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