - \frac{6log_{ln(x)}^{lg(x)}*-3}{x^{3}ln^{4}(x)(x)ln^{2}(ln(x))} - \frac{6log_{ln(x)}^{lg(x)}*-2}{x^{3}ln^{3}(x)ln^{3}(ln(x))(ln(x))(x)} + \frac{-3}{x^{4}ln{10}ln^{2}(ln(x))ln^{2}(x)lg(x)} + \frac{-0}{x^{3}ln^{2}{10}ln^{2}(ln(x))ln^{2}(x)lg(x)} + \frac{-2}{x^{3}ln{10}ln^{3}(ln(x))(ln(x))(x)ln^{2}(x)lg(x)} + \frac{-2}{x^{3}ln{10}ln^{2}(ln(x))ln^{3}(x)(x)lg(x)} + \frac{-1}{x^{3}ln{10}ln^{2}(ln(x))ln^{2}(x)lg^{2}(x)ln{10}(x)} - \frac{2*-3log_{ln(x)}^{lg(x)}}{x^{4}ln^{3}(x)ln(ln(x))} - \frac{2(\frac{(\frac{(\frac{1}{ln{10}(x)})}{(lg(x))} - \frac{(\frac{1}{(x)})log_{ln(x)}^{lg(x)}}{(ln(x))})}{(ln(ln(x)))})}{x^{3}ln^{3}(x)ln(ln(x))} - \frac{2log_{ln(x)}^{lg(x)}*-3}{x^{3}ln^{4}(x)(x)ln(ln(x))} - \frac{2log_{ln(x)}^{lg(x)}*-1}{x^{3}ln^{3}(x)ln^{2}(ln(x))(ln(x))(x)}\\=&\frac{-6}{x^{4}ln{10}ln(ln(x))lg(x)} - \frac{11}{x^{4}ln{10}ln(x)ln^{2}(ln(x))lg(x)} - \frac{11}{x^{4}ln(ln(x))ln^{2}{10}lg^{2}(x)} - \frac{6}{x^{4}ln^{2}(x)ln{10}ln^{2}(ln(x))lg(x)} - \frac{24}{x^{4}ln{10}ln^{3}(ln(x))ln^{2}(x)lg(x)} - \frac{6}{x^{4}ln^{2}{10}ln(x)ln^{2}(ln(x))lg^{2}(x)} - \frac{6}{x^{4}ln^{2}(ln(x))ln(x)ln^{2}{10}lg^{2}(x)} - \frac{12}{x^{4}ln^{3}{10}ln(ln(x))lg^{3}(x)} - \frac{2}{x^{4}ln^{3}(x)ln{10}ln^{2}(ln(x))lg(x)} - \frac{2}{x^{4}ln^{3}(x)ln^{3}(ln(x))ln{10}lg(x)} - \frac{2}{x^{4}ln^{2}{10}ln^{2}(ln(x))ln^{2}(x)lg^{2}(x)} - \frac{12}{x^{4}ln{10}ln^{3}(x)ln^{4}(ln(x))lg(x)} - \frac{8}{x^{4}ln^{3}(ln(x))ln{10}ln^{3}(x)lg(x)} - \frac{6}{x^{4}ln^{2}{10}ln^{3}(ln(x))ln^{2}(x)lg^{2}(x)} - \frac{1}{x^{4}ln^{2}(x)ln^{2}{10}ln^{2}(ln(x))lg^{2}(x)} - \frac{6}{x^{4}ln^{3}{10}ln(x)ln^{2}(ln(x))lg^{3}(x)} - \frac{2}{x^{4}ln^{3}(ln(x))ln^{2}(x)ln^{2}{10}lg^{2}(x)} - \frac{1}{x^{4}ln^{2}(x)ln^{2}(ln(x))ln^{2}{10}lg^{2}(x)} - \frac{6}{x^{4}ln(ln(x))ln^{4}{10}lg^{4}(x)} + \frac{6log_{ln(x)}^{lg(x)}}{x^{4}ln(x)ln(ln(x))} - \frac{11}{x^{4}ln{10}ln^{2}(ln(x))ln(x)lg(x)} + \frac{22log_{ln(x)}^{lg(x)}}{x^{4}ln^{2}(x)ln^{2}(ln(x))} + \frac{11log_{ln(x)}^{lg(x)}}{x^{4}ln^{2}(x)ln(ln(x))} - \frac{12}{x^{4}ln{10}ln^{2}(x)ln^{3}(ln(x))lg(x)} - \frac{6}{x^{4}ln^{2}(ln(x))ln{10}ln^{2}(x)lg(x)} - \frac{6}{x^{4}ln^{2}{10}ln^{2}(ln(x))ln(x)lg^{2}(x)} + \frac{36log_{ln(x)}^{lg(x)}}{x^{4}ln^{3}(x)ln^{3}(ln(x))} + \frac{36log_{ln(x)}^{lg(x)}}{x^{4}ln^{3}(x)ln^{2}(ln(x))} - \frac{6}{x^{4}ln{10}ln^{2}(ln(x))ln^{2}(x)lg(x)} + \frac{12log_{ln(x)}^{lg(x)}}{x^{4}ln^{3}(x)ln(ln(x))} - \frac{4}{x^{4}ln^{3}(x)ln{10}ln^{3}(ln(x))lg(x)} - \frac{12}{x^{4}ln{10}ln^{4}(ln(x))ln^{3}(x)lg(x)} - \frac{4}{x^{4}ln^{2}{10}ln^{2}(x)ln^{3}(ln(x))lg^{2}(x)} - \frac{2}{x^{4}ln^{3}(ln(x))ln^{3}(x)ln{10}lg(x)} - \frac{4}{x^{4}ln{10}ln^{2}(ln(x))ln^{3}(x)lg(x)} - \frac{1}{x^{4}ln^{2}{10}ln^{2}(x)ln^{2}(ln(x))lg^{2}(x)} - \frac{1}{x^{4}ln^{2}(ln(x))ln^{2}{10}ln^{2}(x)lg^{2}(x)} - \frac{2}{x^{4}ln^{3}{10}ln^{2}(ln(x))ln(x)lg^{3}(x)} + \frac{24log_{ln(x)}^{lg(x)}}{x^{4}ln^{4}(x)ln^{4}(ln(x))} + \frac{36log_{ln(x)}^{lg(x)}}{x^{4}ln^{4}(x)ln^{3}(ln(x))} - \frac{6}{x^{4}ln{10}ln^{3}(ln(x))ln^{3}(x)lg(x)} + \frac{22log_{ln(x)}^{lg(x)}}{x^{4}ln^{4}(x)ln^{2}(ln(x))} - \frac{2}{x^{4}ln{10}ln^{3}(x)ln^{3}(ln(x))lg(x)} - \frac{2}{x^{4}ln^{2}(ln(x))ln{10}ln^{3}(x)lg(x)} + \frac{6log_{ln(x)}^{lg(x)}}{x^{4}ln^{4}(x)ln(ln(x))}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ log_{ln(x)}^{lg(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( log_{ln(x)}^{lg(x)}\right)}{dx}\\=&(\frac{(\frac{(\frac{1}{ln{10}(x)})}{(lg(x))} - \frac{(\frac{1}{(x)})log_{ln(x)}^{lg(x)}}{(ln(x))})}{(ln(ln(x)))})\\=&\frac{1}{xln{10}ln(ln(x))lg(x)} - \frac{log_{ln(x)}^{lg(x)}}{xln(x)ln(ln(x))}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{1}{xln{10}ln(ln(x))lg(x)} - \frac{log_{ln(x)}^{lg(x)}}{xln(x)ln(ln(x))}\right)}{dx}\\=&\frac{-1}{x^{2}ln{10}ln(ln(x))lg(x)} + \frac{-0}{xln^{2}{10}ln(ln(x))lg(x)} + \frac{-1}{xln{10}ln^{2}(ln(x))(ln(x))(x)lg(x)} + \frac{-1}{xln{10}ln(ln(x))lg^{2}(x)ln{10}(x)} - \frac{-log_{ln(x)}^{lg(x)}}{x^{2}ln(x)ln(ln(x))} - \frac{(\frac{(\frac{(\frac{1}{ln{10}(x)})}{(lg(x))} - \frac{(\frac{1}{(x)})log_{ln(x)}^{lg(x)}}{(ln(x))})}{(ln(ln(x)))})}{xln(x)ln(ln(x))} - \frac{log_{ln(x)}^{lg(x)}*-1}{xln^{2}(x)(x)ln(ln(x))} - \frac{log_{ln(x)}^{lg(x)}*-1}{xln(x)ln^{2}(ln(x))(ln(x))(x)}\\=&\frac{-1}{x^{2}ln{10}ln(ln(x))lg(x)} - \frac{1}{x^{2}ln{10}ln(x)ln^{2}(ln(x))lg(x)} - \frac{1}{x^{2}ln(ln(x))ln^{2}{10}lg^{2}(x)} + \frac{log_{ln(x)}^{lg(x)}}{x^{2}ln(x)ln(ln(x))} - \frac{1}{x^{2}ln{10}ln^{2}(ln(x))ln(x)lg(x)} + \frac{2log_{ln(x)}^{lg(x)}}{x^{2}ln^{2}(x)ln^{2}(ln(x))} + \frac{log_{ln(x)}^{lg(x)}}{x^{2}ln^{2}(x)ln(ln(x))}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-1}{x^{2}ln{10}ln(ln(x))lg(x)} - \frac{1}{x^{2}ln{10}ln(x)ln^{2}(ln(x))lg(x)} - \frac{1}{x^{2}ln(ln(x))ln^{2}{10}lg^{2}(x)} + \frac{log_{ln(x)}^{lg(x)}}{x^{2}ln(x)ln(ln(x))} - \frac{1}{x^{2}ln{10}ln^{2}(ln(x))ln(x)lg(x)} + \frac{2log_{ln(x)}^{lg(x)}}{x^{2}ln^{2}(x)ln^{2}(ln(x))} + \frac{log_{ln(x)}^{lg(x)}}{x^{2}ln^{2}(x)ln(ln(x))}\right)}{dx}\\=&\frac{--2}{x^{3}ln{10}ln(ln(x))lg(x)} - \frac{-0}{x^{2}ln^{2}{10}ln(ln(x))lg(x)} - \frac{-1}{x^{2}ln{10}ln^{2}(ln(x))(ln(x))(x)lg(x)} - \frac{-1}{x^{2}ln{10}ln(ln(x))lg^{2}(x)ln{10}(x)} - \frac{-2}{x^{3}ln{10}ln(x)ln^{2}(ln(x))lg(x)} - \frac{-0}{x^{2}ln^{2}{10}ln(x)ln^{2}(ln(x))lg(x)} - \frac{-1}{x^{2}ln{10}ln^{2}(x)(x)ln^{2}(ln(x))lg(x)} - \frac{-2}{x^{2}ln{10}ln(x)ln^{3}(ln(x))(ln(x))(x)lg(x)} - \frac{-1}{x^{2}ln{10}ln(x)ln^{2}(ln(x))lg^{2}(x)ln{10}(x)} - \frac{-2}{x^{3}ln(ln(x))ln^{2}{10}lg^{2}(x)} - \frac{-1}{x^{2}ln^{2}(ln(x))(ln(x))(x)ln^{2}{10}lg^{2}(x)} - \frac{-2*0}{x^{2}ln(ln(x))ln^{3}{10}lg^{2}(x)} - \frac{-2}{x^{2}ln(ln(x))ln^{2}{10}lg^{3}(x)ln{10}(x)} + \frac{-2log_{ln(x)}^{lg(x)}}{x^{3}ln(x)ln(ln(x))} + \frac{(\frac{(\frac{(\frac{1}{ln{10}(x)})}{(lg(x))} - \frac{(\frac{1}{(x)})log_{ln(x)}^{lg(x)}}{(ln(x))})}{(ln(ln(x)))})}{x^{2}ln(x)ln(ln(x))} + \frac{log_{ln(x)}^{lg(x)}*-1}{x^{2}ln^{2}(x)(x)ln(ln(x))} + \frac{log_{ln(x)}^{lg(x)}*-1}{x^{2}ln(x)ln^{2}(ln(x))(ln(x))(x)} - \frac{-2}{x^{3}ln{10}ln^{2}(ln(x))ln(x)lg(x)} - \frac{-0}{x^{2}ln^{2}{10}ln^{2}(ln(x))ln(x)lg(x)} - \frac{-2}{x^{2}ln{10}ln^{3}(ln(x))(ln(x))(x)ln(x)lg(x)} - \frac{-1}{x^{2}ln{10}ln^{2}(ln(x))ln^{2}(x)(x)lg(x)} - \frac{-1}{x^{2}ln{10}ln^{2}(ln(x))ln(x)lg^{2}(x)ln{10}(x)} + \frac{2*-2log_{ln(x)}^{lg(x)}}{x^{3}ln^{2}(x)ln^{2}(ln(x))} + \frac{2(\frac{(\frac{(\frac{1}{ln{10}(x)})}{(lg(x))} - \frac{(\frac{1}{(x)})log_{ln(x)}^{lg(x)}}{(ln(x))})}{(ln(ln(x)))})}{x^{2}ln^{2}