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.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ log_{lg(x)}^{log_{2}^{x}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( log_{lg(x)}^{log_{2}^{x}}\right)}{dx}\\=&(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{(ln(2))}))}{(log_{2}^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{log_{2}^{x}}}{(lg(x))})}{(ln(lg(x)))})\\=& - \frac{log_{lg(x)}^{log_{2}^{x}}}{xln{10}ln(lg(x))lg(x)} + \frac{1}{xlog(2, x)ln(2)ln(lg(x))}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( - \frac{log_{lg(x)}^{log_{2}^{x}}}{xln{10}ln(lg(x))lg(x)} + \frac{1}{xlog(2, x)ln(2)ln(lg(x))}\right)}{dx}\\=& - \frac{-log_{lg(x)}^{log_{2}^{x}}}{x^{2}ln{10}ln(lg(x))lg(x)} - \frac{(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{(ln(2))}))}{(log_{2}^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{log_{2}^{x}}}{(lg(x))})}{(ln(lg(x)))})}{xln{10}ln(lg(x))lg(x)} - \frac{log_{lg(x)}^{log_{2}^{x}}*-0}{xln^{2}{10}ln(lg(x))lg(x)} - \frac{log_{lg(x)}^{log_{2}^{x}}*-1}{xln{10}ln^{2}(lg(x))(lg(x))ln{10}(x)lg(x)} - \frac{log_{lg(x)}^{log_{2}^{x}}*-1}{xln{10}ln(lg(x))lg^{2}(x)ln{10}(x)} + \frac{-1}{x^{2}log(2, x)ln(2)ln(lg(x))} + \frac{(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{{\left(log(2, x)^{2}(ln(2))})}{xln(2)ln(lg(x))} + \frac{-0}{xlog(2, x)ln^{2}(2)(2)ln(lg(x))} + \frac{-1}{xlog(2, x)ln(2)ln^{2}(lg(x))(lg(x))ln{10}(x)}\\=&\frac{log_{lg(x)}^{log_{2}^{x}}}{x^{2}ln{10}ln(lg(x))lg(x)} - \frac{2}{x^{2}log(2, x)ln(2)ln^{2}(lg(x))ln{10}lg(x)} + \frac{2log_{lg(x)}^{log_{2}^{x}}}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{log_{lg(x)}^{log_{2}^{x}}}{x^{2}ln(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{1}{x^{2}{\left(log(2, x)^{2}ln^{2}(2)ln(lg(x))} - \frac{1}{x^{2}log(2, x)ln(2)ln(lg(x))}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{log_{lg(x)}^{log_{2}^{x}}}{x^{2}ln{10}ln(lg(x))lg(x)} - \frac{2}{x^{2}log(2, x)ln(2)ln^{2}(lg(x))ln{10}lg(x)} + \frac{2log_{lg(x)}^{log_{2}^{x}}}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{log_{lg(x)}^{log_{2}^{x}}}{x^{2}ln(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{1}{x^{2}{\left(log(2, x)^{2}ln^{2}(2)ln(lg(x))} - \frac{1}{x^{2}log(2, x)ln(2)ln(lg(x))}\right)}{dx}\\=&\frac{-2log_{lg(x)}^{log_{2}^{x}}}{x^{3}ln{10}ln(lg(x))lg(x)} + \frac{(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{(ln(2))}))}{(log_{2}^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{log_{2}^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{2}ln{10}ln(lg(x))lg(x)} + \frac{log_{lg(x)}^{log_{2}^{x}}*-0}{x^{2}ln^{2}{10}ln(lg(x))lg(x)} + \frac{log_{lg(x)}^{log_{2}^{x}}*-1}{x^{2}ln{10}ln^{2}(lg(x))(lg(x))ln{10}(x)lg(x)} + \frac{log_{lg(x)}^{log_{2}^{x}}*-1}{x^{2}ln{10}ln(lg(x))lg^{2}(x)ln{10}(x)} - \frac{2*-2}{x^{3}log(2, x)ln(2)ln^{2}(lg(x))ln{10}lg(x)} - \frac{2(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{{\left(log(2, x)^{2}(ln(2))})}{x^{2}ln(2)ln^{2}(lg(x))ln{10}lg(x)} - \frac{2*-0}{x^{2}log(2, x)ln^{2}(2)(2)ln^{2}(lg(x))ln{10}lg(x)} - \frac{2*-2}{x^{2}log(2, x)ln(2)ln^{3}(lg(x))(lg(x))ln{10}(x)ln{10}lg(x)} - \frac{2*-0}{x^{2}log(2, x)ln(2)ln^{2}(lg(x))ln^{2}{10}lg(x)} - \frac{2*-1}{x^{2}log(2, x)ln(2)ln^{2}(lg(x))ln{10}lg^{2}(x)ln{10}(x)} + \frac{2*-2log_{lg(x)}^{log_{2}^{x}}}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{2(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{(ln(2))}))}{(log_{2}^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{log_{2}^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{2log_{lg(x)}^{log_{2}^{x}}*-2*0}{x^{2}ln^{3}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{2log_{lg(x)}^{log_{2}^{x}}*-2}{x^{2}ln^{2}{10}ln^{3}(lg(x))(lg(x))ln{10}(x)lg^{2}(x)} + \frac{2log_{lg(x)}^{log_{2}^{x}}*-2}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg^{3}(x)ln{10}(x)} + \frac{-2log_{lg(x)}^{log_{2}^{x}}}{x^{3}ln(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{(ln(2))}))}{(log_{2}^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{log_{2}^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{2}ln(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{log_{lg(x)}^{log_{2}^{x}}*-1}{x^{2}ln^{2}(lg(x))(lg(x))ln{10}(x)ln^{2}{10}lg^{2}(x)} + \frac{log_{lg(x)}^{log_{2}^{x}}*-2*0}{x