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} \]
\[ \begin{equation}\begin{split}[2/2]Find\ the\ 4th\ derivative\ of\ function\ e^{xln(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( e^{xln(x)}\right)}{dx}\\=&e^{xln(x)}(ln(x) + \frac{x}{(x)})\\=&e^{xln(x)}ln(x) + e^{xln(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( e^{xln(x)}ln(x) + e^{xln(x)}\right)}{dx}\\=&e^{xln(x)}(ln(x) + \frac{x}{(x)})ln(x) + \frac{e^{xln(x)}}{(x)} + e^{xln(x)}(ln(x) + \frac{x}{(x)})\\=&e^{xln(x)}ln^{2}(x) + 2e^{xln(x)}ln(x) + \frac{e^{xln(x)}}{x} + e^{xln(x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( e^{xln(x)}ln^{2}(x) + 2e^{xln(x)}ln(x) + \frac{e^{xln(x)}}{x} + e^{xln(x)}\right)}{dx}\\=&e^{xln(x)}(ln(x) + \frac{x}{(x)})ln^{2}(x) + \frac{e^{xln(x)}*2ln(x)}{(x)} + 2e^{xln(x)}(ln(x) + \frac{x}{(x)})ln(x) + \frac{2e^{xln(x)}}{(x)} + \frac{-e^{xln(x)}}{x^{2}} + \frac{e^{xln(x)}(ln(x) + \frac{x}{(x)})}{x} + e^{xln(x)}(ln(x) + \frac{x}{(x)})\\=&e^{xln(x)}ln^{3}(x) + 3e^{xln(x)}ln^{2}(x) + \frac{3e^{xln(x)}ln(x)}{x} + 3e^{xln(x)}ln(x) + \frac{3e^{xln(x)}}{x} - \frac{e^{xln(x)}}{x^{2}} + e^{xln(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( e^{xln(x)}ln^{3}(x) + 3e^{xln(x)}ln^{2}(x) + \frac{3e^{xln(x)}ln(x)}{x} + 3e^{xln(x)}ln(x) + \frac{3e^{xln(x)}}{x} - \frac{e^{xln(x)}}{x^{2}} + e^{xln(x)}\right)}{dx}\\=&e^{xln(x)}(ln(x) + \frac{x}{(x)})ln^{3}(x) + \frac{e^{xln(x)}*3ln^{2}(x)}{(x)} + 3e^{xln(x)}(ln(x) + \frac{x}{(x)})ln^{2}(x) + \frac{3e^{xln(x)}*2ln(x)}{(x)} + \frac{3*-e^{xln(x)}ln(x)}{x^{2}} + \frac{3e^{xln(x)}(ln(x) + \frac{x}{(x)})ln(x)}{x} + \frac{3e^{xln(x)}}{x(x)} + 3e^{xln(x)}(ln(x) + \frac{x}{(x)})ln(x) + \frac{3e^{xln(x)}}{(x)} + \frac{3*-e^{xln(x)}}{x^{2}} + \frac{3e^{xln(x)}(ln(x) + \frac{x}{(x)})}{x} - \frac{-2e^{xln(x)}}{x^{3}} - \frac{e^{xln(x)}(ln(x) + \frac{x}{(x)})}{x^{2}} + e^{xln(x)}(ln(x) + \frac{x}{(x)})\\=&e^{xln(x)}ln^{4}(x) + 4e^{xln(x)}ln^{3}(x) + \frac{6e^{xln(x)}ln^{2}(x)}{x} + 6e^{xln(x)}ln^{2}(x) + \frac{12e^{xln(x)}ln(x)}{x} - \frac{4e^{xln(x)}ln(x)}{x^{2}} - \frac{e^{xln(x)}}{x^{2}} + 4e^{xln(x)}ln(x) + \frac{6e^{xln(x)}}{x} + \frac{2e^{xln(x)}}{x^{3}} + e^{xln(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} \]
