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.
\[ \begin{equation}\begin{split}[1/2]Find\ the\ 4th\ derivative\ of\ function\ log_{x}^{10}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( log_{x}^{10}\right)}{dx}\\=&(\frac{(\frac{(0)}{(10)} - \frac{(1)log_{x}^{10}}{(x)})}{(ln(x))})\\=&\frac{-log_{x}^{10}}{xln(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-log_{x}^{10}}{xln(x)}\right)}{dx}\\=&\frac{--log_{x}^{10}}{x^{2}ln(x)} - \frac{(\frac{(\frac{(0)}{(10)} - \frac{(1)log_{x}^{10}}{(x)})}{(ln(x))})}{xln(x)} - \frac{log_{x}^{10}*-1}{xln^{2}(x)(x)}\\=&\frac{log_{x}^{10}}{x^{2}ln(x)} + \frac{2log_{x}^{10}}{x^{2}ln^{2}(x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{log_{x}^{10}}{x^{2}ln(x)} + \frac{2log_{x}^{10}}{x^{2}ln^{2}(x)}\right)}{dx}\\=&\frac{-2log_{x}^{10}}{x^{3}ln(x)} + \frac{(\frac{(\frac{(0)}{(10)} - \frac{(1)log_{x}^{10}}{(x)})}{(ln(x))})}{x^{2}ln(x)} + \frac{log_{x}^{10}*-1}{x^{2}ln^{2}(x)(x)} + \frac{2*-2log_{x}^{10}}{x^{3}ln^{2}(x)} + \frac{2(\frac{(\frac{(0)}{(10)} - \frac{(1)log_{x}^{10}}{(x)})}{(ln(x))})}{x^{2}ln^{2}(x)} + \frac{2log_{x}^{10}*-2}{x^{2}ln^{3}(x)(x)}\\=&\frac{-2log_{x}^{10}}{x^{3}ln(x)} - \frac{6log_{x}^{10}}{x^{3}ln^{2}(x)} - \frac{6log_{x}^{10}}{x^{3}ln^{3}(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2log_{x}^{10}}{x^{3}ln(x)} - \frac{6log_{x}^{10}}{x^{3}ln^{2}(x)} - \frac{6log_{x}^{10}}{x^{3}ln^{3}(x)}\right)}{dx}\\=&\frac{-2*-3log_{x}^{10}}{x^{4}ln(x)} - \frac{2(\frac{(\frac{(0)}{(10)} - \frac{(1)log_{x}^{10}}{(x)})}{(ln(x))})}{x^{3}ln(x)} - \frac{2log_{x}^{10}*-1}{x^{3}ln^{2}(x)(x)} - \frac{6*-3log_{x}^{10}}{x^{4}ln^{2}(x)} - \frac{6(\frac{(\frac{(0)}{(10)} - \frac{(1)log_{x}^{10}}{(x)})}{(ln(x))})}{x^{3}ln^{2}(x)} - \frac{6log_{x}^{10}*-2}{x^{3}ln^{3}(x)(x)} - \frac{6*-3log_{x}^{10}}{x^{4}ln^{3}(x)} - \frac{6(\frac{(\frac{(0)}{(10)} - \frac{(1)log_{x}^{10}}{(x)})}{(ln(x))})}{x^{3}ln^{3}(x)} - \frac{6log_{x}^{10}*-3}{x^{3}ln^{4}(x)(x)}\\=&\frac{6log_{x}^{10}}{x^{4}ln(x)} + \frac{22log_{x}^{10}}{x^{4}ln^{2}(x)} + \frac{36log_{x}^{10}}{x^{4}ln^{3}(x)} + \frac{24log_{x}^{10}}{x^{4}ln^{4}(x)}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/2]Find\ the\ 4th\ derivative\ of\ function\ \frac{1}{lg(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{1}{lg(x)}\right)}{dx}\\=&\frac{-1}{lg^{2}(x)ln{10}(x)}\\=&\frac{-1}{xln{10}lg^{2}(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-1}{xln{10}lg^{2}(x)}\right)}{dx}\\=&\frac{--1}{x^{2}ln{10}lg^{2}(x)} - \frac{-0}{xln^{2}{10}lg^{2}(x)} - \frac{-2}{xln{10}lg^{3}(x)ln{10}(x)}\\=&\frac{1}{x^{2}ln{10}lg^{2}(x)} + \frac{2}{x^{2}ln^{2}{10}lg^{3}(x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{1}{x^{2}ln{10}lg^{2}(x)} + \frac{2}{x^{2}ln^{2}{10}lg^{3}(x)}\right)}{dx}\\=&\frac{-2}{x^{3}ln{10}lg^{2}(x)} + \frac{-0}{x^{2}ln^{2}{10}lg^{2}(x)} + \frac{-2}{x^{2}ln{10}lg^{3}(x)ln{10}(x)} + \frac{2*-2}{x^{3}ln^{2}{10}lg^{3}(x)} + \frac{2*-2*0}{x^{2}ln^{3}{10}lg^{3}(x)} + \frac{2*-3}{x^{2}ln^{2}{10}lg^{4}(x)ln{10}(x)}\\=&\frac{-2}{x^{3}ln{10}lg^{2}(x)} - \frac{6}{x^{3}ln^{2}{10}lg^{3}(x)} - \frac{6}{x^{3}ln^{3}{10}lg^{4}(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2}{x^{3}ln{10}lg^{2}(x)} - \frac{6}{x^{3}ln^{2}{10}lg^{3}(x)} - \frac{6}{x^{3}ln^{3}{10}lg^{4}(x)}\right)}{dx}\\=&\frac{-2*-3}{x^{4}ln{10}lg^{2}(x)} - \frac{2*-0}{x^{3}ln^{2}{10}lg^{2}(x)} - \frac{2*-2}{x^{3}ln{10}lg^{3}(x)ln{10}(x)} - \frac{6*-3}{x^{4}ln^{2}{10}lg^{3}(x)} - \frac{6*-2*0}{x^{3}ln^{3}{10}lg^{3}(x)} - \frac{6*-3}{x^{3}ln^{2}{10}lg^{4}(x)ln{10}(x)} - \frac{6*-3}{x^{4}ln^{3}{10}lg^{4}(x)} - \frac{6*-3*0}{x^{3}ln^{4}{10}lg^{4}(x)} - \frac{6*-4}{x^{3}ln^{3}{10}lg^{5}(x)ln{10}(x)}\\=&\frac{6}{x^{4}ln{10}lg^{2}(x)} + \frac{22}{x^{4}ln^{2}{10}lg^{3}(x)} + \frac{36}{x^{4}ln^{3}{10}lg^{4}(x)} + \frac{24}{x^{4}ln^{4}{10}lg^{5}(x)}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/2]Find\ the\ 4th\ derivative\ of\ function\ log_{x}^{10}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( log_{x}^{10}\right)}{dx}\\=&(\frac{(\frac{(0)}{(10)} - \frac{(1)log_{x}^{10}}{(x)})}{(ln(x))})\\=&\frac{-log_{x}^{10}}{xln(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-log_{x}^{10}}{xln(x)}\right)}{dx}\\=&\frac{--log_{x}^{10}}{x^{2}ln(x)} - \frac{(\frac{(\frac{(0)}{(10)} - \frac{(1)log_{x}^{10}}{(x)})}{(ln(x))})}{xln(x)} - \frac{log_{x}^{10}*-1}{xln^{2}(x)(x)}\\=&\frac{log_{x}^{10}}{x^{2}ln(x)} + \frac{2log_{x}^{10}}{x^{2}ln^{2}(x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{log_{x}^{10}}{x^{2}ln(x)} + \frac{2log_{x}^{10}}{x^{2}ln^{2}(x)}\right)}{dx}\\=&\frac{-2log_{x}^{10}}{x^{3}ln(x)} + \frac{(\frac{(\frac{(0)}{(10)} - \frac{(1)log_{x}^{10}}{(x)})}{(ln(x))})}{x^{2}ln(x)} + \frac{log_{x}^{10}*-1}{x^{2}ln^{2}(x)(x)} + \frac{2*-2log_{x}^{10}}{x^{3}ln^{2}(x)} + \frac{2(\frac{(\frac{(0)}{(10)} - \frac{(1)log_{x}^{10}}{(x)})}{(ln(x))})}{x^{2}ln^{2}(x)} + \frac{2log_{x}^{10}*-2}{x^{2}ln^{3}(x)(x)}\\=&\frac{-2log_{x}^{10}}{x^{3}ln(x)} - \frac{6log_{x}^{10}}{x^{3}ln^{2}(x)} - \frac{6log_{x}^{10}}{x^{3}ln^{3}(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2log_