^{2}ln(lg(x))ln^{3}{10}lg^{2}(x)} + \frac{log_{lg(x)}^{log_{2}^{x}}*-2}{x^{2}ln(lg(x))ln^{2}{10}lg^{3}(x)ln{10}(x)} - \frac{-2}{x^{3}{\left(log(2, x)^{2}ln^{2}(2)ln(lg(x))} - \frac{(\frac{-2(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{{\left(log(2, x)^{3}(ln(2))})}{x^{2}ln^{2}(2)ln(lg(x))} - \frac{-2*0}{x^{2}{\left(log(2, x)^{2}ln^{3}(2)(2)ln(lg(x))} - \frac{-1}{x^{2}{\left(log(2, x)^{2}ln^{2}(2)ln^{2}(lg(x))(lg(x))ln{10}(x)} - \frac{-2}{x^{3}log(2, x)ln(2)ln(lg(x))} - \frac{(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{{\left(log(2, x)^{2}(ln(2))})}{x^{2}ln(2)ln(lg(x))} - \frac{-0}{x^{2}log(2, x)ln^{2}(2)(2)ln(lg(x))} - \frac{-1}{x^{2}log(2, x)ln(2)ln^{2}(lg(x))(lg(x))ln{10}(x)}\\=& - \frac{2log_{lg(x)}^{log_{2}^{x}}}{x^{3}ln{10}ln(lg(x))lg(x)} + \frac{6}{x^{3}log(2, x)ln(2)ln^{2}(lg(x))ln{10}lg(x)} - \frac{6log_{lg(x)}^{log_{2}^{x}}}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} - \frac{3log_{lg(x)}^{log_{2}^{x}}}{x^{3}ln(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{6log_{lg(x)}^{log_{2}^{x}}}{x^{3}ln^{3}{10}ln^{3}(lg(x))lg^{3}(x)} + \frac{6}{x^{3}log(2, x)ln(2)ln^{3}(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{2}{x^{3}log(2, x)ln^{2}{10}ln^{2}(lg(x))ln(2)lg^{2}(x)} - \frac{2log_{lg(x)}^{log_{2}^{x}}}{x^{3}ln^{3}{10}ln^{2}(lg(x))lg^{3}(x)} - \frac{4log_{lg(x)}^{log_{2}^{x}}}{x^{3}ln^{2}(lg(x))ln^{3}{10}lg^{3}(x)} + \frac{1}{x^{3}log(2, x)ln(2)ln^{2}(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{3}{x^{3}{\left(log(2, x)^{2}ln^{2}(2)ln^{2}(lg(x))ln{10}lg(x)} - \frac{2log_{lg(x)}^{log_{2}^{x}}}{x^{3}ln^{3}{10}ln(lg(x))lg^{3}(x)} + \frac{2}{x^{3}{\left(log(2, x)^{3}ln^{3}(2)ln(lg(x))} + \frac{3}{x^{3}{\left(log(2, x)^{2}ln^{2}(2)ln(lg(x))} + \frac{2}{x^{3}log(2, x)ln(2)ln(lg(x))}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( - \frac{2log_{lg(x)}^{log_{2}^{x}}}{x^{3}ln{10}ln(lg(x))lg(x)} + \frac{6}{x^{3}log(2, x)ln(2)ln^{2}(lg(x))ln{10}lg(x)} - \frac{6log_{lg(x)}^{log_{2}^{x}}}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} - \frac{3log_{lg(x)}^{log_{2}^{x}}}{x^{3}ln(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{6log_{lg(x)}^{log_{2}^{x}}}{x^{3}ln^{3}{10}ln^{3}(lg(x))lg^{3}(x)} + \frac{6}{x^{3}log(2, x)ln(2)ln^{3}(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{2}{x^{3}log(2, x)ln^{2}{10}ln^{2}(lg(x))ln(2)lg^{2}(x)} - \frac{2log_{lg(x)}^{log_{2}^{x}}}{x^{3}ln^{3}{10}ln^{2}(lg(x))lg^{3}(x)} - \frac{4log_{lg(x)}^{log_{2}^{x}}}{x^{3}ln^{2}(lg(x))ln^{3}{10}lg^{3}(x)} + \frac{1}{x^{3}log(2, x)ln(2)ln^{2}(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{3}{x^{3}{\left(log(2, x)^{2}ln^{2}(2)ln^{2}(lg(x))ln{10}lg(x)} - \frac{2log_{lg(x)}^{log_{2}^{x}}}{x^{3}ln^{3}{10}ln(lg(x))lg^{3}(x)} + \frac{2}{x^{3}{\left(log(2, x)^{3}ln^{3}(2)ln(lg(x))} + \frac{3}{x^{3}{\left(log(2, x)^{2}ln^{2}(2)ln(lg(x))} + \frac{2}{x^{3}log(2, x)ln(2)ln(lg(x))}\right)}{dx}\\=& - \frac{2*-3log_{lg(x)}^{log_{2}^{x}}}{x^{4}ln{10}ln(lg(x))lg(x)} - \frac{2(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{(ln(2))}))}{(log_{2}^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{log_{2}^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln{10}ln(lg(x))lg(x)} - \frac{2log_{lg(x)}^{log_{2}^{x}}*-0}{x^{3}ln^{2}{10}ln(lg(x))lg(x)} - \frac{2log_{lg(x)}^{log_{2}^{x}}*-1}{x^{3}ln{10}ln^{2}(lg(x))(lg(x))ln{10}(x)lg(x)} - \frac{2log_{lg(x)}^{log_{2}^{x}}*-1}{x^{3}ln{10}ln(lg(x))lg^{2}(x)ln{10}(x)} + \frac{6*-3}{x^{4}log(2, x)ln(2)ln^{2}(lg(x))ln{10}lg(x)} + \frac{6(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{{\left(log(2, x)^{2}(ln(2))})}{x^{3}ln(2)ln^{2}(lg(x))ln{10}lg(x)} + \frac{6*-0}{x^{3}log(2, x)ln^{2}(2)(2)ln^{2}(lg(x))ln{10}lg(x)} + \frac{6*-2}{x^{3}log(2, x)ln(2)ln^{3}(lg(x))(lg(x))ln{10}(x)ln{10}lg(x)} + \frac{6*-0}{x^{3}log(2, x)ln(2)ln^{2}(lg(x))ln^{2}{10}lg(x)} + \frac{6*-1}{x^{3}log(2, x)ln(2)ln^{2}(lg(x))ln{10}lg^{2}(x)ln{10}(x)} - \frac{6*-3log_{lg(x)}^{log_{2}^{x}}}{x^{4}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} - \frac{6(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{(ln(2))}))}{(log_{2}^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{log_{2}^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} - \frac{6log_{lg(x)}^{log_{2}^{x}}*-2*0}{x^{3}ln