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.
\[ \begin{equation}\begin{split}[1/4]Find\ the\ 4th\ derivative\ of\ function\ {x}^{x}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \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} \]
\[ \begin{equation}\begin{split}[2/4]Find\ the\ 4th\ derivative\ of\ function\ {x}^{(2x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( {x}^{(2x)}\right)}{dx}\\=&({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))\\=&2{x}^{(2x)}ln(x) + 2{x}^{(2x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( 2{x}^{(2x)}ln(x) + 2{x}^{(2x)}\right)}{dx}\\=&2({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))ln(x) + \frac{2{x}^{(2x)}}{(x)} + 2({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))\\=&4{x}^{(2x)}ln^{2}(x) + 8{x}^{(2x)}ln(x) + \frac{2{x}^{(2x)}}{x} + 4{x}^{(2x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 4{x}^{(2x)}ln^{2}(x) + 8{x}^{(2x)}ln(x) + \frac{2{x}^{(2x)}}{x} + 4{x}^{(2x)}\right)}{dx}\\=&4({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))ln^{2}(x) + \frac{4{x}^{(2x)}*2ln(x)}{(x)} + 8({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))ln(x) + \frac{8{x}^{(2x)}}{(x)} + \frac{2*-{x}^{(2x)}}{x^{2}} + \frac{2({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))}{x} + 4({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))\\=&8{x}^{(2x)}ln^{3}(x) + 24{x}^{(2x)}ln^{2}(x) + \frac{12{x}^{(2x)}ln(x)}{x} + 24{x}^{(2x)}ln(x) + \frac{12{x}^{(2x)}}{x} - \frac{2{x}^{(2x)}}{x^{2}} + 8{x}^{(2x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 8{x}^{(2x)}ln^{3}(x) + 24{x}^{(2x)}ln^{2}(x) + \frac{12{x}^{(2x)}ln(x)}{x} + 24{x}^{(2x)}ln(x) + \frac{12{x}^{(2x)}}{x} - \frac{2{x}^{(2x)}}{x^{2}} + 8{x}^{(2x)}\right)}{dx}\\=&8({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))ln^{3}(x) + \frac{8{x}^{(2x)}*3ln^{2}(x)}{(x)} + 24({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))ln^{2}(x) + \frac{24{x}^{(2x)}*2ln(x)}{(x)} + \frac{12*-{x}^{(2x)}ln(x)}{x^{2}} + \frac{12({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))ln(x)}{x} + \frac{12{x}^{(2x)}}{x(x)} + 24({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))ln(x) + \frac{24{x}^{(2x)}}{(x)} + \frac{12*-{x}^{(2x)}}{x^{2}} + \frac{12({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))}{x} - \frac{2*-2{x}^{(2x)}}{x^{3}} - \frac{2({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))}{x^{2}} + 8({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))\\=&16{x}^{(2x)}ln^{4}(x) + 64{x}^{(2x)}ln^{3}(x) + \frac{48{x}^{(2x)}ln^{2}(x)}{x} + 