(x)ln^{2}(ln(x))} + \frac{2log_{ln(x)}^{lg(x)}*-2}{x^{2}ln^{3}(x)(x)ln^{2}(ln(x))} + \frac{2log_{ln(x)}^{lg(x)}*-2}{x^{2}ln^{2}(x)ln^{3}(ln(x))(ln(x))(x)} + \frac{-2log_{ln(x)}^{lg(x)}}{x^{3}ln^{2}(x)ln(ln(x))} + \frac{(\frac{(\frac{(\frac{1}{ln{10}(x)})}{(lg(x))} - \frac{(\frac{1}{(x)})log_{ln(x)}^{lg(x)}}{(ln(x))})}{(ln(ln(x)))})}{x^{2}ln^{2}(x)ln(ln(x))} + \frac{log_{ln(x)}^{lg(x)}*-2}{x^{2}ln^{3}(x)(x)ln(ln(x))} + \frac{log_{ln(x)}^{lg(x)}*-1}{x^{2}ln^{2}(x)ln^{2}(ln(x))(ln(x))(x)}\\=&\frac{2}{x^{3}ln{10}ln(ln(x))lg(x)} + \frac{3}{x^{3}ln{10}ln(x)ln^{2}(ln(x))lg(x)} + \frac{3}{x^{3}ln(ln(x))ln^{2}{10}lg^{2}(x)} + \frac{1}{x^{3}ln^{2}(x)ln{10}ln^{2}(ln(x))lg(x)} + \frac{4}{x^{3}ln{10}ln^{3}(ln(x))ln^{2}(x)lg(x)} + \frac{1}{x^{3}ln^{2}{10}ln(x)ln^{2}(ln(x))lg^{2}(x)} + \frac{1}{x^{3}ln^{2}(ln(x))ln(x)ln^{2}{10}lg^{2}(x)} + \frac{2}{x^{3}ln^{3}{10}ln(ln(x))lg^{3}(x)} - \frac{2log_{ln(x)}^{lg(x)}}{x^{3}ln(x)ln(ln(x))} + \frac{3}{x^{3}ln{10}ln^{2}(ln(x))ln(x)lg(x)} - \frac{6log_{ln(x)}^{lg(x)}}{x^{3}ln^{2}(x)ln^{2}(ln(x))} - \frac{3log_{ln(x)}^{lg(x)}}{x^{3}ln^{2}(x)ln(ln(x))} + \frac{2}{x^{3}ln{10}ln^{2}(x)ln^{3}(ln(x))lg(x)} + \frac{1}{x^{3}ln^{2}(ln(x))ln{10}ln^{2}(x)lg(x)} + \frac{1}{x^{3}ln^{2}{10}ln^{2}(ln(x))ln(x)lg^{2}(x)} - \frac{6log_{ln(x)}^{lg(x)}}{x^{3}ln^{3}(x)ln^{3}(ln(x))} - \frac{6log_{ln(x)}^{lg(x)}}{x^{3}ln^{3}(x)ln^{2}(ln(x))} + \frac{1}{x^{3}ln{10}ln^{2}(ln(x))ln^{2}(x)lg(x)} - \frac{2log_{ln(x)}^{lg(x)}}{x^{3}ln^{3}(x)ln(ln(x))}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{2}{x^{3}ln{10}ln(ln(x))lg(x)} + \frac{3}{x^{3}ln{10}ln(x)ln^{2}(ln(x))lg(x)} + \frac{3}{x^{3}ln(ln(x))ln^{2}{10}lg^{2}(x)} + \frac{1}{x^{3}ln^{2}(x)ln{10}ln^{2}(ln(x))lg(x)} + \frac{4}{x^{3}ln{10}ln^{3}(ln(x))ln^{2}(x)lg(x)} + \frac{1}{x^{3}ln^{2}{10}ln(x)ln^{2}(ln(x))lg^{2}(x)} + \frac{1}{x^{3}ln^{2}(ln(x))ln(x)ln^{2}{10}lg^{2}(x)} + \frac{2}{x^{3}ln^{3}{10}ln(ln(x))lg^{3}(x)} - \frac{2log_{ln(x)}^{lg(x)}}{x^{3}ln(x)ln(ln(x))} + \frac{3}{x^{3}ln{10}ln^{2}(ln(x))ln(x)lg(x)} - \frac{6log_{ln(x)}^{lg(x)}}{x^{3}ln^{2}(x)ln^{2}(ln(x))} - \frac{3log_{ln(x)}^{lg(x)}}{x^{3}ln^{2}(x)ln(ln(x))} + \frac