^{3}{10}ln^{2}(lg(x))lg^{2}(x)} - \frac{6log_{lg(x)}^{log_{2}^{x}}*-2}{x^{3}ln^{2}{10}ln^{3}(lg(x))(lg(x))ln{10}(x)lg^{2}(x)} - \frac{6log_{lg(x)}^{log_{2}^{x}}*-2}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{3}(x)ln{10}(x)} - \frac{3*-3log_{lg(x)}^{log_{2}^{x}}}{x^{4}ln(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{3(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{(ln(2))}))}{(log_{2}^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{log_{2}^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{3log_{lg(x)}^{log_{2}^{x}}*-1}{x^{3}ln^{2}(lg(x))(lg(x))ln{10}(x)ln^{2}{10}lg^{2}(x)} - \frac{3log_{lg(x)}^{log_{2}^{x}}*-2*0}{x^{3}ln(lg(x))ln^{3}{10}lg^{2}(x)} - \frac{3log_{lg(x)}^{log_{2}^{x}}*-2}{x^{3}ln(lg(x))ln^{2}{10}lg^{3}(x)ln{10}(x)} - \frac{6*-3log_{lg(x)}^{log_{2}^{x}}}{x^{4}ln^{3}{10}ln^{3}(lg(x))lg^{3}(x)} - \frac{6(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{(ln(2))}))}{(log_{2}^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{log_{2}^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln^{3}{10}ln^{3}(lg(x))lg^{3}(x)} - \frac{6log_{lg(x)}^{log_{2}^{x}}*-3*0}{x^{3}ln^{4}{10}ln^{3}(lg(x))lg^{3}(x)} - \frac{6log_{lg(x)}^{log_{2}^{x}}*-3}{x^{3}ln^{3}{10}ln^{4}(lg(x))(lg(x))ln{10}(x)lg^{3}(x)} - \frac{6log_{lg(x)}^{log_{2}^{x}}*-3}{x^{3}ln^{3}{10}ln^{3}(lg(x))lg^{4}(x)ln{10}(x)} + \frac{6*-3}{x^{4}log(2, x)ln(2)ln^{3}(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{6(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{{\left(log(2, x)^{2}(ln(2))})}{x^{3}ln(2)ln^{3}(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{6*-0}{x^{3}log(2, x)ln^{2}(2)(2)ln^{3}(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{6*-3}{x^{3}log(2, x)ln(2)ln^{4}(lg(x))(lg(x))ln{10}(x)ln^{2}{10}lg^{2}(x)} + \frac{6*-2*0}{x^{3}log(2, x)ln(2)ln^{3}(lg(x))ln^{3}{10}lg^{2}(x)} + \frac{6*-2}{x^{3}log(2, x)ln(2)ln^{3}(lg(x))ln^{2}{10}lg^{3}(x)ln{10}(x)} + \frac{2*-3}{x^{4}log(2, x)ln^{2}{10}ln^{2}(lg(x))ln(2)lg^{2}(x)} + \frac{2(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{{\left(log(2, x)^{2}(ln(2))})}{x^{3}ln^{2}{10}ln^{2}(lg(x))ln(2)lg^{2}(x)} + \frac{2*-2*0}{x^{3}log(2, x)ln^{3}{10}ln^{2}(lg(x))ln(2)lg^{2}(x)} + \frac{2*-2}{x^{3}log(2, x)ln^{2}{10}ln^{3}(lg(x))(lg(x))ln{10}(x)ln(2)lg^{2}(x)} + \frac{2*-0}{x^{3}log(2, x)ln^{2}{10}ln^{2}(lg(x))ln^{2}(2)(2)lg^{2}(x)} + \frac{2*-2}{x^{3}log(2, x)ln^{2}{10}ln^{2}(lg(x))ln(2)lg^{3}(x)ln{10}(x)} - \frac{2*-3log_{lg(x)}^{log_{2}^{x}}}{x^{4}ln^{3}{10}ln^{2}(lg(x))lg^{3}(x)} - \frac{2(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{(ln(2))}))}{(log_{2}^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{log_{2}^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln^{3}{10}ln^{2}(lg(x))lg^{3}(x)} - \frac{2log_{lg(x)}^{log_{2}^{x}}*-3*0}{x^{3}ln^{4}{10}ln^{2}(lg(x))lg^{3}(x)} - \frac{2log_{lg(x)}^{log_{2}^{x}}*-2}{x^{3}ln^{3}{10}ln^{3}(lg(x))(lg(x))ln{10}(x)lg^{3}(x)} - \frac{2log_{lg(x)}^{log_{2}^{x}}*-3}{x^{3}ln^{3}{10}ln^{2}(lg(x))lg^{4}(x)ln{10}(x)} - \frac{4*-3log_{lg(x)}^{log_{2}^{x}}}{x^{4}ln^{2}(lg(x))ln^{3}{10}lg^{3}(x)} - \frac{4(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{(ln(2))}))}{(log_{2}^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{log_{2}^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln^{2}(lg(x))ln^{3}{10}lg^{3}(x)} - \frac{4log_{lg(x)}^{log_{2}^{x}}*-2}{x^{3}ln^{3}(lg(x))(lg(x))ln{10}(x)ln^{3}{10}lg^{3}(x)} - \frac{4log_{lg(x)}^{log_{2}^{x}}*-3*0}{x^{3}ln^{2}(lg(x))ln^{4}{10}lg^{3}(x)} - \frac{4log_{lg(x)}^{log_{2}^{x}}*-3}{x^{3}ln^{2}(lg(x))ln^{3}{10}lg^{4}(x)ln{10}(x)} + \frac{-3}{x^{4}log(2, x)ln(2)ln^{2}(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{{\left(log(2, x)^{2}(ln(2))})}{x^{3}ln(2)ln^{2}(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{-0}{x^{3}log(2, x)ln^{2}(2)(2)ln^{2}(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{-2}{x^{3}log(2, x)ln(2)ln^{3}(lg(x))(lg(x))ln{10}(x)ln^{2}{10}lg^{2}(x)} + \frac{-2*0}{x^{3}log(2, x)ln(2)ln^{2}(lg(x))ln^{3}{10}lg^{2}(x)} + \frac{-2}{x^{3}log(2, x)ln(2)ln^{2}(lg(x))ln^{2}{10}lg^{3}(x)ln{10}(x)} + \frac{3*-3}{x^{4}{\left(log(2, x)^{2}ln^{2}(2)ln^{2}(lg(x))ln{10}lg(x)} + \frac{3(\frac{-2(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{{\left