96{x}^{(2x)}ln^{2}(x) + \frac{96{x}^{(2x)}ln(x)}{x} - \frac{16{x}^{(2x)}ln(x)}{x^{2}} + 64{x}^{(2x)}ln(x) + \frac{48{x}^{(2x)}}{x} + \frac{4{x}^{(2x)}}{x^{3}} - \frac{4{x}^{(2x)}}{x^{2}} + 16{x}^{(2x)}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[3/4]Find\ the\ 4th\ derivative\ of\ function\ {(2x)}^{x}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = (2x)^{x}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( (2x)^{x}\right)}{dx}\\=&((2x)^{x}((1)ln(2x) + \frac{(x)(2)}{(2x)}))\\=&(2x)^{x}ln(2x) + (2x)^{x}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( (2x)^{x}ln(2x) + (2x)^{x}\right)}{dx}\\=&((2x)^{x}((1)ln(2x) + \frac{(x)(2)}{(2x)}))ln(2x) + \frac{(2x)^{x}*2}{(2x)} + ((2x)^{x}((1)ln(2x) + \frac{(x)(2)}{(2x)}))\\=&(2x)^{x}ln^{2}(2x) + 2(2x)^{x}ln(2x) + \frac{(2x)^{x}}{x} + (2x)^{x}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( (2x)^{x}ln^{2}(2x) + 2(2x)^{x}ln(2x) + \frac{(2x)^{x}}{x} + (2x)^{x}\right)}{dx}\\=&((2x)^{x}((1)ln(2x) + \frac{(x)(2)}{(2x)}))ln^{2}(2x) + \frac{(2x)^{x}*2ln(2x)*2}{(2x)} + 2((2x)^{x}((1)ln(2x) + \frac{(x)(2)}{(2x)}))ln(2x) + \frac{2(2x)^{x}*2}{(2x)} + \frac{-(2x)^{x}}{x^{2}} + \frac{((2x)^{x}((1)ln(2x) + \frac{(x)(2)}{(2x)}))}{x} + ((2x)^{x}((1)ln(2x) + \frac{(x)(2)}{(2x)}))\\=&(2x)^{x}ln^{3}(2x) + 3(2x)^{x}ln^{2}(2x) + \frac{3(2x)^{x}ln(2x)}{x} + 3(2x)^{x}ln(2x) + \frac{3(2x)^{x}}{x} - \frac{(2x)^{x}}{x^{2}} + (2x)^{x}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( (2x)^{x}ln^{3}(2x) + 3(2x)^{x}ln^{2}(2x) + \frac{3(2x)^{x}ln(2x)}{x} + 3(2x)^{x}ln(2x) + \frac{3(2x)^{x}}{x} - \frac{(2x)^{x}}{x^{2}} + (2x)^{x}\right)}{dx}\\=&((2x)^{x}((1)ln(2x) + \frac{(x)(2)}{(2x)}))ln^{3}(2x) + \frac{(2x)^{x}*3ln^{2}(2x)*2}{(2x)} + 3((2x)^{x}((1)ln(2x) + \frac{(x)(2)}{(2x)}))ln^{2}(2x) + \frac{3(2x)^{x}*2ln(2x)*2}{(2x)} + \frac{3*-(2x)^{x}ln(2x)}{x^{2}} + \frac{3((2x)^{x}((1)ln(2x) + \frac{(x)(2)}{(2x)}))ln(2x)}{x} + \frac{3(2x)^{x}*2}{x(2x)} + 3((2x)^{x}((1)ln(2x) + \frac{(x)(2)}{(2x)}))ln(2x) + \frac{3(2x)^{x}*2}{(2x)} + \frac{3*-(2x)^{x}}{x^{2}} + \frac{3((2x)^{x}((1)ln(2x) + \frac{(x)(2)}{(2x)}))}{x} - \frac{-2(2x)^{x}}{x^{3}} - \frac{((2x)^{x}((1)ln(2x) + \frac{(x)(2)}{(2x)}))}{x^{2}} + ((2x)^{x}((1)ln(2x) + \frac{(x)(2)}{(2x)}))\\=&(2x)^{x}ln^{4}(2x) + 4(2x)^{x}ln^{3}(2x) + \frac{6(2x)^{x}ln^{2}(2x)}{x} + 6(2x)^{x}ln^{2}(2x) + \frac{12(2x)^{x}ln(2x)}{x} - \frac{4(2x)^{x}ln(2x)}{x^{2}} + 4(2x)^{x}ln(2x) + \frac{6(2x)^{x}}{x} + \frac{2(2x)^{x}}{x^{3}} - \frac{(2x)^{x}}{x^{2}} + (2x)^{x}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[4/4]Find\ the\ 4th\ derivative\ of\ function\ {(2x)}^{(2x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = (2x)^{(2x)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( (2x)^{(2x)}\right)}{dx}\\=&((2x)^{(2x)}((2)ln(2x) + \frac{(2x)(2)}{(2x)}))\\=&2(2x)^{(2x)}ln(2x) + 2(2x)^{(2x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( 2(2x)^{(2x)}ln(2x) + 2(2x)^{(2x)}\right)}{dx}\\=&2((2x)^{(2x)}((2)ln(2x) + \frac{(2x)(2)}{(2x)}))ln(2x) + \frac{2(2x)^{(2x)}*2}{(2x)} + 2((2x)^{(2x)}((2)ln(2x) + \frac{(2x)(2)}{(2x)}))\\=&4(2x)^{(2x)}ln^{2}(2x) + 8(2x)^{(2x)}ln(2x) + \frac{2(2x)^{(2x)}}{x} + 4(2x)^{(2x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 4(2x)^{(2x)}ln^{2}(2x) + 8(2x)^{(2x)}ln(2x) + \frac{2(2x)^{(2x)}}{x} + 4(2x)^{(2x)}\right)}{dx}\\=&4((2x)^{(2x)}((2)ln(2x) + \frac{(2x)(2)}{(2x)}))ln^{2}(2x) + \frac{4(2x)^{(2x)}*2ln(2x)*2}{(2x)} + 8((2x)^{(2x)}((2)ln(2x) + \frac{(2x)(2)}{(2x)}))ln(2x) + \frac{8(2x)^{(2x)}*2}{(2x)} + \frac{2*-(2x)^{(2x)}}{x^{2}} + \frac{2((2x)^{(2x)}((2)ln(2x) + \frac{(2x)(2)}{(2x)}))}{x} + 4((2x)^{(2x)}((2)ln(2x) + \frac{(2x)(2)}{(2x)}))\\=&8(2x)^{(2x)}ln^{3}(2x) + 24(2x)^{(2x)}ln^{2}(2x) + \frac{12(2x)^{(2x)}ln(2x)}{x} + 24(2x)^{(2x)}ln(2x) + \frac{12(2x)^{(2x)}}{x} - \frac{2(2x)^{(2x)}}{x^{2}} + 8(2x)^{(2x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 8(2x)^{(2x)}ln^{3}(2x) + 24(2x)^{(2x)}ln^{2}(2x) + \frac{12(2x)^{(2x)}ln(2x)}{x} + 24(2x)^{(2x)}ln(2x) + \frac{12(2x)^{(2x)}}{x} - \frac{2(2x)^{(2x)}}{x^{2}} + 8(2x)^{(2x)}\right)}{dx}\\=&8((2x)^{(2x)}((2)ln(2x) + \frac{(2x)(2)}{(2x)}))ln^{3}(2x) + \frac{8(2x)^{(2x)}*3ln^{2}(2x)*2}{(2x)} + 24((2x)^{(2x)}((2)ln(2x) + \frac{(2x)(2)}{(2x)}))ln^{2}(2x) + \frac{24(2x)^{(2x)}*2ln(2x)*2}{(2x)} + \frac{12*-(2x)^{(2x)}ln(2x)}{x^{2}} + \frac{12((2x)^{(2x)}((2)ln(2x) + \frac{(2x)(2)}{(2x)}))ln(2x)}{x} + \frac{12(2x)^{(2x)}*2}{x(2x)} + 24((2x)^{(2x)}((2)ln(2x) + \frac{(2x)(2)}{(2x)}))ln(2x) + \frac{24(2x)^{(2x)}*2}{(2x)} + \frac{12*-(2x)^{(2x)}}{x^{2}} + \frac{12((2x)^{(2x)}((2)ln(2x) + \frac{(2x)(2)}{(2x)}))}{x} - \frac{2*-2(2x)^{(2x)}}{x^{3}} - \frac{2((2x)^{(2x)}((2)ln(2x) + \frac{(2x)(2)}{(2x)}))}{x^{2}} + 8((2x)^{(2x)}((2)ln(2x) + \frac{(2x)(2)}{(2x)}))\\=&16(2x)^{(2x)}ln^{4}(2x) + 64(2x)^{(2x)}ln^{3}(2x) + \frac{48(2x)^{(2x)}ln^{2}(2x)}{x} + 96(2x)^{(2x)}ln^{2}(2x) + \frac{96(2x)^{(2x)}ln(2x)}{x} - \frac{16(2x)^{(2x)}ln(2x)}{x^{2}} + 64(2x)^{(2x)}ln(2x) + \frac{48(2x)^{(2x)}}{x} + \frac{4(2x)^{(2x)}}{x^{3}} - \frac{4(2x)^{(2x)}}{x^{2}} + 16(2x)^{(2x)}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/4]Find\ the\ 4th\ derivative\ of\ function\ {x}^{x}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \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} \]
