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.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ log_{lg(x)}^{e^{x}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( log_{lg(x)}^{e^{x}}\right)}{dx}\\=&(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})\\=&\frac{1}{ln(lg(x))} - \frac{log_{lg(x)}^{e^{x}}}{xln{10}ln(lg(x))lg(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{1}{ln(lg(x))} - \frac{log_{lg(x)}^{e^{x}}}{xln{10}ln(lg(x))lg(x)}\right)}{dx}\\=&\frac{-1}{ln^{2}(lg(x))(lg(x))ln{10}(x)} - \frac{-log_{lg(x)}^{e^{x}}}{x^{2}ln{10}ln(lg(x))lg(x)} - \frac{(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{xln{10}ln(lg(x))lg(x)} - \frac{log_{lg(x)}^{e^{x}}*-0}{xln^{2}{10}ln(lg(x))lg(x)} - \frac{log_{lg(x)}^{e^{x}}*-1}{xln{10}ln^{2}(lg(x))(lg(x))ln{10}(x)lg(x)} - \frac{log_{lg(x)}^{e^{x}}*-1}{xln{10}ln(lg(x))lg^{2}(x)ln{10}(x)}\\=&\frac{-1}{xln{10}ln^{2}(lg(x))lg(x)} + \frac{log_{lg(x)}^{e^{x}}}{x^{2}ln{10}ln(lg(x))lg(x)} - \frac{1}{xln^{2}(lg(x))ln{10}lg(x)} + \frac{2log_{lg(x)}^{e^{x}}}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{log_{lg(x)}^{e^{x}}}{x^{2}ln(lg(x))ln^{2}{10}lg^{2}(x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-1}{xln{10}ln^{2}(lg(x))lg(x)} + \frac{log_{lg(x)}^{e^{x}}}{x^{2}ln{10}ln(lg(x))lg(x)} - \frac{1}{xln^{2}(lg(x))ln{10}lg(x)} + \frac{2log_{lg(x)}^{e^{x}}}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{log_{lg(x)}^{e^{x}}}{x^{2}ln(lg(x))ln^{2}{10}lg^{2}(x)}\right)}{dx}\\=&\frac{--1}{x^{2}ln{10}ln^{2}(lg(x))lg(x)} - \frac{-0}{xln^{2}{10}ln^{2}(lg(x))lg(x)} - \frac{-2}{xln{10}ln^{3}(lg(x))(lg(x))ln{10}(x)lg(x)} - \frac{-1}{xln{10}ln^{2}(lg(x))lg^{2}(x)ln{10}(x)} + \frac{-2log_{lg(x)}^{e^{x}}}{x^{3}ln{10}ln(lg(x))lg(x)} + \frac{(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{2}ln{10}ln(lg(x))lg(x)} + \frac{log_{lg(x)}^{e^{x}}*-0}{x^{2}ln^{2}{10}ln(lg(x))lg(x)} + \frac{log_{lg(x)}^{e^{x}}*-1}{x^{2}ln{10}ln^{2}(lg(x))(lg(x))ln{10}(x)lg(x)} + \frac{log_{lg(x)}^{e^{x}}*-1}{x^{2}ln{10}ln(lg(x))lg^{2}(x)ln{10}(x)} - \frac{-1}{x^{2}ln^{2}(lg(x))ln{10}lg(x)} - \frac{-2}{xln^{3}(lg(x))(lg(x))ln{10}(x)ln{10}lg(x)} - \frac{-0}{xln^{2}(lg(x))ln^{2}{10}lg(x)} - \frac{-1}{xln^{2}(lg(x))ln{10}lg^{2}(x)ln{10}(x)} + \frac{2*-2log_{lg(x)}^{e^{x}}}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{2(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{2log_{lg(x)}^{e^{x}}*-2*0}{x^{2}ln^{3}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{2log_{lg(x)}^{e^{x}}*-2}{x^{2}ln^{2}{10}ln^{3}(lg(x))(lg(x))ln{10}(x)lg^{2}(x)} + \frac{2log_{lg(x)}^{e^{x}}*-2}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg^{3}(x)ln{10}(x)} + \frac{-2log_{lg(x)}^{e^{x}}}{x^{3}ln(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{2}ln(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{log_{lg(x)}^{e^{x}}*-1}{x^{2}ln^{2}(lg(x))(lg(x))ln{10}(x)ln^{2}{10}lg^{2}(x)} + \frac{log_{lg(x)}^{e^{x}}*-2*0}{x^{2}ln(lg(x))ln^{3}{10}lg^{2}(x)} + \frac{log_{lg(x)}^{e^{x}}*-2}{x^{2}ln(lg(x))ln^{2}{10}lg^{3}(x)ln{10}(x)}\\=&\frac{1}{x^{2}ln{10}ln^{2}(lg(x))lg(x)} + \frac{4}{x^{2}ln^{2}{10}ln^{3}(lg(x))lg^{2}(x)} + \frac{2}{x^{2}ln^{2}(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{2log_{lg(x)}^{e^{x}}}{x^{3}ln{10}ln(lg(x))lg(x)} + \frac{2}{x^{2}ln^{2}(lg(x))ln{10}lg(x)} - \frac{6log_{lg(x)}^{e^{x}}}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} - \frac{3log_{lg(x)}^{e^{x}}}{x^{3}ln(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{1}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{2}{x^{2}ln^{3}(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{6log_{lg(x)}^{e^{x}}}{x^{3}ln^{3}{10}ln^{3}(lg(x))lg^{3}(x)} - \frac{4log_{lg(x)}^{e^{x}}}{x^{3}ln^{2}(lg(x))ln^{3}{10}lg^{3}(x)} - \frac{2log_{lg(x)}^{e^{x}}}{x^{3}ln^{3}{10}ln^{2}(lg(x))lg^{3}(x)} - \frac{2log_{lg(x)}^{e^{x}}}{x^{3}ln^{3}{10}ln(lg(x))lg^{3}(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{1}{x^{2}ln{10}ln^{2}(lg(x))lg(x)} + \frac{4}{x^{2}ln^{2}{10}ln^{3}(lg(x))lg^{2}(x)} + \frac{2}{x^{2}ln^{2}(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{2log_{lg(x)}^{e^{x}}}{x^{3}ln{10}ln(lg(x))lg(x)} + \frac{2}{x^{2}ln^{2}(lg(x))ln{10}lg(x)} - \frac{6log_{lg(x)}^{e^{x}}}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} - \frac{3log_{lg(x)}^{e^{x}}}{x^{3}ln(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{1}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{2}{x^{2}ln^{3}(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{6log_{lg(x)}^{e^{x}}}{x^{3}ln^{3}{10}ln^{3}(lg(x))lg^{3}(x)} - \frac{4log_{lg(x)}^{e^{x}}}{x^{3}ln^{2}(lg(x))ln^{3}{10}lg^{3}(x)} - \frac{2log_{lg(x)}^{e^{x}}}{x^{3}ln^{3}{10}ln^{2}(lg(x))lg^{3}(x)} - \frac{2log_{lg(x)}^{e^{x}}}{x^{3}ln^{3}{10}ln(lg(x))lg^{3}(x)}\right)}{dx}\\=&\frac{-2}{x^{3}ln{10}ln^{2}(lg(x))lg(x)} + \frac{-0}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg(x)} + \frac{-2}{x^{2}ln{10}ln^{3}(lg(x))(lg(x))ln{10}(x)lg(x)} + \frac{-1}{x^{2}ln{10}ln^{2}(lg(x))lg^{2}(x)ln{10}(x)} + \frac{4*-2}{x^{3}ln^{2}{10}ln^{3}(lg(x))lg^{2}(x)} + \frac{4*-2*0}{x^{2}ln^{3}{10}ln^{3}(lg(x))lg^{2}(x)} + \frac{4*-3}{x^{2}ln^{2}{10}ln^{4}(lg(x))(lg(x))ln{10}(x)lg^{2}(x)} + \frac{4*-2}{x^{2}ln^{2}{10}ln^{3}(lg(x))lg^{3}(x)ln{10}(x)} + \frac{2*-2}{x^{3}ln^{2}(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{2*-2