(log(2, x)^{3}(ln(2))})}{x^{3}ln^{2}(2)ln^{2}(lg(x))ln{10}lg(x)} + \frac{3*-2*0}{x^{3}{\left(log(2, x)^{2}ln^{3}(2)(2)ln^{2}(lg(x))ln{10}lg(x)} + \frac{3*-2}{x^{3}{\left(log(2, x)^{2}ln^{2}(2)ln^{3}(lg(x))(lg(x))ln{10}(x)ln{10}lg(x)} + \frac{3*-0}{x^{3}{\left(log(2, x)^{2}ln^{2}(2)ln^{2}(lg(x))ln^{2}{10}lg(x)} + \frac{3*-1}{x^{3}{\left(log(2, x)^{2}ln^{2}(2)ln^{2}(lg(x))ln{10}lg^{2}(x)ln{10}(x)} - \frac{2*-3log_{lg(x)}^{log_{2}^{x}}}{x^{4}ln^{3}{10}ln(lg(x))lg^{3}(x)} - \frac{2(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{(ln(2))}))}{(log_{2}^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{log_{2}^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln^{3}{10}ln(lg(x))lg^{3}(x)} - \frac{2log_{lg(x)}^{log_{2}^{x}}*-3*0}{x^{3}ln^{4}{10}ln(lg(x))lg^{3}(x)} - \frac{2log_{lg(x)}^{log_{2}^{x}}*-1}{x^{3}ln^{3}{10}ln^{2}(lg(x))(lg(x))ln{10}(x)lg^{3}(x)} - \frac{2log_{lg(x)}^{log_{2}^{x}}*-3}{x^{3}ln^{3}{10}ln(lg(x))lg^{4}(x)ln{10}(x)} + \frac{2*-3}{x^{4}{\left(log(2, x)^{3}ln^{3}(2)ln(lg(x))} + \frac{2(\frac{-3(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{{\left(log(2, x)^{4}(ln(2))})}{x^{3}ln^{3}(2)ln(lg(x))} + \frac{2*-3*0}{x^{3}{\left(log(2, x)^{3}ln^{4}(2)(2)ln(lg(x))} + \frac{2*-1}{x^{3}{\left(log(2, x)^{3}ln^{3}(2)ln^{2}(lg(x))(lg(x))ln{10}(x)} + \frac{3*-3}{x^{4}{\left(log(2, x)^{2}ln^{2}(2)ln(lg(x))} + \frac{3(\frac{-2(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{{\left(log(2, x)^{3}(ln(2))})}{x^{3}ln^{2}(2)ln(lg(x))} + \frac{3*-2*0}{x^{3}{\left(log(2, x)^{2}ln^{3}(2)(2)ln(lg(x))} + \frac{3*-1}{x^{3}{\left(log(2, x)^{2}ln^{2}(2)ln^{2}(lg(x))(lg(x))ln{10}(x)} + \frac{2*-3}{x^{4}log(2, x)ln(2)ln(lg(x))} + \frac{2(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{{\left(log(2, x)^{2}(ln(2))})}{x^{3}ln(2)ln(lg(x))} + \frac{2*-0}{x^{3}log(2, x)ln^{2}(2)(2)ln(lg(x))} + \frac{2*-1}{x^{3}log(2, x)ln(2)ln^{2}(lg(x))(lg(x))ln{10}(x)}\\=&\frac{6log_{lg(x)}^{log_{2}^{x}}}{x^{4}ln{10}ln(lg(x))lg(x)} - \frac{22}{x^{4}log(2, x)ln(2)ln^{2}(lg(x))ln{10}lg(x)} + \frac{22log_{lg(x)}^{log_{2}^{x}}}{x^{4}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{11log_{lg(x)}^{log_{2}^{x}}}{x^{4}ln(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{36log_{lg(x)}^{log_{2}^{x}}}{x^{4}ln^{3}{10}ln^{3}(lg(x))lg^{3}(x)} - \frac{36}{x^{4}log(2, x)ln(2)ln^{3}(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{12}{x^{4}log(2, x)ln^{2}{10}ln^{2}(lg(x))ln(2)lg^{2}(x)} + \frac{12log_{lg(x)}^{log_{2}^{x}}}{x^{4}ln^{3}{10}ln^{2}(lg(x))lg^{3}(x)} + \frac{24log_{lg(x)}^{log_{2}^{x}}}{x^{4}ln^{2}(lg(x))ln^{3}{10}lg^{3}(x)} - \frac{6}{x^{4}log(2, x)ln(2)ln^{2}(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{24log_{lg(x)}^{log_{2}^{x}}}{x^{4}ln^{4}{10}ln^{4}(lg(x))lg^{4}(x)} + \frac{12log_{lg(x)}^{log_{2}^{x}}}{x^{4}ln^{3}{10}ln(lg(x))lg^{3}(x)} - \frac{24}{x^{4}log(2, x)ln(2)ln^{4}(lg(x))ln^{3}{10}lg^{3}(x)} + \frac{18log_{lg(x)}^{log_{2}^{x}}}{x^{4}ln^{4}{10}ln^{3}(lg(x))lg^{4}(x)} + \frac{18log_{lg(x)}^{log_{2}^{x}}}{x^{4}ln^{3}(lg(x))ln^{4}{10}lg^{4}(x)} + \frac{16log_{lg(x)}^{log_{2}^{x}}}{x^{4}ln^{4}{10}ln^{2}(lg(x))lg^{4}(x)} - \frac{16}{x^{4}log(2, x)ln^{3}{10}ln^{3}(lg(x))ln(2)lg^{3}(x)} - \frac{2}{x^{4}{\left(log(2, x)^{2}ln^{2}(2)ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} - \frac{6}{x^{4}log(2, x)ln^{3}{10}ln^{2}(lg(x))ln(2)lg^{3}(x)} - \frac{8}{x^{4}log(2, x)ln(2)ln^{3}(lg(x))ln^{3}{10}lg^{3}(x)} - \frac{18}{x^{4}{\left(log(2, x)^{2}ln^{2}(2)ln^{2}(lg(x))ln{10}lg(x)} + \frac{6log_{lg(x)}^{log_{2}^{x}}}{x^{4}ln^{2}(lg(x))ln^{4}{10}lg^{4}(x)} - \frac{1}{x^{4}{\left(log(2, x)^{2}ln^{2}(2)ln^{2}(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{8}{x^{4}{\left(log(2, x)^{3}ln^{3}(2)ln^{2}(lg(x))ln{10}lg(x)} - \frac{12}{x^{4}{\left(log(2, x)^{2}ln^{2}(2)ln^{3}(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{3}{x^{4}{\left(log(2, x)^{2}ln^{2}{10}ln^{2}(lg(x))ln^{2}(2)lg^{2}(x)} - \frac{2}{x^{4}log(2, x)ln(2)ln^{2}(lg(x))ln^{3}{10}lg^{3}(x)} + \frac{6log_{lg(x)}^{log_{2}^{x}}}{x^{4}ln(lg(x))ln^{4}{10}lg^{4}(x)} - \frac{6}{x^{4}{\left(log(2, x)^{4}ln^{4}(2)ln(lg(x))} - \frac{12}{x^{4}{\left(log(2, x)^{3}ln^{3}(2)ln(lg(x))} - \frac{11}{x^{4}{\left(log(2, x)^{2}ln^{2}(2)ln(lg(x))} - \frac{6}{x^{4}log(2, x)ln(2)ln(lg(x))}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ log_{lg(x)}^{log_{2}^{x}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( log_{lg(x)}^{log_{2}^{x}}\right)}{dx}\\=&(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{(ln(2))}))}{(log_{2}^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{log_{2}^{x}}}{(lg(x))})}{(ln(lg(x)))})\\=& - \frac{log_{lg(x)}^{log_{2}^{x}}}{xln{10}ln(lg(x))lg(x)} + \frac{1}{xlog(2, x)ln(2)ln(lg(x))}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( - \frac{log_{lg(x)}^{log_{2}^{x}}}{xln{10}ln(lg(x))lg(x)} + \frac{1}{xlog(2, x)ln(2)ln(lg(x))}\right)}{dx}\\=& - \frac{-log_{lg(x)}^{log_{2}^{x}}}{x^{2}ln{10}ln(lg(x))lg(x)} - \frac{(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{(ln(2))}))}{(log_{2}^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{log_{2}^{x}}}{(lg(x))})}{(ln(lg(x)))})}{xln{10}ln(lg(x))lg(x)} - \frac{log_{lg(x)}^{log_{2}^{x}}*-0}{xln^{2}{10}ln(lg(x))lg(x)} - \frac{log_{lg(x)}^{log_{2}^{x}}*-1}{xln{10}ln^{2}(lg(x))(lg(x))ln{10}(x)lg(x)} - \frac{log_{lg(x)}^{log_{2}^{x}}*-1}{xln{10}ln(lg(x))lg^{2}(x)ln{10}(x)} + \frac{-1}{x^{2}log(2, x)ln(2)ln(lg(x))} + \frac{(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{{\left(log(2, x)^{2}(ln(2))})}{xln(2)ln(lg(x))} + \frac{-0}{xlog(2, x)ln^{2}(2)(2)ln(lg(x))} + \frac{-1}{xlog(2, x)ln(2)ln^{2}(lg(x))(lg(x))ln{10}(x)}\\=&\frac{log_{lg(x)}^{log_{2}^{x}}}{x^{2}ln{10}ln(lg(x))lg(x)} - \frac{2}{x^{2}log(2, x)ln(2)ln^{2}(lg(x))ln{10}lg(x)} + \frac{2log_{lg(x)}^{log_{2}^{x}}}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{log_{lg(x)}^{log_{2}^{x}}}{x^{2}ln(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{1}{x^{2}{\left(log(2, x)^{2}ln^{2}(2)ln(lg(x))} - \frac{1}{x^{2}log(2, x)ln(2)ln(lg(x))}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{log_{lg(x)}^{log_{2}^{x}}}{x^{2}ln{10}ln(lg(x))lg(x)} - \frac{2}{x^{2}log(2, x)ln(2)ln^{2}(lg(x))ln{10}lg(x)} + \frac{2log_{lg(x)}^{log_{2}^{x}}}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{log_{lg(x)}^{log_{2}^{x}}}{x^{2}ln(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{1}{x^{2}{\left(log(2, x)^{2}ln^{2}(2)ln(lg(x))} - \frac{1}{x^{2}log(2, x)ln(2)ln(lg(x))}\right)}{dx}\\=&\frac{-2log_{lg(x)}^{log_{2}^{x}}}{x^{3}ln{10}ln(lg(x))lg(x)} + \frac{(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{(ln(2))}))}{(log_{2}^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{log_{2}^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{2}ln{10}ln(lg(x))lg(x)} + \frac{log_{lg(x)}^{log_{2}^{x}}*-0}{x^{2}ln^{2}{10}ln(lg(x))lg(x)} + \frac{log_{lg(x)}^{log_{2}^{x}}*-1}{x^{2}ln{10}ln^{2}(lg(x))(lg(x))ln{10}(x)lg(x)} + \frac{log_{lg(x)}^{log_{2}^{x}}*-1}{x^{2}ln{10}ln(lg(x))lg^{2}(x)ln{10}(x)} - \frac{2*-2}{x^{3}log(2, x)ln(2)ln^{2}(lg(x))ln{10}lg(x)} - \frac{2(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{{\left(log(2, x)^{2}(ln(2))})}{x^{2}ln(2)ln^{2}(lg(x))ln{10}lg(x)} - \frac{2*-0}{x^{2}log(2, x)ln^{2}(2)(2)ln^{2}(lg(x))ln{10}lg(x)} - \frac{2*-2}{x^{2}log(2, x)ln(2)ln^{3}(lg(x))(lg(x))ln{10}(x)ln{10}lg(x)} - \frac{2*-0}{x^{2}log(2, x)ln(2)ln^{2}(lg(x))ln^{2}{10}lg(x)} - \frac{2*-1}{x^{2}log(2, x)ln(2)ln^{2}(lg(x))ln{10}lg^{2}(x)ln{10}(x)} + \frac{2*-2log_{lg(x)}^{log_{2}^{x}}}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{2(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{(ln(2))}))}{(log_{2}^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{log_{2}^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{2log_{lg(x)}^{log_{2}^{x}}*-2*0}{x^{2}ln^{3}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{2log_{lg(x)}^{log_{2}^{x}}*-2}{x^{2}ln^{2}{10}ln^{3}(lg(x))(lg(x))ln{10}(x)lg^{2}(x)} + \frac{2log_{lg(x)}^{log_{2}^{x}}*-2}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg^{3}(x)ln{10}(x)} + \frac{-2log_{lg(x)}^{log_{2}^{x}}}{x^{3}ln(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{(ln(2))}))}{(log_{2}^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{log_{2}^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{2}ln(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{log_{lg(x)}^{log_{2}^{x}}*-1}{x^{2}ln^{2}(lg(x))(lg(x))ln{10}(x)ln^{2}{10}lg^{2}(x)} + \frac{log_{lg(x)}^{log_{2}^{x}}*-2*0}{x^{2}ln(lg(x))ln^{3}{10}lg^{2}(x)} + \frac{log