{2}{x^{3}ln{10}ln^{2}(x)ln^{3}(ln(x))lg(x)} + \frac{1}{x^{3}ln^{2}(ln(x))ln{10}ln^{2}(x)lg(x)} + \frac{1}{x^{3}ln^{2}{10}ln^{2}(ln(x))ln(x)lg^{2}(x)} - \frac{6log_{ln(x)}^{lg(x)}}{x^{3}ln^{3}(x)ln^{3}(ln(x))} - \frac{6log_{ln(x)}^{lg(x)}}{x^{3}ln^{3}(x)ln^{2}(ln(x))} + \frac{1}{x^{3}ln{10}ln^{2}(ln(x))ln^{2}(x)lg(x)} - \frac{2log_{ln(x)}^{lg(x)}}{x^{3}ln^{3}(x)ln(ln(x))}\right)}{dx}\\=&\frac{2*-3}{x^{4}ln{10}ln(ln(x))lg(x)} + \frac{2*-0}{x^{3}ln^{2}{10}ln(ln(x))lg(x)} + \frac{2*-1}{x^{3}ln{10}ln^{2}(ln(x))(ln(x))(x)lg(x)} + \frac{2*-1}{x^{3}ln{10}ln(ln(x))lg^{2}(x)ln{10}(x)} + \frac{3*-3}{x^{4}ln{10}ln(x)ln^{2}(ln(x))lg(x)} + \frac{3*-0}{x^{3}ln^{2}{10}ln(x)ln^{2}(ln(x))lg(x)} + \frac{3*-1}{x^{3}ln{10}ln^{2}(x)(x)ln^{2}(ln(x))lg(x)} + \frac{3*-2}{x^{3}ln{10}ln(x)ln^{3}(ln(x))(ln(x))(x)lg(x)} + \frac{3*-1}{x^{3}ln{10}ln(x)ln^{2}(ln(x))lg^{2}(x)ln{10}(x)} + \frac{3*-3}{x^{4}ln(ln(x))ln^{2}{10}lg^{2}(x)} + \frac{3*-1}{x^{3}ln^{2}(ln(x))(ln(x))(x)ln^{2}{10}lg^{2}(x)} + \frac{3*-2*0}{x^{3}ln(ln(x))ln^{3}{10}lg^{2}(x)} + \frac{3*-2}{x^{3}ln(ln(x))ln^{2}{10}lg^{3}(x)ln{10}(x)} + \frac{-3}{x^{4}ln^{2}(x)ln{10}ln^{2}(ln(x))lg(x)} + \frac{-2}{x^{3}ln^{3}(x)(x)ln{10}ln^{2}(ln(x))lg(x)} + \frac{-0}{x^{3}ln^{2}(x)ln^{2}{10}ln^{2}(ln(x))lg(x)} + \frac{-2}{x^{3}ln^{2}(x)ln{10}ln^{3}(ln(x))(ln(x))(x)lg(x)} + \frac{-1}{x^{3}ln^{2}(x)ln{10}ln^{2}(ln(x))lg^{2}(x)ln{10}(x)} + \frac{4*-3}{x^{4}ln{10}ln^{3}(ln(x))ln^{2}(x)lg(x)} + \frac{4*-0}{x^{3}ln^{2}{10}ln^{3}(ln(x))ln^{2}(x)lg(x)} + \frac{4*-3}{x^{3}ln{10}ln^{4}(ln(x))(ln(x))(x)ln^{2}(x)lg(x)} + \frac{4*-2}{x^{3}ln{10}ln^{3}(ln(x))ln^{3}(x)(x)lg(x)} + \frac{4*-1}{x^{3}ln{10}ln^{3}(ln(x))ln^{2}(x)lg^{2}(x)ln{10}(x)} + \frac{-3}{x^{4}ln^{2}{10}ln(x)ln^{2}(ln(x))lg^{2}(x)} + \frac{-2*0}{x^{3}ln^{3}{10}ln(x)ln^{2}(ln(x))lg^{2}(x)} + \frac{-1}{x^{3}ln^{2}{10}ln^{2}(x)(x)ln^{2}(ln(x))lg^{2}(x)} + \frac{-2}{x^{3}ln^{2}{10}ln(x)ln^{3}(ln(x))(ln(x))(x)lg^{2}(x)} + \frac{-2}{x^{3}ln^{2}{10}ln(x)ln^{2}(ln(x))lg^{3}(x)ln{10}(x)} + \frac{-3}{x^{4}ln^{2}(ln(x))ln(x)ln^{2}{10}lg^{2}(x)} + \frac{-2}{x^{3}ln^{3}(ln(x))(ln(x))(x)ln(x)ln^{2}{10}lg^{2}(x)} + \frac{-1}{x^{3}ln