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