_{lg(x)}^{log_{2}^{x}}*-2}{x^{2}ln(lg(x))ln^{2}{10}lg^{3}(x)ln{10}(x)} - \frac{-2}{x^{3}{\left(log(2, x)^{2}ln^{2}(2)ln(lg(x))} - \frac{(\frac{-2(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{{\left(log(2, x)^{3}(ln(2))})}{x^{2}ln^{2}(2)ln(lg(x))} - \frac{-2*0}{x^{2}{\left(log(2, x)^{2}ln^{3}(2)(2)ln(lg(x))} - \frac{-1}{x^{2}{\left(log(2, x)^{2}ln^{2}(2)ln^{2}(lg(x))(lg(x))ln{10}(x)} - \frac{-2}{x^{3}log(2, x)ln(2)ln(lg(x))} - \frac{(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{{\left(log(2, x)^{2}(ln(2))})}{x^{2}ln(2)ln(lg(x))} - \frac{-0}{x^{2}log(2, x)ln^{2}(2)(2)ln(lg(x))} - \frac{-1}{x^{2}log(2, x)ln(2)ln^{2}(lg(x))(lg(x))ln{10}(x)}\\=& - \frac{2log_{lg(x)}^{log_{2}^{x}}}{x^{3}ln{10}ln(lg(x))lg(x)} + \frac{6}{x^{3}log(2, x)ln(2)ln^{2}(lg(x))ln{10}lg(x)} - \frac{6log_{lg(x)}^{log_{2}^{x}}}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} - \frac{3log_{lg(x)}^{log_{2}^{x}}}{x^{3}ln(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{6log_{lg(x)}^{log_{2}^{x}}}{x^{3}ln^{3}{10}ln^{3}(lg(x))lg^{3}(x)} + \frac{6}{x^{3}log(2, x)ln(2)ln^{3}(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{2}{x^{3}log(2, x)ln^{2}{10}ln^{2}(lg(x))ln(2)lg^{2}(x)} - \frac{2log_{lg(x)}^{log_{2}^{x}}}{x^{3}ln^{3}{10}ln^{2}(lg(x))lg^{3}(x)} - \frac{4log_{lg(x)}^{log_{2}^{x}}}{x^{3}ln^{2}(lg(x))ln^{3}{10}lg^{3}(x)} + \frac{1}{x^{3}log(2, x)ln(2)ln^{2}(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{3}{x^{3}{\left(log(2, x)^{2}ln^{2}(2)ln^{2}(lg(x))ln{10}lg(x)} - \frac{2log_{lg(x)}^{log_{2}^{x}}}{x^{3}ln^{3}{10}ln(lg(x))lg^{3}(x)} + \frac{2}{x^{3}{\left(log(2, x)^{3}ln^{3}(2)ln(lg(x))} + \frac{3}{x^{3}{\left(log(2, x)^{2}ln^{2}(2)ln(lg(x))} + \frac{2}{x^{3}log(2, x)ln(2)ln(lg(x))}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( - \frac{2log_{lg(x)}^{log_{2}^{x}}}{x^{3}ln{10}ln(lg(x))lg(x)} + \frac{6}{x^{3}log(2, x)ln(2)ln^{2}(lg(x))ln{10}lg(x)} - \frac{6log_{lg(x)}^{log_{2}^{x}}}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} - \frac{3log_{lg(x)}^{log_{2}^{x}}}{x^{3}ln(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{6log_{lg(x)}^{log_{2}^{x}}}{x^{3}ln^{3}{10}ln^{3}(lg(x))lg^{3}(x)} + \frac{6}{x^{3}log(2, x)ln(2)ln^{3}(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{2}{x^{3}log(2, x)ln^{2}{10}ln^{2}(lg(x))ln(2)lg^{2}(x)} - \frac{2log_{lg(x)}^{log_{2}^{x}}}{x^{3}ln^{3}{10}ln^{2}(lg(x))lg^{3}(x)} - \frac{4log_{lg(x)}^{log_{2}^{x}}}{x^{3}ln^{2}(lg(x))ln^{3}{10}lg^{3}(x)} + \frac{1}{x^{3}log(2, x)ln(2)ln^{2}(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{3}{x^{3}{\left(log(2, x)^{2}ln^{2}(2)ln^{2}(lg(x))ln{10}lg(x)} - \frac{2log_{lg(x)}^{log_{2}^{x}}}{x^{3}ln^{3}{10}ln(lg(x))lg^{3}(x)} + \frac{2}{x^{3}{\left(log(2, x)^{3}ln^{3}(2)ln(lg(x))} + \frac{3}{x^{3}{\left(log(2, x)^{2}ln^{2}(2)ln(lg(x))} + \frac{2}{x^{3}log(2, x)ln(2)ln(lg(x))}\right)}{dx}\\=& - \frac{2*-3log_{lg(x)}^{log_{2}^{x}}}{x^{4}ln{10}ln(lg(x))lg(x)} - \frac{2(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{(ln(2))}))}{(log_{2}^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{log_{2}^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln{10}ln(lg(x))lg(x)} - \frac{2log_{lg(x)}^{log_{2}^{x}}*-0}{x^{3}ln^{2}{10}ln(lg(x))lg(x)} - \frac{2log_{lg(x)}^{log_{2}^{x}}*-1}{x^{3}ln{10}ln^{2}(lg(x))(lg(x))ln{10}(x)lg(x)} - \frac{2log_{lg(x)}^{log_{2}^{x}}*-1}{x^{3}ln{10}ln(lg(x))lg^{2}(x)ln{10}(x)} + \frac{6*-3}{x^{4}log(2, x)ln(2)ln^{2}(lg(x))ln{10}lg(x)} + \frac{6(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{{\left(log(2, x)^{2}(ln(2))})}{x^{3}ln(2)ln^{2}(lg(x))ln{10}lg(x)} + \frac{6*-0}{x^{3}log(2, x)ln^{2}(2)(2)ln^{2}(lg(x))ln{10}lg(x)} + \frac{6*-2}{x^{3}log(2, x)ln(2)ln^{3}(lg(x))(lg(x))ln{10}(x)ln{10}lg(x)} + \frac{6*-0}{x^{3}log(2, x)ln(2)ln^{2}(lg(x))ln^{2}{10}lg(x)} + \frac{6*-1}{x^{3}log(2, x)ln(2)ln^{2}(lg(x))ln{10}lg^{2}(x)ln{10}(x)} - \frac{6*-3log_{lg(x)}^{log_{2}^{x}}}{x^{4}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} - \frac{6(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{(ln(2))}))}{(log_{2}^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{log_{2}^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} - \frac{6log_{lg(x)}^{log_{2}^{x}}*-2*0}{x^{3}ln^{3}{10}ln^{2}(lg(x))lg^{2}(x)} - \frac{6