^{2}(ln(x))ln^{2}(x)(x)ln^{2}{10}lg^{2}(x)} + \frac{-2*0}{x^{3}ln^{2}(ln(x))ln(x)ln^{3}{10}lg^{2}(x)} + \frac{-2}{x^{3}ln^{2}(ln(x))ln(x)ln^{2}{10}lg^{3}(x)ln{10}(x)} + \frac{2*-3}{x^{4}ln^{3}{10}ln(ln(x))lg^{3}(x)} + \frac{2*-3*0}{x^{3}ln^{4}{10}ln(ln(x))lg^{3}(x)} + \frac{2*-1}{x^{3}ln^{3}{10}ln^{2}(ln(x))(ln(x))(x)lg^{3}(x)} + \frac{2*-3}{x^{3}ln^{3}{10}ln(ln(x))lg^{4}(x)ln{10}(x)} - \frac{2*-3log_{ln(x)}^{lg(x)}}{x^{4}ln(x)ln(ln(x))} - \frac{2(\frac{(\frac{(\frac{1}{ln{10}(x)})}{(lg(x))} - \frac{(\frac{1}{(x)})log_{ln(x)}^{lg(x)}}{(ln(x))})}{(ln(ln(x)))})}{x^{3}ln(x)ln(ln(x))} - \frac{2log_{ln(x)}^{lg(x)}*-1}{x^{3}ln^{2}(x)(x)ln(ln(x))} - \frac{2log_{ln(x)}^{lg(x)}*-1}{x^{3}ln(x)ln^{2}(ln(x))(ln(x))(x)} + \frac{3*-3}{x^{4}ln{10}ln^{2}(ln(x))ln(x)lg(x)} + \frac{3*-0}{x^{3}ln^{2}{10}ln^{2}(ln(x))ln(x)lg(x)} + \frac{3*-2}{x^{3}ln{10}ln^{3}(ln(x))(ln(x))(x)ln(x)lg(x)} + \frac{3*-1}{x^{3}ln{10}ln^{2}(ln(x))ln^{2}(x)(x)lg(x)} + \frac{3*-1}{x^{3}ln{10}ln^{2}(ln(x))ln(x)lg^{2}(x)ln{10}(x)} - \frac{6*-3log_{ln(x)}^{lg(x)}}{x^{4}ln^{2}(x)ln^{2}(ln(x))} - \frac{6(\frac{(\frac{(\frac{1}{ln{10}(x)})}{(lg(x))} - \frac{(\frac{1}{(x)})log_{ln(x)}^{lg(x)}}{(ln(x))})}{(ln(ln(x)))})}{x^{3}ln^{2}(x)ln^{2}(ln(x))} - \frac{6log_{ln(x)}^{lg(x)}*-2}{x^{3}ln^{3}(x)(x)ln^{2}(ln(x))} - \frac{6log_{ln(x)}^{lg(x)}*-2}{x^{3}ln^{2}(x)ln^{3}(ln(x))(ln(x))(x)} - \frac{3*-3log_{ln(x)}^{lg(x)}}{x^{4}ln^{2}(x)ln(ln(x))} - \frac{3(\frac{(\frac{(\frac{1}{ln{10}(x)})}{(lg(x))} - \frac{(\frac{1}{(x)})log_{ln(x)}^{lg(x)}}{(ln(x))})}{(ln(ln(x)))})}{x^{3}ln^{2}(x)ln(ln(x))} - \frac{3log_{ln(x)}^{lg(x)}*-2}{x^{3}ln^{3}(x)(x)ln(ln(x))} - \frac{3log_{ln(x)}^{lg(x)}*-1}{x^{3}ln^{2}(x)ln^{2}(ln(x))(ln(x))(x)} + \frac{2*-3}{x^{4}ln{10}ln^{2}(x)ln^{3}(ln(x))lg(x)} + \frac{2*-0}{x^{3}ln^{2}{10}ln^{2}(x)ln^{3}(ln(x))lg(x)} + \frac{2*-2}{x^{3}ln{10}ln^{3}(x)(x)ln^{3}(ln(x))lg(x)} + \frac{2*-3}{x^{3}ln{10}ln^{2}(x)ln^{4}(ln(x))(ln(x))(x)lg(x)} + \frac{2*-1}{x^{3}ln{10}ln^{2}(x)ln^{3}(ln(x))lg^{2}(x)ln{10}(x)} + \frac{-3}{x^{4}ln^{2}(ln(x))ln{10}ln^{2}(x)lg(x)} + \frac{-2}{x^{3}ln^{3}(ln(x))(ln(x))(x)ln{10}ln^{2}(x)lg(x)} + \frac{-0}{x^{3}ln