{x}^{10}}{x^{3}ln(x)} - \frac{6log_{x}^{10}}{x^{3}ln^{2}(x)} - \frac{6log_{x}^{10}}{x^{3}ln^{3}(x)}\right)}{dx}\\=&\frac{-2*-3log_{x}^{10}}{x^{4}ln(x)} - \frac{2(\frac{(\frac{(0)}{(10)} - \frac{(1)log_{x}^{10}}{(x)})}{(ln(x))})}{x^{3}ln(x)} - \frac{2log_{x}^{10}*-1}{x^{3}ln^{2}(x)(x)} - \frac{6*-3log_{x}^{10}}{x^{4}ln^{2}(x)} - \frac{6(\frac{(\frac{(0)}{(10)} - \frac{(1)log_{x}^{10}}{(x)})}{(ln(x))})}{x^{3}ln^{2}(x)} - \frac{6log_{x}^{10}*-2}{x^{3}ln^{3}(x)(x)} - \frac{6*-3log_{x}^{10}}{x^{4}ln^{3}(x)} - \frac{6(\frac{(\frac{(0)}{(10)} - \frac{(1)log_{x}^{10}}{(x)})}{(ln(x))})}{x^{3}ln^{3}(x)} - \frac{6log_{x}^{10}*-3}{x^{3}ln^{4}(x)(x)}\\=&\frac{6log_{x}^{10}}{x^{4}ln(x)} + \frac{22log_{x}^{10}}{x^{4}ln^{2}(x)} + \frac{36log_{x}^{10}}{x^{4}ln^{3}(x)} + \frac{24log_{x}^{10}}{x^{4}ln^{4}(x)}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/2]Find\ the\ 4th\ derivative\ of\ function\ \frac{1}{lg(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{1}{lg(x)}\right)}{dx}\\=&\frac{-1}{lg^{2}(x)ln{10}(x)}\\=&\frac{-1}{xln{10}lg^{2}(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-1}{xln{10}lg^{2}(x)}\right)}{dx}\\=&\frac{--1}{x^{2}ln{10}lg^{2}(x)} - \frac{-0}{xln^{2}{10}lg^{2}(x)} - \frac{-2}{xln{10}lg^{3}(x)ln{10}(x)}\\=&\frac{1}{x^{2}ln{10}lg^{2}(x)} + \frac{2}{x^{2}ln^{2}{10}lg^{3}(x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{1}{x^{2}ln{10}lg^{2}(x)} + \frac{2}{x^{2}ln^{2}{10}lg^{3}(x)}\right)}{dx}\\=&\frac{-2}{x^{3}ln{10}lg^{2}(x)} + \frac{-0}{x^{2}ln^{2}{10}lg^{2}(x)} + \frac{-2}{x^{2}ln{10}lg^{3}(x)ln{10}(x)} + \frac{2*-2}{x^{3}ln^{2}{10}lg^{3}(x)} + \frac{2*-2*0}{x^{2}ln^{3}{10}lg^{3}(x)} + \frac{2*-3}{x^{2}ln^{2}{10}lg^{4}(x)ln{10}(x)}\\=&\frac{-2}{x^{3}ln{10}lg^{2}(x)} - \frac{6}{x^{3}ln^{2}{10}lg^{3}(x)} - \frac{6}{x^{3}ln^{3}{10}lg^{4}(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2}{x^{3}ln{10}lg^{2}(x)} - \frac{6}{x^{3}ln^{2}{10}lg^{3}(x)} - \frac{6}{x^{3}ln^{3}{10}lg^{4}(x)}\right)}{dx}\\=&\frac{-2*-3}{x^{4}ln{10}lg^{2}(x)} - \frac{2*-0}{x^{3}ln^{2}{10}lg^{2}(x)} - \frac{2*-2}{x^{3}ln{10}lg^{3}(x)ln{10}(x)} - \frac{6*-3}{x^{4}ln^{2}{10}lg^{3}(x)} - \frac{6*-2*0}{x^{3}ln^{3}{10}lg^{3}(x)} - \frac{6*-3}{x^{3}ln^{2}{10}lg^{4}(x)ln{10}(x)} - \frac{6*-3}{x^{4}ln^{3}{10}lg^{4}(x)} - \frac{6*-3*0}{x^{3}ln^{4}{10}lg^{4}(x)} - \frac{6*-4}{x^{3}ln^{3}{10}lg^{5}(x)ln{10}(x)}\\=&\frac{6}{x^{4}ln{10}lg^{2}(x)} + \frac{22}{x^{4}ln^{2}{10}lg^{3}(x)} + \frac{36}{x^{4}ln^{3}{10}lg^{4}(x)} + \frac{24}{x^{4}ln^{4}{10}lg^{5}(x)}\\ \end{split}\end{equation} \]