log_{lg(x)}^{log_{2}^{x}}*-2}{x^{3}ln^{2}{10}ln^{3}(lg(x))(lg(x))ln{10}(x)lg^{2}(x)} - \frac{6log_{lg(x)}^{log_{2}^{x}}*-2}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{3}(x)ln{10}(x)} - \frac{3*-3log_{lg(x)}^{log_{2}^{x}}}{x^{4}ln(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{3(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{(ln(2))}))}{(log_{2}^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{log_{2}^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{3log_{lg(x)}^{log_{2}^{x}}*-1}{x^{3}ln^{2}(lg(x))(lg(x))ln{10}(x)ln^{2}{10}lg^{2}(x)} - \frac{3log_{lg(x)}^{log_{2}^{x}}*-2*0}{x^{3}ln(lg(x))ln^{3}{10}lg^{2}(x)} - \frac{3log_{lg(x)}^{log_{2}^{x}}*-2}{x^{3}ln(lg(x))ln^{2}{10}lg^{3}(x)ln{10}(x)} - \frac{6*-3log_{lg(x)}^{log_{2}^{x}}}{x^{4}ln^{3}{10}ln^{3}(lg(x))lg^{3}(x)} - \frac{6(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{(ln(2))}))}{(log_{2}^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{log_{2}^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln^{3}{10}ln^{3}(lg(x))lg^{3}(x)} - \frac{6log_{lg(x)}^{log_{2}^{x}}*-3*0}{x^{3}ln^{4}{10}ln^{3}(lg(x))lg^{3}(x)} - \frac{6log_{lg(x)}^{log_{2}^{x}}*-3}{x^{3}ln^{3}{10}ln^{4}(lg(x))(lg(x))ln{10}(x)lg^{3}(x)} - \frac{6log_{lg(x)}^{log_{2}^{x}}*-3}{x^{3}ln^{3}{10}ln^{3}(lg(x))lg^{4}(x)ln{10}(x)} + \frac{6*-3}{x^{4}log(2, x)ln(2)ln^{3}(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{6(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{{\left(log(2, x)^{2}(ln(2))})}{x^{3}ln(2)ln^{3}(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{6*-0}{x^{3}log(2, x)ln^{2}(2)(2)ln^{3}(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{6*-3}{x^{3}log(2, x)ln(2)ln^{4}(lg(x))(lg(x))ln{10}(x)ln^{2}{10}lg^{2}(x)} + \frac{6*-2*0}{x^{3}log(2, x)ln(2)ln^{3}(lg(x))ln^{3}{10}lg^{2}(x)} + \frac{6*-2}{x^{3}log(2, x)ln(2)ln^{3}(lg(x))ln^{2}{10}lg^{3}(x)ln{10}(x)} + \frac{2*-3}{x^{4}log(2, x)ln^{2}{10}ln^{2}(lg(x))ln(2)lg^{2}(x)} + \frac{2(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{{\left(log(2, x)^{2}(ln(2))})}{x^{3}ln^{2}{10}ln^{2}(lg(x))ln(2)lg^{2}(x)} + \frac{2*-2*0}{x^{3}log(2, x)ln^{3}{10}ln^{2}(lg(x))ln(2)lg^{2}(x)} + \frac{2*-2}{x^{3}log(2, x)ln^{2}{10}ln^{3}(lg(x))(lg(x))ln{10}(x)ln(2)lg^{2}(x)} + \frac{2*-0}{x^{3}log(2, x)ln^{2}{10}ln^{2}(lg(x))ln^{2}(2)(2)lg^{2}(x)} + \frac{2*-2}{x^{3}log(2, x)ln^{2}{10}ln^{2}(lg(x))ln(2)lg^{3}(x)ln{10}(x)} - \frac{2*-3log_{lg(x)}^{log_{2}^{x}}}{x^{4}ln^{3}{10}ln^{2}(lg(x))lg^{3}(x)} - \frac{2(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{(ln(2))}))}{(log_{2}^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{log_{2}^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln^{3}{10}ln^{2}(lg(x))lg^{3}(x)} - \frac{2log_{lg(x)}^{log_{2}^{x}}*-3*0}{x^{3}ln^{4}{10}ln^{2}(lg(x))lg^{3}(x)} - \frac{2log_{lg(x)}^{log_{2}^{x}}*-2}{x^{3}ln^{3}{10}ln^{3}(lg(x))(lg(x))ln{10}(x)lg^{3}(x)} - \frac{2log_{lg(x)}^{log_{2}^{x}}*-3}{x^{3}ln^{3}{10}ln^{2}(lg(x))lg^{4}(x)ln{10}(x)} - \frac{4*-3log_{lg(x)}^{log_{2}^{x}}}{x^{4}ln^{2}(lg(x))ln^{3}{10}lg^{3}(x)} - \frac{4(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{(ln(2))}))}{(log_{2}^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{log_{2}^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln^{2}(lg(x))ln^{3}{10}lg^{3}(x)} - \frac{4log_{lg(x)}^{log_{2}^{x}}*-2}{x^{3}ln^{3}(lg(x))(lg(x))ln{10}(x)ln^{3}{10}lg^{3}(x)} - \frac{4log_{lg(x)}^{log_{2}^{x}}*-3*0}{x^{3}ln^{2}(lg(x))ln^{4}{10}lg^{3}(x)} - \frac{4log_{lg(x)}^{log_{2}^{x}}*-3}{x^{3}ln^{2}(lg(x))ln^{3}{10}lg^{4}(x)ln{10}(x)} + \frac{-3}{x^{4}log(2, x)ln(2)ln^{2}(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{{\left(log(2, x)^{2}(ln(2))})}{x^{3}ln(2)ln^{2}(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{-0}{x^{3}log(2, x)ln^{2}(2)(2)ln^{2}(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{-2}{x^{3}log(2, x)ln(2)ln^{3}(lg(x))(lg(x))ln{10}(x)ln^{2}{10}lg^{2}(x)} + \frac{-2*0}{x^{3}log(2, x)ln(2)ln^{2}(lg(x))ln^{3}{10}lg^{2}(x)} + \frac{-2}{x^{3}log(2, x)ln(2)ln^{2}(lg(x))ln^{2}{10}lg^{3}(x)ln{10}(x)} + \frac{3*-3}{x^{4}{\left(log(2, x)^{2}ln^{2}(2)ln^{2}(lg(x))ln{10}lg(x)} + \frac{3(\frac{-2(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{{\left(log(2, x)^{3}(ln(2))})}{x^{3}ln^{2}(2)ln