\[ \begin{equation}\begin{split}[2/4]Find\ the\ 4th\ derivative\ of\ function\ {x}^{(2x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( {x}^{(2x)}\right)}{dx}\\=&({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))\\=&2{x}^{(2x)}ln(x) + 2{x}^{(2x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( 2{x}^{(2x)}ln(x) + 2{x}^{(2x)}\right)}{dx}\\=&2({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))ln(x) + \frac{2{x}^{(2x)}}{(x)} + 2({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))\\=&4{x}^{(2x)}ln^{2}(x) + 8{x}^{(2x)}ln(x) + \frac{2{x}^{(2x)}}{x} + 4{x}^{(2x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 4{x}^{(2x)}ln^{2}(x) + 8{x}^{(2x)}ln(x) + \frac{2{x}^{(2x)}}{x} + 4{x}^{(2x)}\right)}{dx}\\=&4({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))ln^{2}(x) + \frac{4{x}^{(2x)}*2ln(x)}{(x)} + 8({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))ln(x) + \frac{8{x}^{(2x)}}{(x)} + \frac{2*-{x}^{(2x)}}{x^{2}} + \frac{2({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))}{x} + 4({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))\\=&8{x}^{(2x)}ln^{3}(x) + 24{x}^{(2x)}ln^{2}(x) + \frac{12{x}^{(2x)}ln(x)}{x} + 24{x}^{(2x)}ln(x) + \frac{12{x}^{(2x)}}{x} - \frac{2{x}^{(2x)}}{x^{2}} + 8{x}^{(2x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 8{x}^{(2x)}ln^{3}(x) + 24{x}^{(2x)}ln^{2}(x) + \frac{12{x}^{(2x)}ln(x)}{x} + 24{x}^{(2x)}ln(x) + \frac{12{x}^{(2x)}}{x} - \frac{2{x}^{(2x)}}{x^{2}} + 8{x}^{(2x)}\right)}{dx}\\=&8({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))ln^{3}(x) + \frac{8{x}^{(2x)}*3ln^{2}(x)}{(x)} + 24({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))ln^{2}(x) + \frac{24{x}^{(2x)}*2ln(x)}{(x)} + \frac{12*-{x}^{(2x)}ln(x)}{x^{2}} + \frac{12({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))ln(x)}{x} + \frac{12{x}^{(2x)}}{x(x)} + 24({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))ln(x) + \frac{24{x}^{(2x)}}{(x)} + \frac{12*-{x}^{(2x)}}{x^{2}} + \frac{12({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))}{x} - \frac{2*-2{x}^{(2x)}}{x^{3}} - \frac{2({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))}{x^{2}} + 8({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))\\=&16{x}^{(2x)}ln^{4}(x) + 64{x}^{(2x)}ln^{3}(x) + \frac{48{x}^{(2x)}ln^{2}(x)}{x} + 