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 2 questions in this calculation: for each question, the 4 derivative of x is calculated.
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/2]Find\ the\ 4th\ derivative\ of\ function\ log_{2}^{log_{2}^{x}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( log_{2}^{log_{2}^{x}}\right)}{dx}\\=&(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{(ln(2))}))}{(log_{2}^{x})} - \frac{(0)log_{2}^{log_{2}^{x}}}{(2)})}{(ln(2))})\\=&\frac{1}{xlog(2, x)ln^{2}(2)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{1}{xlog(2, x)ln^{2}(2)}\right)}{dx}\\=&\frac{-1}{x^{2}log(2, x)ln^{2}(2)} + \frac{(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{{\left(log(2, x)^{2}(ln(2))})}{xln^{2}(2)} + \frac{-2*0}{xlog(2, x)ln^{3}(2)(2)}\\=&\frac{-1}{x^{2}log(2, x)ln^{2}(2)} - \frac{1}{x^{2}{\left(log(2, x)^{2}ln^{3}(2)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-1}{x^{2}log(2, x)ln^{2}(2)} - \frac{1}{x^{2}{\left(log(2, x)^{2}ln^{3}(2)}\right)}{dx}\\=&\frac{--2}{x^{3}log(2, x)ln^{2}(2)} - \frac{(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{{\left(log(2, x)^{2}(ln(2))})}{x^{2}ln^{2}(2)} - \frac{-2*0}{x^{2}log(2, x)ln^{3}(2)(2)} - \frac{-2}{x^{3}{\left(log(2, x)^{2}ln^{3}(2)} - \frac{(\frac{-2(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{{\left(log(2, x)^{3}(ln(2))})}{x^{2}ln^{3}(2)} - \frac{-3*0}{x^{2}{\left(log(2, x)^{2}ln^{4}(2)(2)}\\=&\frac{2}{x^{3}log(2, x)ln^{2}(2)} + \frac{3}{x^{3}{\left(log(2, x)^{2}ln^{3}(2)} + \frac{2}{x^{3}{\left(log(2, x)^{3}ln^{4}(2)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{2}{x^{3}log(2, x)ln^{2}(2)} + \frac{3}{x^{3}{\left(log(2, x)^{2}ln^{3}(2)} + \frac{2}{x^{3}{\left(log(2, x)^{3}ln^{4}(2)}\right)}{dx}\\=&\frac{2*-3}{x^{4}log(2, x)ln^{2}(2)} + \frac{2(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{{\left(log(2, x)^{2}(ln(2))})}{x^{3}ln^{2}(2)} + \frac{2*-2*0}{x^{3}log(2, x)ln^{3}(2)(2)} + \frac{3*-3}{x^{4}{\left(log(2, x)^{2}ln^{3}(2)} + \frac{3(\frac{-2(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{{\left(log(2, x)^{3}(ln(2))})}{x^{3}ln^{3}(2)} + \frac{3*-3*0}{x^{3}{\left(log(2, x)^{2}ln^{4}(2)(2)} + \frac{2*-3}{x^{4}{\left(log(2, x)^{3}ln^{4}(2)} + \frac{2(\frac{-3(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{{\left(log(2, x)^{4}(ln(2))})}{x^{3}ln^{4}(2)} + \frac{2*-4*0}{x^{3}{\left(log(2, x)^{3}ln^{5}(2)(2)}\\=&\frac{-6}{x^{4}log(2, x)ln^{2}(2)} - \frac{11}{x^{4}{\left(log(2, x)^{2}ln^{3}(2)} - \frac{12}{x^{4}{\left(log(2, x)^{3}ln^{4}(2)} - \frac{6}{x^{4}{\left(log(2, x)^{4}ln^{5}(2)}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/2]Find\ the\ 4th\ derivative\ of\ function\ lg(lg(x))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( lg(lg(x))\right)}{dx}\\=&\frac{1}{ln{10}(lg(x))ln{10}(x)}\\=&\frac{1}{xln^{2}{10}lg(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{1}{xln^{2}{10}lg(x)}\right)}{dx}\\=&\frac{-1}{x^{2}ln^{2}{10}lg(x)} + \frac{-2*0}{xln^{3}{10}lg(x)} + \frac{-1}{xln^{2}{10}lg^{2}(x)ln{10}(x)}\\=&\frac{-1}{x^{2}ln^{2}{10}lg(x)} - \frac{1}{x^{2}ln^{3}{10}lg^{2}(x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-1}{x^{2}ln^{2}{10}lg(x)} - \frac{1}{x^{2}ln^{3}{10}lg^{2}(x)}\right)}{dx}\\=&\frac{--2}{x^{3}ln^{2}{10}lg(x)} - \frac{-2*0}{x^{2}ln^{3}{10}lg(x)} - \frac{-1}{x^{2}ln^{2}{10}lg^{2}(x)ln{10}(x)} - \frac{-2}{x^{3}ln^{3}{10}lg^{2}(x)} - \frac{-3*0}{x^{2}ln^{4}{10}lg^{2}(x)} - \frac{-2}{x^{2}ln^{3}{10}lg^{3}(x)ln{10}(x)}\\=&\frac{2}{x^{3}ln^{2}{10}lg(x)} + \frac{3}{x^{3}ln^{3}{10}lg^{2}(x)} + \frac{2}{x^{3}ln^{4}{10}lg^{3}(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{2}{x^{3}ln^{2}{10}lg(x)} + \frac{3}{x^{3}ln^{3}{10}lg^{2}(x)} + \frac{2}{x^{3}ln^{4}{10}lg^{3}(x)}\right)}{dx}\\=&\frac{2*-3}{x^{4}ln^{2}{10}lg(x)} + \frac{2*-2*0}{x^{3}ln^{3}{10}lg(x)} + \frac{2*-1}{x^{3}ln^{2}{10}lg^{2}(x)ln{10}(x)} + \frac{3*-3}{x^{4}ln^{3}{10}lg^{2}(x)} + \frac{3*-3*0}{x^{3}ln^{4}{10}lg^{2}(x)} + \frac{3*-2}{x^{3}ln^{3}{10}lg^{3}(x)ln{10}(x)} + \frac{2*-3}{x^{4}ln^{4}{10}lg^{3}(x)} + \frac{2*-4*0}{x^{3}ln^{5}{10}lg^{3}(x)} + \frac{2*-3}{x^{3}ln^{4}{10}lg^{4}(x)ln{10}(x)}\\=&\frac{-6}{x^{4}ln^{2}{10}lg(x)} - \frac{11}{x^{4}ln^{3}{10}lg^{2}(x)} - \frac{12}{x^{4}ln^{4}{10}lg^{3}(x)} - \frac{6}{x^{4}ln^{5}{10}lg^{4}(x)}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/2]Find\ the\ 4th\ derivative\ of\ function\ log_{2}^{log_{2}^{x}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( log_{2}^{log_{2}^{x}}\right)}{dx}\\=&(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{(ln(2))}))}{(log_{2}^{x})} - \frac{(0)log_{2}^{log_{2}^{x}}}{(2)})}{(ln(2))})\\=&\frac{1}{xlog(2, x)ln^{2}(2)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{1}{xlog(2, x)ln^{2}(2)}\right)}{dx}\\=&\frac{-1}{x^{2}log(2, x)ln^{2}(2)} + \frac{(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{{\left(log(2, x)^{2}(ln(2))})}{xln^{2}(2)} + \frac{-2*0}{xlog(2, x)ln^{3}(2)(2)}\\=&\frac{-1}{x^{2}log(2, x)ln^{2}(2)} - \frac{1}{x^{2}{\left(log(2, x)^{2}ln^{3}(2)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-1}{x^{2}log(2, x)ln^{2}(2)} - \frac{1}{x^{2}{\left(log(2, x)^{2}ln^{3}(2)}\right)}{dx}\\=&\frac{--2}{x^{3}log(2, x)ln^{2}(2)} - \frac{(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{{\left(log(2, x)^{2}(ln(2))})}{x^{2}ln^{2}(2)} - \frac{-2*0}{x^{2}log(2, x)ln^{3}(2)(2)} - \frac{-2}{x^{3}{\left(log(2, x)^{2}ln^{3}(2)} - \frac{(\frac{-2(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{{\left(log(2, x)^{3}(ln(2))})}{x^{2}ln^{3}(2)} - \frac{-3*0}{x^{2}{\left(log(2, x)^{2}ln^{4}(2)(2)}\\=&\frac{2}{x^{3}log(2, x)ln^{2}(2)} + \frac{3}{x^{3}{\left