}{x^{2}ln^{3}(lg(x))(lg(x))ln{10}(x)ln^{2}{10}lg^{2}(x)} + \frac{2*-2*0}{x^{2}ln^{2}(lg(x))ln^{3}{10}lg^{2}(x)} + \frac{2*-2}{x^{2}ln^{2}(lg(x))ln^{2}{10}lg^{3}(x)ln{10}(x)} - \frac{2*-3log_{lg(x)}^{e^{x}}}{x^{4}ln{10}ln(lg(x))lg(x)} - \frac{2(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln{10}ln(lg(x))lg(x)} - \frac{2log_{lg(x)}^{e^{x}}*-0}{x^{3}ln^{2}{10}ln(lg(x))lg(x)} - \frac{2log_{lg(x)}^{e^{x}}*-1}{x^{3}ln{10}ln^{2}(lg(x))(lg(x))ln{10}(x)lg(x)} - \frac{2log_{lg(x)}^{e^{x}}*-1}{x^{3}ln{10}ln(lg(x))lg^{2}(x)ln{10}(x)} + \frac{2*-2}{x^{3}ln^{2}(lg(x))ln{10}lg(x)} + \frac{2*-2}{x^{2}ln^{3}(lg(x))(lg(x))ln{10}(x)ln{10}lg(x)} + \frac{2*-0}{x^{2}ln^{2}(lg(x))ln^{2}{10}lg(x)} + \frac{2*-1}{x^{2}ln^{2}(lg(x))ln{10}lg^{2}(x)ln{10}(x)} - \frac{6*-3log_{lg(x)}^{e^{x}}}{x^{4}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} - \frac{6(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} - \frac{6log_{lg(x)}^{e^{x}}*-2*0}{x^{3}ln^{3}{10}ln^{2}(lg(x))lg^{2}(x)} - \frac{6log_{lg(x)}^{e^{x}}*-2}{x^{3}ln^{2}{10}ln^{3}(lg(x))(lg(x))ln{10}(x)lg^{2}(x)} - \frac{6log_{lg(x)}^{e^{x}}*-2}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{3}(x)ln{10}(x)} - \frac{3*-3log_{lg(x)}^{e^{x}}}{x^{4}ln(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{3(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{3log_{lg(x)}^{e^{x}}*-1}{x^{3}ln^{2}(lg(x))(lg(x))ln{10}(x)ln^{2}{10}lg^{2}(x)} - \frac{3log_{lg(x)}^{e^{x}}*-2*0}{x^{3}ln(lg(x))ln^{3}{10}lg^{2}(x)} - \frac{3log_{lg(x)}^{e^{x}}*-2}{x^{3}ln(lg(x))ln^{2}{10}lg^{3}(x)ln{10}(x)} + \frac{-2}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{-2*0}{x^{2}ln^{3}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{-2}{x^{2}ln^{2}{10}ln^{3}(lg(x))(lg(x))ln{10}(x)lg^{2}(x)} + \frac{-2}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg^{3}(x)ln{10}(x)} + \frac{2*-2}{x^{3}ln^{3}(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{2*-3}{x^{2}ln^{4}(lg(x))(lg(x))ln{10}(x)ln^{2}{10}lg^{2}(x)} + \frac{2*-2*0}{x^{2}ln^{3}(lg(x))ln^{3}{10}lg^{2}(x)} + \frac{2*-2}{x^{2}ln^{3}(lg(x))ln^{2}{10}lg^{3}(x)ln{10}(x)} - \frac{6*-3log_{lg(x)}^{e^{x}}}{x^{4}ln^{3}{10}ln^{3}(lg(x))lg^{3}(x)} - \frac{6(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln^{3}{10}ln^{3}(lg(x))lg^{3}(x)} - \frac{6log_{lg(x)}^{e^{x}}*-3*0}{x^{3}ln^{4}{10}ln^{3}(lg(x))lg^{3}(x)} - \frac{6log_{lg(x)}^{e^{x}}*-3}{x^{3}ln^{3}{10}ln^{4}(lg(x))(lg(x))ln{10}(x)lg^{3}(x)} - \frac{6log_{lg(x)}^{e^{x}}*-3}{x^{3}ln^{3}{10}ln^{3}(lg(x))lg^{4}(x)ln{10}(x)} - \frac{4*-3log_{lg(x)}^{e^{x}}}{x^{4}ln^{2}(lg(x))ln^{3}{10}lg^{3}(x)} - \frac{4(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln^{2}(lg(x))ln^{3}{10}lg^{3}(x)} - \frac{4log_{lg(x)}^{e^{x}}*-2}{x^{3}ln^{3}(lg(x))(lg(x))ln{10}(x)ln^{3}{10}lg^{3}(x)} - \frac{4log_{lg(x)}^{e^{x}}*-3*0}{x^{3}ln^{2}(lg(x))ln^{4}{10}lg^{3}(x)} - \frac{4log_{lg(x)}^{e^{x}}*-3}{x^{3}ln^{2}(lg(x))ln^{3}{10}lg^{4}(x)ln{10}(x)} - \frac{2*-3log_{lg(x)}^{e^{x}}}{x^{4}ln^{3}{10}ln^{2}(lg(x))lg^{3}(x)} - \frac{2(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln^{3}{10}ln^{2}(lg(x))lg^{3}(x)} - \frac{2log_{lg(x)}^{e^{x}}*-3*0}{x^{3}ln^{4}{10}ln^{2}(lg(x))lg^{3}(x)} - \frac{2log_{lg(x)}^{e^{x}}*-2}{x^{3}ln^{3}{10}ln^{3}(lg(x))(lg(x))ln{10}(x)lg^{3}(x)} - \frac{2log_{lg(x)}^{e^{x}}*-3}{x^{3}ln^{3}{10}ln^{2}(lg(x))lg^{4}(x)ln{10}(x)} - \frac{2*-3log_{lg(x)}^{e^{x}}}{x^{4}ln^{3}{10}ln(lg(x))lg^{3}(x)} - \frac{2(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln^{3}{10}ln(lg(x))lg^{3}(x)} - \frac{2log_{lg(x)}^{e^{x}}*-3*0}{x^{3}ln^{4}{10}ln(lg(x))lg^{3}(x)} - \frac{2log_{lg(x)}^{e^{x}}*-1}{x^{3}ln^{3}{10}ln^{2}(lg(x))(lg(x))ln{10}(x)lg^{3}(x)} - \frac{2log_{lg(x)}^{e^{x}}*-3}{x^{3}ln^{3}{10}ln(lg(x))lg^{4}(x)ln{10}(x)}\\=&\frac{-2}{x^{3}ln{10}ln^{2}(lg(x))lg(x)} - \frac{14}{x^{3}ln^{2}{10}ln^{3}(lg(x))lg^{2}(x)} - \frac{8}{x^{3}ln^{2}(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{18}{x^{3}ln^{3}{10}ln^{4}(lg(x))lg^{3}(x)} - \frac{14}{x^{3}ln^{3}(lg(x))ln^{3}{10}lg^{3}(x)} - \frac{10}{x^{3}ln^{3}{10}ln^{3}(lg(x))lg^{3}(x)} - \frac{4}{x^{3}ln^{3}{10}ln^{2}(lg(x))lg^{3}(x)} + \frac{6log_{lg(x)}^{e^{x}}}{x^{4}ln{10}ln(lg(x))lg(x)} - \frac{6}{x^{3}ln^{2}(lg(x))ln{10}lg(x)} + \frac{22log_{lg(x)}^{e^{x}}}{x^{4}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{11log_{lg(x)}^{e^{x}}}{x^{4}ln(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{4}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} - \frac{10}{x^{3}ln^{3}(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{36log_{lg(x)}^{e^{x}}}{x^{4}ln^{3}{10}ln^{3}(lg(x))lg^{3}(x)} + \frac{24log_{lg(x)}^{e^{x}}}{x^{4}ln^{2}(lg(x))ln^{3}{10}lg^{3}(x)} + \frac{12log_{lg(x)}^{e^{x}}}{x^{4}ln^{3}{10}ln^{2}(lg(x))lg^{3}(x)} + \frac{12log_{lg(x)}^{e^{x}}}{x^{4}ln^{3}{10}ln(lg(x))lg^{3}(x)} - \frac{4}{x^{3}ln^{2}(lg(x))ln^{3}{10}lg^{3}(x)} - \frac{6}{x^{3}ln^{4}(lg(x))ln^{3}{10}lg^{3}(x)} + \frac{24log_{lg(x)}^{e^{x}}}{x^{4}ln^{4}{10}ln^{4}(lg(x))lg^{4}(x)} + \frac{18log_{lg(x)}^{e^{x}}}{x^{4}ln^{3}(lg(x))ln^{4}{10}lg^{4}(x)} + \frac{18log_{lg(x)}^{e^{x}}}{x^{4}ln^{4}{10}ln^{3}(lg(x))lg^{4}(x)} + \frac{16log_{lg(x)}^{e^{x}}}{x^{4}ln^{4}{10}ln^{2}(lg(x))lg^{4}(x)} + \frac{6log_{lg(x)}^{e^{x}}}{x^{4}ln^{2}(lg(x))ln^{4}{10}lg^{4}(x)} + \frac{6log_{lg(x)}^{e^{x}}}{x^{4}ln(lg(x))ln^{4}{10}lg^{4}(x)}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ log_{lg(x)}^{e^{x}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( log_{lg(x)}^{e^{x}}\right)}{dx}\\=&(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})\\=&\frac{1}{ln(lg(x))} - \frac{log_{lg(x)}^{e^{x}}}{xln{10}ln(lg(x))lg(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{1}{ln(lg(x))} - \frac{log_{lg(x)}^{e^{x}}}{xln{10}ln(lg(x))lg(x)}\right)}{dx}\\=&\frac{-1}{ln^{2}(lg(x))(lg(x))ln{10}(x)} - \frac{-log_{lg(x)}^{e^{x}}}{x^{2}ln{10}ln(lg(x))lg(x)} - \frac{(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{xln{10}ln(lg(x))lg(x)} - \frac{log_{lg(x)}^{e^{x}}*-0}{xln^{2}{10}ln(lg(x))lg(x)} - \frac{log_{lg(x)}^{e^{x}}*-1}{xln{10}ln^{2}(lg(x))(lg(x))ln{10}(x)lg(x)} - \frac{log_{lg(x)}^{e^{x}}*-1}{xln{10}ln(lg(x))lg^{2}(x)ln{10}(x)}\\=&\frac{-1}{xln{10}ln^{2}(lg(x))lg(x)} + \frac{log_{lg(x)}^{e^{x}}}{x^{2}ln{10}ln(lg(x))lg(x)} - \frac{1}{xln^{2}(lg(x))ln{10}lg(x)} + \frac{2log_{lg(x)}^{e^{x}}}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{log_{lg(x)}^{e^{x}}}{x^{2}ln(lg(x))ln^{2}{10}lg^{2}(x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-1}{xln{10}ln^{2}(lg(x))lg(x)} + \frac{log_{lg(x)}^{e^{x}}}{x^{2}ln{10}ln(lg(x))lg(x)} - \frac{1}{xln^{2}(lg(x))ln{10}lg(x)} + \frac{2log_{lg(x)}^{e^{x}}}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{log_{lg(x)}^{e^{x}}}{x^{2}ln(lg(x))ln^{2}{10}lg^{2}(x)}\right)}{dx}\\=&\frac{--1}{x^{2}ln{10}ln^{2}(lg(x))lg(x)} - \frac{-0}{xln^{2}{10}ln^{2}(lg(x))lg(x)} - \frac{-2}{xln{10}ln^{3}(lg(x))(lg(x))ln{10}(x)lg(x)} - \frac{-1}{xln{10}ln^{2}(lg(x))lg^{2}(x)ln{10}(x)} + \frac{-2log_{lg(x)}^{e^{x}}}{x^{3}ln{10}ln(lg(x))lg(x)} + \frac{(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{2}ln{10}ln(lg(x))lg(x)} + \frac{log_{lg(x)}^{e^{x}}*-0}{x^{2}ln^{2}{10}ln(lg(x))lg(x)} + \frac{log_{lg(x)}^{e^{x}}*-1}{x^{2}ln{10}ln^{2}(lg(x))(lg(x))ln{10}(x)lg(x)} + \frac{log_{lg(x)}^{e^{x}}*-1}{x^{2}ln{10}ln(lg(x))lg^{2}(x)ln{10}(x)} - \frac{-1}{x^{2}ln^{2}(lg(x))ln{10}lg(x)} - \frac{-2}{xln^{3}(lg(x))(lg(x))ln{10}(x)ln{10}lg(x)} - \frac{-0}{xln^{2}(lg(x))ln^{2}{10}lg(x)} - \frac{-1}{xln^{2}(lg(x))ln{10}lg^{2}(x)ln{10}(x)} + \frac{2*-2log_{lg(x)}^{e^{x}}}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{2(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{2log_{lg(x)}^{e^{x}}*-2*0}{x^{2}ln^{3}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{2log_{lg(x)}^{e^{x}}*-2}{x^{2}ln^{2}{10}ln^{3}(lg(x))(lg(x))ln{10}(x)lg^{2}(x)} + \frac{2log_{lg(x)}^{e^{x}}*-2}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg^{3}(x)ln{10}(x)} + \frac{-2log_{lg(x)}^{e^{x}}}{x^{3}ln(