^{2}(ln(x))ln^{2}{10}ln^{2}(x)lg(x)} + \frac{-2}{x^{3}ln^{2}(ln(x))ln{10}ln^{3}(x)(x)lg(x)} + \frac{-1}{x^{3}ln^{2}(ln(x))ln{10}ln^{2}(x)lg^{2}(x)ln{10}(x)} + \frac{-3}{x^{4}ln^{2}{10}ln^{2}(ln(x))ln(x)lg^{2}(x)} + \frac{-2*0}{x^{3}ln^{3}{10}ln^{2}(ln(x))ln(x)lg^{2}(x)} + \frac{-2}{x^{3}ln^{2}{10}ln^{3}(ln(x))(ln(x))(x)ln(x)lg^{2}(x)} + \frac{-1}{x^{3}ln^{2}{10}ln^{2}(ln(x))ln^{2}(x)(x)lg^{2}(x)} + \frac{-2}{x^{3}ln^{2}{10}ln^{2}(ln(x))ln(x)lg^{3}(x)ln{10}(x)} - \frac{6*-3log_{ln(x)}^{lg(x)}}{x^{4}ln^{3}(x)ln^{3}(ln(x))} - \frac{6(\frac{(\frac{(\frac{1}{ln{10}(x)})}{(lg(x))} - \frac{(\frac{1}{(x)})log_{ln(x)}^{lg(x)}}{(ln(x))})}{(ln(ln(x)))})}{x^{3}ln^{3}(x)ln^{3}(ln(x))} - \frac{6log_{ln(x)}^{lg(x)}*-3}{x^{3}ln^{4}(x)(x)ln^{3}(ln(x))} - \frac{6log_{ln(x)}^{lg(x)}*-3}{x^{3}ln^{3}(x)ln^{4}(ln(x))(ln(x))(x)} - \frac{6*-3log_{ln(x)}^{lg(x)}}{x^{4}ln^{3}(x)ln^{2}(ln(x))} - \frac{6(\frac{(\frac{(\frac{1}{ln{10}(x)})}{(lg(x))} - \frac{(\frac{1}{(x)})log_{ln(x)}^{lg(x)}}{(ln(x))})}{(ln(ln(x)))})}{x^{3}ln^{3}(x)ln^{2}(ln(x))} - \frac{6log_{ln(x)}^{lg(x)}*-3}{x^{3}ln^{4}(x)(x)ln^{2}(ln(x))} - \frac{6log_{ln(x)}^{lg(x)}*-2}{x^{3}ln^{3}(x)ln^{3}(ln(x))(ln(x))(x)} + \frac{-3}{x^{4}ln{10}ln^{2}(ln(x))ln^{2}(x)lg(x)} + \frac{-0}{x^{3}ln^{2}{10}ln^{2}(ln(x))ln^{2}(x)lg(x)} + \frac{-2}{x^{3}ln{10}ln^{3}(ln(x))(ln(x))(x)ln^{2}(x)lg(x)} + \frac{-2}{x^{3}ln{10}ln^{2}(ln(x))ln^{3}(x)(x)lg(x)} + \frac{-1}{x^{3}ln{10}ln^{2}(ln(x))ln^{2}(x)lg^{2}(x)ln{10}(x)} - \frac{2*-3log_{ln(x)}^{lg(x)}}{x^{4}ln^{3}(x)ln(ln(x))} - \frac{2(\frac{(\frac{(\frac{1}{ln{10}(x)})}{(lg(x))} - \frac{(\frac{1}{(x)})log_{ln(x)}^{lg(x)}}{(ln(x))})}{(ln(ln(x)))})}{x^{3}ln^{3}(x)ln(ln(x))} - \frac{2log_{ln(x)}^{lg(x)}*-3}{x^{3}ln^{4}(x)(x)ln(ln(x))} - \frac{2log_{ln(x)}^{lg(x)}*-1}{x^{3}ln^{3}(x)ln^{2}(ln(x))(ln(x))(x)}\\=&\frac{-6}{x^{4}ln{10}ln(ln(x))lg(x)} - \frac{11}{x^{4}ln{10}ln(x)ln^{2}(ln(x))lg(x)} - \frac{11}{x^{4}ln(ln(x))ln^{2}{10}lg^{2}(x)} - \frac{6}{x^{4}ln^{2}(x)ln{10}ln^{2}(ln(x))lg(x)} - \frac{24}{x^{4}ln{10}ln^{3}(ln(x))ln^{2}(x)lg(x)} - \frac{6}{x^{4}ln^{2}{10}ln(x)ln^{2}(ln(x))lg^{2}(x)} - \frac{6}{x^{4}ln^{2}(ln(x))ln