(log(2, x)^{2}ln^{3}(2)} + \frac{2}{x^{3}{\left(log(2, x)^{3}ln^{4}(2)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{2}{x^{3}log(2, x)ln^{2}(2)} + \frac{3}{x^{3}{\left(log(2, x)^{2}ln^{3}(2)} + \frac{2}{x^{3}{\left(log(2, x)^{3}ln^{4}(2)}\right)}{dx}\\=&\frac{2*-3}{x^{4}log(2, x)ln^{2}(2)} + \frac{2(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{{\left(log(2, x)^{2}(ln(2))})}{x^{3}ln^{2}(2)} + \frac{2*-2*0}{x^{3}log(2, x)ln^{3}(2)(2)} + \frac{3*-3}{x^{4}{\left(log(2, x)^{2}ln^{3}(2)} + \frac{3(\frac{-2(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{{\left(log(2, x)^{3}(ln(2))})}{x^{3}ln^{3}(2)} + \frac{3*-3*0}{x^{3}{\left(log(2, x)^{2}ln^{4}(2)(2)} + \frac{2*-3}{x^{4}{\left(log(2, x)^{3}ln^{4}(2)} + \frac{2(\frac{-3(\frac{(1)}{(x)} - \frac{(0)log_{2}^{x}}{(2)})}{{\left(log(2, x)^{4}(ln(2))})}{x^{3}ln^{4}(2)} + \frac{2*-4*0}{x^{3}{\left(log(2, x)^{3}ln^{5}(2)(2)}\\=&\frac{-6}{x^{4}log(2, x)ln^{2}(2)} - \frac{11}{x^{4}{\left(log(2, x)^{2}ln^{3}(2)} - \frac{12}{x^{4}{\left(log(2, x)^{3}ln^{4}(2)} - \frac{6}{x^{4}{\left(log(2, x)^{4}ln^{5}(2)}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/2]Find\ the\ 4th\ derivative\ of\ function\ lg(lg(x))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( lg(lg(x))\right)}{dx}\\=&\frac{1}{ln{10}(lg(x))ln{10}(x)}\\=&\frac{1}{xln^{2}{10}lg(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{1}{xln^{2}{10}lg(x)}\right)}{dx}\\=&\frac{-1}{x^{2}ln^{2}{10}lg(x)} + \frac{-2*0}{xln^{3}{10}lg(x)} + \frac{-1}{xln^{2}{10}lg^{2}(x)ln{10}(x)}\\=&\frac{-1}{x^{2}ln^{2}{10}lg(x)} - \frac{1}{x^{2}ln^{3}{10}lg^{2}(x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-1}{x^{2}ln^{2}{10}lg(x)} - \frac{1}{x^{2}ln^{3}{10}lg^{2}(x)}\right)}{dx}\\=&\frac{--2}{x^{3}ln^{2}{10}lg(x)} - \frac{-2*0}{x^{2}ln^{3}{10}lg(x)} - \frac{-1}{x^{2}ln^{2}{10}lg^{2}(x)ln{10}(x)} - \frac{-2}{x^{3}ln^{3}{10}lg^{2}(x)} - \frac{-3*0}{x^{2}ln^{4}{10}lg^{2}(x)} - \frac{-2}{x^{2}ln^{3}{10}lg^{3}(x)ln{10}(x)}\\=&\frac{2}{x^{3}ln^{2}{10}lg(x)} + \frac{3}{x^{3}ln^{3}{10}lg^{2}(x)} + \frac{2}{x^{3}ln^{4}{10}lg^{3}(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{2}{x^{3}ln^{2}{10}lg(x)} + \frac{3}{x^{3}ln^{3}{10}lg^{2}(x)} + \frac{2}{x^{3}ln^{4}{10}lg^{3}(x)}\right)}{dx}\\=&\frac{2*-3}{x^{4}ln^{2}{10}lg(x)} + \frac{2*-2*0}{x^{3}ln^{3}{10}lg(x)} + \frac{2*-1}{x^{3}ln^{2}{10}lg^{2}(x)ln{10}(x)} + \frac{3*-3}{x^{4}ln^{3}{10}lg^{2}(x)} + \frac{3*-3*0}{x^{3}ln^{4}{10}lg^{2}(x)} + \frac{3*-2}{x^{3}ln^{3}{10}lg^{3}(x)ln{10}(x)} + \frac{2*-3}{x^{4}ln^{4}{10}lg^{3}(x)} + \frac{2*-4*0}{x^{3}ln^{5}{10}lg^{3}(x)} + \frac{2*-3}{x^{3}ln^{4}{10}lg^{4}(x)ln{10}(x)}\\=&\frac{-6}{x^{4}ln^{2}{10}lg(x)} - \frac{11}{x^{4}ln^{3}{10}lg^{2}(x)} - \frac{12}{x^{4}ln^{4}{10}lg^{3}(x)} - \frac{6}{x^{4}ln^{5}{10}lg^{4}(x)}\\ \end{split}\end{equation} \]