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{2}ln(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{log_{lg(x)}^{e^{x}}*-1}{x^{2}ln^{2}(lg(x))(lg(x))ln{10}(x)ln^{2}{10}lg^{2}(x)} + \frac{log_{lg(x)}^{e^{x}}*-2*0}{x^{2}ln(lg(x))ln^{3}{10}lg^{2}(x)} + \frac{log_{lg(x)}^{e^{x}}*-2}{x^{2}ln(lg(x))ln^{2}{10}lg^{3}(x)ln{10}(x)}\\=&\frac{1}{x^{2}ln{10}ln^{2}(lg(x))lg(x)} + \frac{4}{x^{2}ln^{2}{10}ln^{3}(lg(x))lg^{2}(x)} + \frac{2}{x^{2}ln^{2}(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{2log_{lg(x)}^{e^{x}}}{x^{3}ln{10}ln(lg(x))lg(x)} + \frac{2}{x^{2}ln^{2}(lg(x))ln{10}lg(x)} - \frac{6log_{lg(x)}^{e^{x}}}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} - \frac{3log_{lg(x)}^{e^{x}}}{x^{3}ln(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{1}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{2}{x^{2}ln^{3}(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{6log_{lg(x)}^{e^{x}}}{x^{3}ln^{3}{10}ln^{3}(lg(x))lg^{3}(x)} - \frac{4log_{lg(x)}^{e^{x}}}{x^{3}ln^{2}(lg(x))ln^{3}{10}lg^{3}(x)} - \frac{2log_{lg(x)}^{e^{x}}}{x^{3}ln^{3}{10}ln^{2}(lg(x))lg^{3}(x)} - \frac{2log_{lg(x)}^{e^{x}}}{x^{3}ln^{3}{10}ln(lg(x))lg^{3}(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{1}{x^{2}ln{10}ln^{2}(lg(x))lg(x)} + \frac{4}{x^{2}ln^{2}{10}ln^{3}(lg(x))lg^{2}(x)} + \frac{2}{x^{2}ln^{2}(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{2log_{lg(x)}^{e^{x}}}{x^{3}ln{10}ln(lg(x))lg(x)} + \frac{2}{x^{2}ln^{2}(lg(x))ln{10}lg(x)} - \frac{6log_{lg(x)}^{e^{x}}}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} - \frac{3log_{lg(x)}^{e^{x}}}{x^{3}ln(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{1}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{2}{x^{2}ln^{3}(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{6log_{lg(x)}^{e^{x}}}{x^{3}ln^{3}{10}ln^{3}(lg(x))lg^{3}(x)} - \frac{4log_{lg(x)}^{e^{x}}}{x^{3}ln^{2}(lg(x))ln^{3}{10}lg^{3}(x)} - \frac{2log_{lg(x)}^{e^{x}}}{x^{3}ln^{3}{10}ln^{2}(lg(x))lg^{3}(x)} - \frac{2log_{lg(x)}^{e^{x}}}{x^{3}ln^{3}{10}ln(lg(x))lg^{3}(x)}\right)}{dx}\\=&\frac{-2}{x^{3}ln{10}ln^{2}(lg(x))lg(x)} + \frac{-0}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg(x)} + \frac{-2}{x^{2}ln{10}ln^{3}(lg(x))(lg(x))ln{10}(x)lg(x)} + \frac{-1}{x^{2}ln{10}ln^{2}(lg(x))lg^{2}(x)ln{10}(x)} + \frac{4*-2}{x^{3}ln^{2}{10}ln^{3}(lg(x))lg^{2}(x)} + \frac{4*-2*0}{x^{2}ln^{3}{10}ln^{3}(lg(x))lg^{2}(x)} + \frac{4*-3}{x^{2}ln^{2}{10}ln^{4}(lg(x))(lg(x))ln{10}(x)lg^{2}(x)} + \frac{4*-2}{x^{2}ln^{2}{10}ln^{3}(lg(x))lg^{3}(x)ln{10}(x)} + \frac{2*-2}{x^{3}ln^{2}(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{2*-2}{x^{2}ln^{3}(lg(x))(lg(x))ln{10}(x)ln^{2}{