96{x}^{(2x)}ln^{2}(x) + \frac{96{x}^{(2x)}ln(x)}{x} - \frac{16{x}^{(2x)}ln(x)}{x^{2}} + 64{x}^{(2x)}ln(x) + \frac{48{x}^{(2x)}}{x} + \frac{4{x}^{(2x)}}{x^{3}} - \frac{4{x}^{(2x)}}{x^{2}} + 16{x}^{(2x)}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[3/4]Find\ the\ 4th\ derivative\ of\ function\ {(2x)}^{x}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = (2x)^{x}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( (2x)^{x}\right)}{dx}\\=&((2x)^{x}((1)ln(2x) + \frac{(x)(2)}{(2x)}))\\=&(2x)^{x}ln(2x) + (2x)^{x}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( (2x)^{x}ln(2x) + (2x)^{x}\right)}{dx}\\=&((2x)^{x}((1)ln(2x) + \frac{(x)(2)}{(2x)}))ln(2x) + \frac{(2x)^{x}*2}{(2x)} + ((2x)^{x}((1)ln(2x) + \frac{(x)(2)}{(2x)}))\\=&(2x)^{x}ln^{2}(2x) + 2(2x)^{x}ln(2x) + \frac{(2x)^{x}}{x} + (2x)^{x}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( (2x)^{x}ln^{2}(2x) + 2(2x)^{x}ln(2x) + \frac{(2x)^{x}}{x} + (2x)^{x}\right)}{dx}\\=&((2x)^{x}((1)ln(2x) + \frac{(x)(2)}{(2x)}))ln^{2}(2x) + \frac{(2x)^{x}*2ln(2x)*2}{(2x)} + 2((2x)^{x}((1)ln(2x) + \frac{(x)(2)}{(2x)}))ln(2x) + \frac{2(2x)^{x}*2}{(2x)} + \frac{-(2x)^{x}}{x^{2}} + \frac{((2x)^{x}((1)ln(2x) + \frac{(x)(2)}{(2x)}))}{x} + ((2x)^{x}((1)ln(2x) + \frac{(x)(2)}{(2x)}))\\=&(2x)^{x}ln^{3}(2x) + 3(2x)^{x}ln^{2}(2x) + \frac{3(2x)^{x}ln(2x)}{x} + 3(2x)^{x}ln(2x) + \frac{3(2x)^{x}}{x} - \frac{(2x)^{x}}{x^{2}} + (2x)^{x}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( (2x)^{x}ln^{3}(2x) + 3(2x)^{x}ln^{2}(2x) + \frac{3(2x)^{x}ln(2x)}{x} + 3(2x)^{x}ln(2x) + \frac{3(2x)^{x}}{x} - \frac{(2x)^{x}}{x^{2}} + (2x)^{x}\right)}{dx}\\=&((2x)^{x}((1)ln(2x) + \frac{(x)(2)}{(2x)}))ln^{3}(2x) + \frac{(2x)^{x}*3ln^{2}(2x)*2}{(2x)} + 3((2x)^{x}((1)ln(2x) + \frac{(x)(2)}{(2x)}))ln^{2}(2x) + \frac{3(2x)^{x}*2ln(2x)*2}{(2x)} + \frac{3*-(2x)^{x}ln(2x)}{x^{2}} + \frac{3((2x)^{x}((1)ln(2x) + \frac{(x)(2)}{(2x)}))ln(2x)}{x} + \frac{3(2x)^{x}*2}{x(2x)} + 3((2x)^{x}((1)ln(2x) + \frac{(x)(2)}{(2x)}))ln(2x) + \frac{3(2x)^{x}*2}{(2x)} + \frac{3*-(2x)^{x}}{x^{2}} + \frac{3((2x)^{x}((1)ln(2x) + \frac{(x)(2)}{(2x)}))}{x} - \frac{-2(2x)^{x}}{x^{3}} - \frac{((2x)^{x}((1)ln(2x) + \frac{(x)(2)}{(2x)}))}{x^{2}} + ((2x)^{x}((1)ln(2x) + \frac{(x)(2)}{(2x)}))\\=&(2x)^{x}ln^{4}(2x) + 4(2x)^{x}ln^{3}(2x) + \frac{6(2x)^{x}ln^{2}(2x)}{x} + 6(2x)^{x}ln^{2}(2x) + \frac{12(2x)^{x}ln(2x)}{x} - \frac{4(2x)^{x}ln(2x)}{x^{2}} + 4(2x)^{x}ln(2x) + \frac{6(2x)^{x}}{x} + \frac{2(2x)^{x}}{x^{3}} - \frac{(2x)^{x}}{x^{2}} + (2x)^{x}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[4/4]Find\ the\ 4th\ derivative\ of\ function\ {(2x)}^{(2x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = (2x)^{(2x)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( (2x)^{(2x)}\right)}{dx}\\=&((2x)^{(2x)}((2)ln(2x) + \frac{(2x)(2)}{(2x)}))\\=&2(2x)^{(2x)}ln(2x) + 2(2x)^{(2x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( 2(2x)^{(2x)}ln(2x) + 2(2x)^{(2x)}\right)}{dx}\\=&2((2x)^{(2x)}((2)ln(2x) + \frac{(2x)(2)}{(2x)}))ln(2x) + \frac{2(2x)^{(2x)}*2}{(2x)} + 2((2x)^{(2x)}((2)ln(2x) + \frac{(2x)(2)}{(2x)}))\\=&4(2x)^{(2x)}ln^{2}(2x) + 8(2x)^{(2x)}ln(2x) + \frac{2(2x)^{(2x)}}{x} + 4(2x)^{(2x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 4(2x)^{(2x)}ln^{2}(2x) + 8(2x)^{(2x)}ln(2x) + \frac{2(2x)^{(2x)}}{x} + 4(2x)^{(2x)}\right)}{dx}\\=&4((2x)^{(2x)}((2)ln(2x) + \frac{(2x)(2)}{(2x)}))ln^{2}(2x) + \frac{4(2x)^{(2x)}*2ln(2x)*2}{(2x)} + 8((2x)^{(2x)}((2)ln(2x) + \frac{(2x)(2)}{(2x)}))ln(2x) + \frac{8(2x)^{(2x)}*2}{(2x)} + \frac{2*-(2x)^{(2x)}}{x^{2}} + \frac{2((2x)^{(2x)}((2)ln(2x) + \frac{(2x)(2)}{(2x)}))}{x} + 4((2x)^{(2x)}((2)ln(2x) + \frac{(2x)(2)}{(2x)}))\\=&8(2x)^{(2x)}ln^{3}(2x) + 24(2x)^{(2x)}ln^{2}(2x) + \frac{12(2x)^{(2x)}ln(2x)}{x} + 24(2x)^{(2x)}ln(2x) + \frac{12(2x)^{(2x)}}{x} - \frac{2(2x)^{(2x)}}{x^{2}} + 8(2x)^{(2x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 8(2x)^{(2x)}ln^{3}(2x) + 24(2x)^{(2x)}ln^{2}(2x) + \frac{12(2x)^{(2x)}ln(2x)}{x} + 24(2x)^{(2x)}ln(2x) + \frac{12(2x)^{(2x)}}{x} - \frac{2(2x)^{(2x)}}{x^{2}} + 8(2x)^{(2x)}\right)}{dx}\\=&8((2x)^{(2x)}((2)ln(2x) + \frac{(2x)(2)}{(2x)}))ln^{3}(2x) + \frac{8(2x)^{(2x)}*3ln^{2}(2x)*2}{(2x)} + 24((2x)^{(2x)}((2)ln(2x) + \frac{(2x)(2)}{(2x)}))ln^{2}(2x) + \frac{24(2x)^{(2x)}*2ln(2x)*2}{(2x)} + \frac{12*-(2x)^{(2x)}ln(2x)}{x^{2}} + \frac{12((2x)^{(2x)}((2)ln(2x) + \frac{(2x)(2)}{(2x)}))ln(2x)}{x} + \frac{12(2x)^{(2x)}*2}{x(2x)} + 24((2x)^{(2x)}((2)ln(2x) + \frac{(2x)(2)}{(2x)}))ln(2x) + \frac{24(2x)^{(2x)}*2}{(2x)} + \frac{12*-(2x)^{(2x)}}{x^{2}} + \frac{12((2x)^{(2x)}((2)ln(2x) + \frac{(2x)(2)}{(2x)}))}{x} - \frac{2*-2(2x)^{(2x)}}{x^{3}} - \frac{2((2x)^{(2x)}((2)ln(2x) + \frac{(2x)(2)}{(2x)}))}{x^{2}} + 8((2x)^{(2x)}((2)ln(2x) + \frac{(2x)(2)}{(2x)}))\\=&16(2x)^{(2x)}ln^{4}(2x) + 64(2x)^{(2x)}ln^{3}(2x) + \frac{48(2x)^{(2x)}ln^{2}(2x)}{x} + 96(2x)^{(2x)}ln^{2}(2x) + \frac{96(2x)^{(2x)}ln(2x)}{x} - \frac{16(2x)^{(2x)}ln(2x)}{x^{2}} + 64(2x)^{(2x)}ln(2x) + \frac{48(2x)^{(2x)}}{x} + \frac{4(2x)^{(2x)}}{x^{3}} - \frac{4(2x)^{(2x)}}{x^{2}} + 16(2x)^{(2x)}\\ \end{split}\end{equation} \]