(x)ln^{2}{10}lg^{2}(x)} - \frac{12}{x^{4}ln^{3}{10}ln(ln(x))lg^{3}(x)} - \frac{2}{x^{4}ln^{3}(x)ln{10}ln^{2}(ln(x))lg(x)} - \frac{2}{x^{4}ln^{3}(x)ln^{3}(ln(x))ln{10}lg(x)} - \frac{2}{x^{4}ln^{2}{10}ln^{2}(ln(x))ln^{2}(x)lg^{2}(x)} - \frac{12}{x^{4}ln{10}ln^{3}(x)ln^{4}(ln(x))lg(x)} - \frac{8}{x^{4}ln^{3}(ln(x))ln{10}ln^{3}(x)lg(x)} - \frac{6}{x^{4}ln^{2}{10}ln^{3}(ln(x))ln^{2}(x)lg^{2}(x)} - \frac{1}{x^{4}ln^{2}(x)ln^{2}{10}ln^{2}(ln(x))lg^{2}(x)} - \frac{6}{x^{4}ln^{3}{10}ln(x)ln^{2}(ln(x))lg^{3}(x)} - \frac{2}{x^{4}ln^{3}(ln(x))ln^{2}(x)ln^{2}{10}lg^{2}(x)} - \frac{1}{x^{4}ln^{2}(x)ln^{2}(ln(x))ln^{2}{10}lg^{2}(x)} - \frac{6}{x^{4}ln(ln(x))ln^{4}{10}lg^{4}(x)} + \frac{6log_{ln(x)}^{lg(x)}}{x^{4}ln(x)ln(ln(x))} - \frac{11}{x^{4}ln{10}ln^{2}(ln(x))ln(x)lg(x)} + \frac{22log_{ln(x)}^{lg(x)}}{x^{4}ln^{2}(x)ln^{2}(ln(x))} + \frac{11log_{ln(x)}^{lg(x)}}{x^{4}ln^{2}(x)ln(ln(x))} - \frac{12}{x^{4}ln{10}ln^{2}(x)ln^{3}(ln(x))lg(x)} - \frac{6}{x^{4}ln^{2}(ln(x))ln{10}ln^{2}(x)lg(x)} - \frac{6}{x^{4}ln^{2}{10}ln^{2}(ln(x))ln(x)lg^{2}(x)} + \frac{36log_{ln(x)}^{lg(x)}}{x^{4}ln^{3}(x)ln^{3}(ln(x))} + \frac{36log_{ln(x)}^{lg(x)}}{x^{4}ln^{3}(x)ln^{2}(ln(x))} - \frac{6}{x^{4}ln{10}ln^{2}(ln(x))ln^{2}(x)lg(x)} + \frac{12log_{ln(x)}^{lg(x)}}{x^{4}ln^{3}(x)ln(ln(x))} - \frac{4}{x^{4}ln^{3}(x)ln{10}ln^{3}(ln(x))lg(x)} - \frac{12}{x^{4}ln{10}ln^{4}(ln(x))ln^{3}(x)lg(x)} - \frac{4}{x^{4}ln^{2}{10}ln^{2}(x)ln^{3}(ln(x))lg^{2}(x)} - \frac{2}{x^{4}ln^{3}(ln(x))ln^{3}(x)ln{10}lg(x)} - \frac{4}{x^{4}ln{10}ln^{2}(ln(x))ln^{3}(x)lg(x)} - \frac{1}{x^{4}ln^{2}{10}ln^{2}(x)ln^{2}(ln(x))lg^{2}(x)} - \frac{1}{x^{4}ln^{2}(ln(x))ln^{2}{10}ln^{2}(x)lg^{2}(x)} - \frac{2}{x^{4}ln^{3}{10}ln^{2}(ln(x))ln(x)lg^{3}(x)} + \frac{24log_{ln(x)}^{lg(x)}}{x^{4}ln^{4}(x)ln^{4}(ln(x))} + \frac{36log_{ln(x)}^{lg(x)}}{x^{4}ln^{4}(x)ln^{3}(ln(x))} - \frac{6}{x^{4}ln{10}ln^{3}(ln(x))ln^{3}(x)lg(x)} + \frac{22log_{ln(x)}^{lg(x)}}{x^{4}ln^{4}(x)ln^{2}(ln(x))} - \frac{2}{x^{4}ln{10}ln^{3}(x)ln^{3}(ln(x))lg(x)} - \frac{2}{x^{4}ln^{2}(ln(x))ln{10}ln^{3}(x)lg(x)} + \frac{6log_{ln(x)}^{lg(x)}}{x^{4}ln^{4}(x)ln(ln(x))}\\ \end{split}\end{equation} \]