10}lg^{2}(x)} + \frac{2*-2*0}{x^{2}ln^{2}(lg(x))ln^{3}{10}lg^{2}(x)} + \frac{2*-2}{x^{2}ln^{2}(lg(x))ln^{2}{10}lg^{3}(x)ln{10}(x)} - \frac{2*-3log_{lg(x)}^{e^{x}}}{x^{4}ln{10}ln(lg(x))lg(x)} - \frac{2(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln{10}ln(lg(x))lg(x)} - \frac{2log_{lg(x)}^{e^{x}}*-0}{x^{3}ln^{2}{10}ln(lg(x))lg(x)} - \frac{2log_{lg(x)}^{e^{x}}*-1}{x^{3}ln{10}ln^{2}(lg(x))(lg(x))ln{10}(x)lg(x)} - \frac{2log_{lg(x)}^{e^{x}}*-1}{x^{3}ln{10}ln(lg(x))lg^{2}(x)ln{10}(x)} + \frac{2*-2}{x^{3}ln^{2}(lg(x))ln{10}lg(x)} + \frac{2*-2}{x^{2}ln^{3}(lg(x))(lg(x))ln{10}(x)ln{10}lg(x)} + \frac{2*-0}{x^{2}ln^{2}(lg(x))ln^{2}{10}lg(x)} + \frac{2*-1}{x^{2}ln^{2}(lg(x))ln{10}lg^{2}(x)ln{10}(x)} - \frac{6*-3log_{lg(x)}^{e^{x}}}{x^{4}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} - \frac{6(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} - \frac{6log_{lg(x)}^{e^{x}}*-2*0}{x^{3}ln^{3}{10}ln^{2}(lg(x))lg^{2}(x)} - \frac{6log_{lg(x)}^{e^{x}}*-2}{x^{3}ln^{2}{10}ln^{3}(lg(x))(lg(x))ln{10}(x)lg^{2}(x)} - \frac{6log_{lg(x)}^{e^{x}}*-2}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{3}(x)ln{10}(x)} - \frac{3*-3log_{lg(x)}^{e^{x}}}{x^{4}ln(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{3(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{3log_{lg(x)}^{e^{x}}*-1}{x^{3}ln^{2}(lg(x))(lg(x))ln{10}(x)ln^{2}{10}lg^{2}(x)} - \frac{3log_{lg(x)}^{e^{x}}*-2*0}{x^{3}ln(lg(x))ln^{3}{10}lg^{2}(x)} - \frac{3log_{lg(x)}^{e^{x}}*-2}{x^{3}ln(lg(x))ln^{2}{10}lg^{3}(x)ln{10}(x)} + \frac{-2}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{-2*0}{x^{2}ln^{3}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{-2}{x^{2}ln^{2}{10}ln^{3}(lg(x))(lg(x))ln{10}(x)lg^{2}(x)} + \frac{-2}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg^{3}(x)ln{10}(x)} + \frac{2*-2}{x^{3}ln^{3}(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{2*-3}{x^{2}ln^{4}(lg(x))(lg(x))ln{10}(x)ln^{2}{10}lg^{2}(x)} + \frac{2*-2*0}{x^{2}ln^{3}(lg(x))ln^{3}{10}lg^{2}(x)} + \frac{2*-2}{x^{2}ln^{3}(lg(x))ln^{2}{10}lg^{3}(x)ln{10}(x)} - \frac{6*-3log_{lg(x)}^{e^{x}}}{x^{4}ln^{3}{10}ln^{3}(lg(x))lg^{3}(x)} - \frac{6(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln^{3}{10}ln^{3}(lg(x))lg^{3}(x)} - \frac{6log_{lg(x)}^{e^{x}}*-3*0}{x^{3}ln^{4}{10}ln^{3}(lg(x))lg^{3}(x)} - \frac{6log_{lg(x)}^{e^{x}}*-3}{x^{3}ln^{3}{10}ln^{4}(lg(x))(lg(x))ln{10}(x)lg^{3}(x)} - \frac{6log_{lg(x)}^{e^{x}}*-3}{x^{3}ln^{3}{10}ln^{3}(lg(x))lg^{4}(x)ln{10}(x)} - \frac{4*-3log_{lg(x)}^{e^{x}}}{x^{4}ln^{2}(lg(x))ln^{3}{10}lg^{3}(x)} - \frac{4(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln^{2}(lg(x))ln^{3}{10}lg^{3}(x)} - \frac{