\[ \begin{equation}\begin{split}[2/2]Find\ the\ 4th\ derivative\ of\ function\ e^{xln(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( e^{xln(x)}\right)}{dx}\\=&e^{xln(x)}(ln(x) + \frac{x}{(x)})\\=&e^{xln(x)}ln(x) + e^{xln(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( e^{xln(x)}ln(x) + e^{xln(x)}\right)}{dx}\\=&e^{xln(x)}(ln(x) + \frac{x}{(x)})ln(x) + \frac{e^{xln(x)}}{(x)} + e^{xln(x)}(ln(x) + \frac{x}{(x)})\\=&e^{xln(x)}ln^{2}(x) + 2e^{xln(x)}ln(x) + \frac{e^{xln(x)}}{x} + e^{xln(x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( e^{xln(x)}ln^{2}(x) + 2e^{xln(x)}ln(x) + \frac{e^{xln(x)}}{x} + e^{xln(x)}\right)}{dx}\\=&e^{xln(x)}(ln(x) + \frac{x}{(x)})ln^{2}(x) + \frac{e^{xln(x)}*2ln(x)}{(x)} + 2e^{xln(x)}(ln(x) + \frac{x}{(x)})ln(x) + \frac{2e^{xln(x)}}{(x)} + \frac{-e^{xln(x)}}{x^{2}} + \frac{e^{xln(x)}(ln(x) + \frac{x}{(x)})}{x} + e^{xln(x)}(ln(x) + \frac{x}{(x)})\\=&e^{xln(x)}ln^{3}(x) + 3e^{xln(x)}ln^{2}(x) + \frac{3e^{xln(x)}ln(x)}{x} + 3e^{xln(x)}ln(x) + \frac{3e^{xln(x)}}{x} - \frac{e^{xln(x)}}{x^{2}} + e^{xln(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( e^{xln(x)}ln^{3}(x) + 3e^{xln(x)}ln^{2}(x) + \frac{3e^{xln(x)}ln(x)}{x} + 3e^{xln(x)}ln(x) + \frac{3e^{xln(x)}}{x} - \frac{e^{xln(x)}}{x^{2}} + e^{xln(x)}\right)}{dx}\\=&e^{xln(x)}(ln(x) + \frac{x}{(x)})ln^{3}(x) + \frac{e^{xln(x)}*3ln^{2}(x)}{(x)} + 3e^{xln(x)}(ln(x) + \frac{x}{(x)})ln^{2}(x) + \frac{3e^{xln(x)}*2ln(x)}{(x)} + \frac{3*-e^{xln(x)}ln(x)}{x^{2}} + \frac{3e^{xln(x)}(ln(x) + \frac{x}{(x)})ln(x)}{x} + \frac{3e^{xln(x)}}{x(x)} + 3e^{xln(x)}(ln(x) + \frac{x}{(x)})ln(x) + \frac{3e^{xln(x)}}{(x)} + \frac{3*-e^{xln(x)}}{x^{2}} + \frac{3e^{xln(x)}(ln(x) + \frac{x}{(x)})}{x} - \frac{-2e^{xln(x)}}{x^{3}} - \frac{e^{xln(x)}(ln(x) + \frac{x}{(x)})}{x^{2}} + e^{xln(x)}(ln(x) + \frac{x}{(x)})\\=&e^{xln(x)}ln^{4}(x) + 4e^{xln(x)}ln^{3}(x) + \frac{6e^{xln(x)}ln^{2}(x)}{x} + 6e^{xln(x)}ln^{2}(x) + \frac{12e^{xln(x)}ln(x)}{x} - \frac{4e^{xln(x)}ln(x)}{x^{2}} - \frac{e^{xln(x)}}{x^{2}} + 4e^{xln(x)}ln(x) + \frac{6e^{xln(x)}}{x} + \frac{2e^{xln(x)}}{x^{3}} + e^{xln(x)}\\ \end{split}\end{equation} \]