^{2}(lg(x))ln{10}lg(x)} + \frac{3*-2*0}{x^{3}{\left(log(2, x)^{2}ln^{3}(2)(2)ln^{2}(lg(x))ln{10}lg(x)} + \frac{3*-2}{x^{3}{\left(log(2, x)^{2}ln^{2}(2)ln^{3}(lg(x))(lg(x))ln{10}(x)ln{10}lg(x)} + \frac{3*-0}{x^{3}{\left(log(2, x)^{2}ln^{2}(2)ln^{2}(lg(x))ln^{2}{10}lg(x)} + \frac{3*-1}{x^{3}{\left(log(2, x)^{2}ln^{2}(2)ln^{2}(lg(x))ln{10}lg^{2}(x)ln{10}(x)} - \frac{2*-3log_{lg(x)}^{log_{2}^{x}}}{x^{4}ln^{3}{10}ln(lg(x))lg^{3}(x)} - \frac{2(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{(ln(2))}))}{(log_{2}^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{log_{2}^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln^{3}{10}ln(lg(x))lg^{3}(x)} - \frac{2log_{lg(x)}^{log_{2}^{x}}*-3*0}{x^{3}ln^{4}{10}ln(lg(x))lg^{3}(x)} - \frac{2log_{lg(x)}^{log_{2}^{x}}*-1}{x^{3}ln^{3}{10}ln^{2}(lg(x))(lg(x))ln{10}(x)lg^{3}(x)} - \frac{2log_{lg(x)}^{log_{2}^{x}}*-3}{x^{3}ln^{3}{10}ln(lg(x))lg^{4}(x)ln{10}(x)} + \frac{2*-3}{x^{4}{\left(log(2, x)^{3}ln^{3}(2)ln(lg(x))} + \frac{2(\frac{-3(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{{\left(log(2, x)^{4}(ln(2))})}{x^{3}ln^{3}(2)ln(lg(x))} + \frac{2*-3*0}{x^{3}{\left(log(2, x)^{3}ln^{4}(2)(2)ln(lg(x))} + \frac{2*-1}{x^{3}{\left(log(2, x)^{3}ln^{3}(2)ln^{2}(lg(x))(lg(x))ln{10}(x)} + \frac{3*-3}{x^{4}{\left(log(2, x)^{2}ln^{2}(2)ln(lg(x))} + \frac{3(\frac{-2(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{{\left(log(2, x)^{3}(ln(2))})}{x^{3}ln^{2}(2)ln(lg(x))} + \frac{3*-2*0}{x^{3}{\left(log(2, x)^{2}ln^{3}(2)(2)ln(lg(x))} + \frac{3*-1}{x^{3}{\left(log(2, x)^{2}ln^{2}(2)ln^{2}(lg(x))(lg(x))ln{10}(x)} + \frac{2*-3}{x^{4}log(2, x)ln(2)ln(lg(x))} + \frac{2(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{{\left(log(2, x)^{2}(ln(2))})}{x^{3}ln(2)ln(lg(x))} + \frac{2*-0}{x^{3}log(2, x)ln^{2}(2)(2)ln(lg(x))} + \frac{2*-1}{x^{3}log(2, x)ln(2)ln^{2}(lg(x))(lg(x))ln{10}(x)}\\=&\frac{6log_{lg(x)}^{log_{2}^{x}}}{x^{4}ln{10}ln(lg(x))lg(x)} - \frac{22}{x^{4}log(2, x)ln(2)ln^{2}(lg(x))ln{10}lg(x)} + \frac{22log_{lg(x)}^{log_{2}^{x}}}{x^{4}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{11log_{lg(x)}^{log_{2}^{x}}}{x^{4}ln(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{36log_{lg(x)}^{log_{2}^{x}}}{x^{4}ln^{3}{10}ln^{3}(lg(x))lg^{3}(x)} - \frac{36}{x^{4}log(2, x)ln(2)ln^{3}(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{12}{x^{4}log(2, x)ln^{2}{10}ln^{2}(lg(x))ln(2)lg^{2}(x)} + \frac{12log_{lg(x)}^{log_{2}^{x}}}{x^{4}ln^{3}{10}ln^{2}(lg(x))lg^{3}(x)} + \frac{24log_{lg(x)}^{log_{2}^{x}}}{x^{4}ln^{2}(lg(x))ln^{3}{10}lg^{3}(x)} - \frac{6}{x^{4}log(2, x)ln(2)ln^{2}(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{24log_{lg(x)}^{log_{2}^{x}}}{x^{4}ln^{4}{10}ln^{4}(lg(x))lg^{4}(x)} + \frac{12log_{lg(x)}^{log_{2}^{x}}}{x^{4}ln^{3}{10}ln(lg(x))lg^{3}(x)} - \frac{24}{x^{4}log(2, x)ln(2)ln^{4}(lg(x))ln^{3}{10}lg^{3}(x)} + \frac{18log_{lg(x)}^{log_{2}^{x}}}{x^{4}ln^{4}{10}ln^{3}(lg(x))lg^{4}(x)} + \frac{18log_{lg(x)}^{log_{2}^{x}}}{x^{4}ln^{3}(lg(x))ln^{4}{10}lg^{4}(x)} + \frac{16log_{lg(x)}^{log_{2}^{x}}}{x^{4}ln^{4}{10}ln^{2}(lg(x))lg^{4}(x)} - \frac{16}{x^{4}log(2, x)ln^{3}{10}ln^{3}(lg(x))ln(2)lg^{3}(x)} - \frac{2}{x^{4}{\left(log(2, x)^{2}ln^{2}(2)ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} - \frac{6}{x^{4}log(2, x)ln^{3}{10}ln^{2}(lg(x))ln(2)lg^{3}(x)} - \frac{8}{x^{4}log(2, x)ln(2)ln^{3}(lg(x))ln^{3}{10}lg^{3}(x)} - \frac{18}{x^{4}{\left(log(2, x)^{2}ln^{2}(2)ln^{2}(lg(x))ln{10}lg(x)} + \frac{6log_{lg(x)}^{log_{2}^{x}}}{x^{4}ln^{2}(lg(x))ln^{4}{10}lg^{4}(x)} - \frac{1}{x^{4}{\left(log(2, x)^{2}ln^{2}(2)ln^{2}(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{8}{x^{4}{\left(log(2, x)^{3}ln^{3}(2)ln^{2}(lg(x))ln{10}lg(x)} - \frac{12}{x^{4}{\left(log(2, x)^{2}ln^{2}(2)ln^{3}(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{3}{x^{4}{\left(log(2, x)^{2}ln^{2}{10}ln^{2}(lg(x))ln^{2}(2)lg^{2}(x)} - \frac{2}{x^{4}log(2, x)ln(2)ln^{2}(lg(x))ln^{3}{10}lg^{3}(x)} + \frac{6log_{lg(x)}^{log_{2}^{x}}}{x^{4}ln(lg(x))ln^{4}{10}lg^{4}(x)} - \frac{6}{x^{4}{\left(log(2, x)^{4}ln^{4}(2)ln(lg(x))} - \frac{12}{x^{4}{\left(log(2, x)^{3}ln^{3}(2)ln(lg(x))} - \frac{11}{x^{4}{\left(log(2, x)^{2}ln^{2}(2)ln(lg(x))} - \frac{6}{x^{4}log(2, x)ln(2)ln(lg(x))}\\ \end{split}\end{equation} \]