4log_{lg(x)}^{e^{x}}*-2}{x^{3}ln^{3}(lg(x))(lg(x))ln{10}(x)ln^{3}{10}lg^{3}(x)} - \frac{4log_{lg(x)}^{e^{x}}*-3*0}{x^{3}ln^{2}(lg(x))ln^{4}{10}lg^{3}(x)} - \frac{4log_{lg(x)}^{e^{x}}*-3}{x^{3}ln^{2}(lg(x))ln^{3}{10}lg^{4}(x)ln{10}(x)} - \frac{2*-3log_{lg(x)}^{e^{x}}}{x^{4}ln^{3}{10}ln^{2}(lg(x))lg^{3}(x)} - \frac{2(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln^{3}{10}ln^{2}(lg(x))lg^{3}(x)} - \frac{2log_{lg(x)}^{e^{x}}*-3*0}{x^{3}ln^{4}{10}ln^{2}(lg(x))lg^{3}(x)} - \frac{2log_{lg(x)}^{e^{x}}*-2}{x^{3}ln^{3}{10}ln^{3}(lg(x))(lg(x))ln{10}(x)lg^{3}(x)} - \frac{2log_{lg(x)}^{e^{x}}*-3}{x^{3}ln^{3}{10}ln^{2}(lg(x))lg^{4}(x)ln{10}(x)} - \frac{2*-3log_{lg(x)}^{e^{x}}}{x^{4}ln^{3}{10}ln(lg(x))lg^{3}(x)} - \frac{2(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln^{3}{10}ln(lg(x))lg^{3}(x)} - \frac{2log_{lg(x)}^{e^{x}}*-3*0}{x^{3}ln^{4}{10}ln(lg(x))lg^{3}(x)} - \frac{2log_{lg(x)}^{e^{x}}*-1}{x^{3}ln^{3}{10}ln^{2}(lg(x))(lg(x))ln{10}(x)lg^{3}(x)} - \frac{2log_{lg(x)}^{e^{x}}*-3}{x^{3}ln^{3}{10}ln(lg(x))lg^{4}(x)ln{10}(x)}\\=&\frac{-2}{x^{3}ln{10}ln^{2}(lg(x))lg(x)} - \frac{14}{x^{3}ln^{2}{10}ln^{3}(lg(x))lg^{2}(x)} - \frac{8}{x^{3}ln^{2}(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{18}{x^{3}ln^{3}{10}ln^{4}(lg(x))lg^{3}(x)} - \frac{14}{x^{3}ln^{3}(lg(x))ln^{3}{10}lg^{3}(x)} - \frac{10}{x^{3}ln^{3}{10}ln^{3}(lg(x))lg^{3}(x)} - \frac{4}{x^{3}ln^{3}{10}ln^{2}(lg(x))lg^{3}(x)} + \frac{6log_{lg(x)}^{e^{x}}}{x^{4}ln{10}ln(lg(x))lg(x)} - \frac{6}{x^{3}ln^{2}(lg(x))ln{10}lg(x)} + \frac{22log_{lg(x)}^{e^{x}}}{x^{4}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{11log_{lg(x)}^{e^{x}}}{x^{4}ln(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{4}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} - \frac{10}{x^{3}ln^{3}(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{36log_{lg(x)}^{e^{x}}}{x^{4}ln^{3}{10}ln^{3}(lg(x))lg^{3}(x)} + \frac{24log_{lg(x)}^{e^{x}}}{x^{4}ln^{2}(lg(x))ln^{3}{10}lg^{3}(x)} + \frac{12log_{lg(x)}^{e^{x}}}{x^{4}ln^{3}{10}ln^{2}(lg(x))lg^{3}(x)} + \frac{12log_{lg(x)}^{e^{x}}}{x^{4}ln^{3}{10}ln(lg(x))lg^{3}(x)} - \frac{4}{x^{3}ln^{2}(lg(x))ln^{3}{10}lg^{3}(x)} - \frac{6}{x^{3}ln^{4}(lg(x))ln^{3}{10}lg^{3}(x)} + \frac{24log_{lg(x)}^{e^{x}}}{x^{4}ln^{4}{10}ln^{4}(lg(x))lg^{4}(x)} + \frac{18log_{lg(x)}^{e^{x}}}{x^{4}ln^{3}(lg(x))ln^{4}{10}lg^{4}(x)} + \frac{18log_{lg(x)}^{e^{x}}}{x^{4}ln^{4}{10}ln^{3}(lg(x))lg^{4}(x)} + \frac{16log_{lg(x)}^{e^{x}}}{x^{4}ln^{4}{10}ln^{2}(lg(x))lg^{4}(x)} + \frac{6log_{lg(x)}^{e^{x}}}{x^{4}ln^{2}(lg(x))ln^{4}{10}lg^{4}(x)} + \frac{6log_{lg(x)}^{e^{x}}}{x^{4}ln(lg(x))ln^{4}{10}lg^{4}(x)}\\ \end{split}\end{equation} \]