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.
\[ \begin{equation}\begin{split}[1/2]Find\ the\ 4th\ derivative\ of\ function\ {10}^{(xlg(x))}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( {10}^{(xlg(x))}\right)}{dx}\\=&({10}^{(xlg(x))}((lg(x) + \frac{x}{ln{10}(x)})ln(10) + \frac{(xlg(x))(0)}{(10)}))\\=&{10}^{(xlg(x))}ln(10)lg(x) + \frac{{10}^{(xlg(x))}ln(10)}{ln{10}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( {10}^{(xlg(x))}ln(10)lg(x) + \frac{{10}^{(xlg(x))}ln(10)}{ln{10}}\right)}{dx}\\=&({10}^{(xlg(x))}((lg(x) + \frac{x}{ln{10}(x)})ln(10) + \frac{(xlg(x))(0)}{(10)}))ln(10)lg(x) + \frac{{10}^{(xlg(x))}*0lg(x)}{(10)} + \frac{{10}^{(xlg(x))}ln(10)}{ln{10}(x)} + \frac{({10}^{(xlg(x))}((lg(x) + \frac{x}{ln{10}(x)})ln(10) + \frac{(xlg(x))(0)}{(10)}))ln(10)}{ln{10}} + \frac{{10}^{(xlg(x))}*0}{(10)ln{10}} + \frac{{10}^{(xlg(x))}ln(10)*-0}{ln^{2}{10}}\\=&{10}^{(xlg(x))}ln^{2}(10)lg^{2}(x) + \frac{2 * {10}^{(xlg(x))}ln^{2}(10)lg(x)}{ln{10}} + \frac{{10}^{(xlg(x))}ln(10)}{xln{10}} + \frac{{10}^{(xlg(x))}ln^{2}(10)}{ln^{2}{10}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( {10}^{(xlg(x))}ln^{2}(10)lg^{2}(x) + \frac{2 * {10}^{(xlg(x))}ln^{2}(10)lg(x)}{ln{10}} + \frac{{10}^{(xlg(x))}ln(10)}{xln{10}} + \frac{{10}^{(xlg(x))}ln^{2}(10)}{ln^{2}{10}}\right)}{dx}\\=&({10}^{(xlg(x))}((lg(x) + \frac{x}{ln{10}(x)})ln(10) + \frac{(xlg(x))(0)}{(10)}))ln^{2}(10)lg^{2}(x) + \frac{{10}^{(xlg(x))}*2ln(10)*0lg^{2}(x)}{(10)} + \frac{{10}^{(xlg(x))}ln^{2}(10)*2lg(x)}{ln{10}(x)} + \frac{2({10}^{(xlg(x))}((lg(x) + \frac{x}{ln{10}(x)})ln(10) + \frac{(xlg(x))(0)}{(10)}))ln^{2}(10)lg(x)}{ln{10}} + \frac{2 * {10}^{(xlg(x))}*2ln(10)*0lg(x)}{(10)ln{10}} + \frac{2 * {10}^{(xlg(x))}ln^{2}(10)*-0lg(x)}{ln^{2}{10}} + \frac{2 * {10}^{(xlg(x))}ln^{2}(10)}{ln{10}ln{10}(x)} + \frac{-{10}^{(xlg(x))}ln(10)}{x^{2}ln{10}} + \frac{({10}^{(xlg(x))}((lg(x) + \frac{x}{ln{10}(x)})ln(10) + \frac{(xlg(x))(0)}{(10)}))ln(10)}{xln{10}} + \frac{{10}^{(xlg(x))}*-0ln(10)}{xln^{2}{10}} + \frac{{10}^{(xlg(x))}*0}{xln{10}(10)} + \frac{({10}^{(xlg(x))}((lg(x) + \frac{x}{ln{10}(x)})ln(10) + \frac{(xlg(x))(0)}{(10)}))ln^{2}(10)}{ln^{2}{10}} + \frac{{10}^{(xlg(x))}*2ln(10)*0}{(10)ln^{2}{10}} + \frac{{10}^{(xlg(x))}ln^{2}(10)*-2*0}{ln^{3}{10}}\\=&{10}^{(xlg(x))}ln^{3}(10)lg^{3}(x) + \frac{3 * {10}^{(xlg(x))}ln^{3}(10)lg^{2}(x)}{ln{10}} + \frac{2 * {10}^{(xlg(x))}ln^{2}(10)lg(x)}{xln{10}} + \frac{3 * {10}^{(xlg(x))}ln^{3}(10)lg(x)}{ln^{2}{10}} + \frac{{10}^{(xlg(x))}ln^{2}(10)lg(x)}{xln{10}} - \frac{{10}^{(xlg(x))}ln(10)}{x^{2}ln{10}} + \frac{2 * {10}^{(xlg(x))}ln^{2}(10)}{xln^{2}{10}} + \frac{{10}^{(xlg(x))}ln^{2}(10)}{xln^{2}{10}} + \frac{{10}^{(xlg(x))}ln^{3}(10)}{ln^{3}{10}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( {10}^{(xlg(x))}ln^{3}(10)lg^{3}(x) + \frac{3 * {10}^{(xlg(x))}ln^{3}(10)lg^{2}(x)}{ln{10}} + \frac{2 * {10}^{(xlg(x))}ln^{2}(10)lg(x)}{xln{10}} + \frac{3 * {10}^{(xlg(x))}ln^{3}(10)lg(x)}{ln^{2}{10}} + \frac{{10}^{(xlg(x))}ln^{2}(10)lg(x)}{xln{10}} - \frac{{10}^{(xlg(x))}ln(10)}{x^{2}ln{10}} + \frac{2 * {10}^{(xlg(x))}ln^{2}(10)}{xln^{2}{10}} + \frac{{10}^{(xlg(x))}ln^{2}(10)}{xln^{2}{10}} + \frac{{10}^{(xlg(x))}ln^{3}(10)}{ln^{3}{10}}\right)}{dx}\\=&({10}^{(xlg(x))}((lg(x) + \frac{x}{ln{10}(x)})ln(10) + \frac{(xlg(x))(0)}{(10)}))ln^{3}(10)lg^{3}(x) + \frac{{10}^{(xlg(x))}*3ln^{2}(10)*0lg^{3}(x)}{(10)} + \frac{{10}^{(xlg(x))}ln^{3}(10)*3lg^{2}(x)}{ln{10}(x)} + \frac{3({10}^{(xlg(x))}((lg(x) + \frac{x}{ln{10}(x)})ln(10) + \frac{(xlg(x))(0)}{(10)}))ln^{3}(10)lg^{2}(x)}{ln{10}} + \frac{3 * {10}^{(xlg(x))}*3ln^{2}(10)*0lg^{2}(x)}{(10)ln{10}} + \frac{3 * {10}^{(xlg(x))}ln^{3}(10)*-0lg^{2}(x)}{ln^{2}{10}} + \frac{3 * {10}^{(xlg(x))}ln^{3}(10)*2lg(x)}{ln{10}ln{10}(x)} + \frac{2*-{10}^{(xlg(x))}ln^{2}(10)lg(x)}{x^{2}ln{10}} + \frac{2({10}^{(xlg(x))}((lg(x) + \frac{x}{ln{10}(x)})ln(10) + \frac{(xlg(x))(0)}{(10)}))ln^{2}(10)lg(x)}{xln{10}} + \frac{2 * {10}^{(xlg(x))}*-0ln^{2}(10)lg(x)}{xln^{2}{10}} + \frac{2 * {10}^{(xlg(x))}*2ln(10)*0lg(x)}{xln{10}(10)} + \frac{2 * {10}^{(xlg(x))}ln^{2}(10)}{xln{10}ln{10}(x)} + \frac{3({10}^{(xlg(x))}((lg(x) + \frac{x}{ln{10}(x)})ln(10) + \frac{(xlg(x))(0)}{(10)}))ln^{3}(10)lg(x)}{ln^{2}{10}} + \frac{3 * {10}^{(xlg(x))}*3ln^{2}(10)*0lg(x)}{(10)ln^{2}{10}} + \frac{3 * {10}^{(xlg(x))}ln^{3}(10)*-2*0lg(x)}{ln^{3}{10}} + \frac{3 * {10}^{(xlg(x))}ln^{3}(10)}{ln^{2}{10}ln{10}(x)} + \frac{-{10}^{(xlg(x))}ln^{2}(10)lg(x)}{x^{2}ln{10}} + \frac{({10}^{(xlg(x))}((lg(x) + \frac{x}{ln{10}(x)})ln(10) + \frac{(xlg(x))(0)}{(10)}))ln^{2}(10)lg(x)}{xln{10}} + \frac{{10}^{(xlg(x))}*2ln(10)*0lg(x)}{x(10)ln{10}} + \frac{{10}^{(xlg(x))}ln^{2}(10)*-0lg(x)}{xln^{2}{10}} + \frac{{10}^{(xlg(x))}ln^{2}(10)}{xln{10}ln{10}(x)} - \frac{-2 * {10}^{(xlg(x))}ln(10)}{x^{3}ln{10}} - \frac{({10}^{(xlg(x))}((lg(x) + \frac{x}{ln{10}(x)})ln(10) + \frac{(xlg(x))(0)}{(10)}))ln(10)}{x^{2}ln{10}} - \frac{{10}^{(xlg(x))}*-0ln(10)}{x^{2}ln^{2}{10}} - \frac{{10}^{(xlg(x))}*0}{x^{2}ln{10}(10)} + \frac{2*-{10}^{(xlg(x))}ln^{2}(10)}{x^{2}ln^{2}{10}} + \frac{2({10}^{(xlg(x))}((lg(x) + \frac{x}{ln{10}(x)})ln(10) + \frac{(xlg(x))(0)}{(10)}))ln^{2}(10)}{xln^{2}{10}} + \frac{2 * {10}^{(xlg(x))}*2ln(10)*0}{x(10)ln^{2}{10}} + \frac{2 * {10}^{(xlg(x))}ln^{2}(10)*-2*0}{xln^{3}{10}} + \frac{-{10}^{(xlg(x))}ln^{2}(10)}{x^{2}ln^{2}{10}} + \frac{({10}^{(xlg(x))}((lg(x) + \frac{x}{ln{10}(x)})ln(10) + \frac{(xlg(x))(0)}{(10)}))ln^{2}(10)}{xln^{2}{10}} + \frac{{10}^{(xlg(x))}*-2*0ln^{2}(10)}{xln^{3}{10}} + \frac{{10}^{(xlg(x))}*2ln(10)*0}{xln^{2}{10}(10)} + \frac{({10}^{(xlg(x))}((lg(x) + \frac{x}{ln{10}(x)})ln(10) + \frac{(xlg(x))(0)}{(10)}))ln^{3}(10)}{ln^{3}{10}} + \frac{{10}^{(xlg(x))}*3ln^{2}(10)*0}{(10)ln^{3}{10}} + \frac{{10}^{(xlg(x))}ln^{3}(10)*-3*0}{ln^{4}{10}}\\=&{10}^{(xlg(x))}ln^{4}(10)lg^{4}(x) + \frac{4 * {10}^{(xlg(x))}ln^{4}(10)lg^{3}(x)}{ln{10}} + \frac{3 * {10}^{(xlg(x))}ln^{3}(10)lg^{2}(x)}{xln{10}} + \frac{6 * {10}^{(xlg(x))}ln^{4}(10)lg^{2}(x)}{ln^{2}{10}} + \frac{9 * {10}^{(xlg(x))}ln^{3}(10)lg(x)}{xln^{2}{10}} - \frac{2 * {10}^{(xlg(x))}ln^{2}(10)lg(x)}{x^{2}ln{10}} + \frac{3 * {10}^{(xlg(x))}ln^{3}(10)lg^{2}(x)}{xln{10}} - \frac{2 * {10}^{(xlg(x))}ln^{2}(10)lg(x)}{x^{2}ln{10}} + \frac{4 * {10}^{(xlg(x))}ln^{4}(10)lg(x)}{ln^{3}{10}} + \frac{3 * {10}^{(xlg(x))}ln^{3}(10)lg(x)}{xln^{2}{10}} - \frac{{10}^{(xlg(x))}ln^{2}(10)}{x^{2}ln^{2}{10}} + \frac{2 * {10}^{(xlg(x))}ln(10)}{x^{3}ln{10}} + \frac{3 * {10}^{(xlg(x))}ln^{3}(10)}{xln^{3}{10}} + \frac{3 * {10}^{(xlg(x))}ln^{3}(10)}{xln^{3}{10}} + \frac{{10}^{(xlg(x))}ln^{4}(10)}{ln^{4}{10}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/2]Find\ the\ 4th\ derivative\ of\ function\ {x}^{x}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \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} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/2]Find\ the\ 4th\ derivative\ of\ function\ {10}^{(xlg(x))}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( {10}^{(xlg(x))}\right)}{dx}\\=&({10}^{(xlg(x))}((lg(x) + \frac{x}{ln{10}(x)})ln(10) + \frac{(xlg(x))(0)}{(10)}))\\=&{10}^{(xlg(x))}ln(10)lg(x) + \frac{{10}^{(xlg(x))}ln(10)}{ln{10}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( {10}^{(xlg(x))}ln(10)lg(x) + \frac{{10}^{(xlg(x))}ln(10)}{ln{10}}\right)}{dx}\\=&({10}^{(xlg(x))}((lg(x) + \frac{x}{ln{10}(x)})ln(10) + \frac{(xlg(x))(0)}{(10)}))ln(10)lg(x) + \frac{{10}^{(xlg(x))}*0lg(x)}{(10)} + \frac{{10}^{(xlg(x))}ln(10)}{ln{10}(x)} + \frac{({10}^{(xlg(x))}((lg(x) + \frac{x}{ln{10}(x)})ln(10) + \frac{(xlg(x))(0)}{(10)}))ln(10)}{ln{10}} + \frac{{10}^{(xlg(x))}*0}{(10)ln{10}} + \frac{{10}^{(xlg(x))}ln(10)*-0}{ln^{2}{10}}\\=&{10}^{(xlg(x))}ln^{2}(10)lg^{2}(x) + \frac{2 * {10}^{(xlg(x))}ln^{2}(10)lg(x)}{ln{10}} + \frac{{10}^{(xlg(x))}ln(10)}{xln{10}} + \frac{{10}^{(xlg(x))}ln^{2}(10)}{ln^{2}{10}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( {10}^{(xlg(x))}ln^{2}(10)lg^{2}(x) + \frac{2 * {10}^{(xlg(x))}ln^{2}(10)lg(x)}{ln{10}} + \frac{{10}^{(xlg(x))}ln(10)}{xln{10}} + \frac{{10}^{(xlg(x))}ln^{2}(10)}{ln^{2}{10}}\right)}{dx}\\=&({10}^{(xlg(x))}((lg(x) + \frac{x}{ln{10}(x)})ln(10) + \frac{(xlg(x))(0)}{(10)}))ln^{2}(10)lg^{2}(x) + \frac{{10}^{(xlg(x))}*2ln(10)*0lg^{2}(x)}{(10)} + \frac{{10}^{(xlg(x))}ln^{2}(10)*2lg(x)}{ln{10}(x)} + \frac{2({10}^{(xlg(x))}((lg(x) + \frac{x}{ln{10}(x)})ln(10) + \frac{(xlg(x))(0)}{(10)}))ln^{2}(10)lg(x)}{ln{10}} + \frac{2 * {10}^{(xlg(x))}*2ln(10)*0lg(x)}{(10)ln{10}} + \frac{2 * {10}^{(xlg(x))}ln^{2}(10)*-0lg(x)}{ln^{2}{10}} + \frac{2 * {10}^{(xlg(x))}ln^{2}(10)}{ln{10}ln{10}(x)} + \frac{-{10}^{(xlg(x))}ln(10)}{x^{2}ln{10}} + \frac{({10}^{(xlg(x))}((lg(x) + \frac{x}{ln{10}(x)})ln(10) + \frac{(xlg(x))(0)}{(10)}))ln(10)}{xln{10}} + \frac{{10}^{(xlg(x))}*-0ln(10)}{xln^{2}{10}} + \frac{{10}^{(xlg(x))}*0}{xln{10}(10)} + \frac{({10}^{(xlg(x))}((lg(x) + \frac{x}{ln{10}(x)})ln(10) + \frac{(xlg(x))(0)}{(10)}))ln^{2}(10)}{ln^{2}{10}} + \frac{{10}^{(xlg(x))}*2ln(10)*0}{(10)ln^{2}{10}} + \frac{{10}^{(xlg(x))}ln^{2}(10)*-2*0}{ln^{3}{10}}\\=&{10}^{(xlg(x))}ln^{3}(10)lg^{3}(x) + \frac{3 * {10}^{(xlg(x))}ln^{3}(10)lg^{2}(x)}{ln{10}} + \frac{2 * {10}^{(xlg(x))}ln^{2}(10)lg(x)}{xln{10}} + \frac{3 * {10}^{(xlg(x))}ln^{3}(10)lg(x)}{ln^{2}{10}} + \frac{{10}^{(xlg(x))}ln^{2}(10)lg(x)}{xln{10}} - \frac{{10}^{(xlg(x))}ln(10)}{x^{2}ln{10}} + \frac{2 * {10}^{(xlg(x))}ln^{2}(10)}{xln^{2}{10}} + \frac{{10}^{(xlg(x))}ln^{2}(10)}{xln^{2}{10}} + \frac{{10}^{(xlg(x))}ln^{3}(10)}{ln^{3}{10}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( {10}^{(xlg(x))}ln^{3}(10)lg^{3}(x) + \frac{3 * {10}^{(xlg(x))}ln^{3}(10)lg^{2}(x)}{ln{10}} + \frac{2 * {10}^{(xlg(x))}ln^{2}(10)lg(x)}{xln{10}} + \frac{3 * {10}^{(xlg(x))}ln^{3}(10)lg(x)}{ln^{2}{10}} + \frac{{10}^{(xlg(x))}ln^{2}(10)lg(x)}{xln{10}} - \frac{{10}^{(xlg(x))}ln(10)}{x^{2}ln{10}} + \frac{2 * {10}^{(xlg(x))}ln^{2}(10)}{xln^{2}{10}} + \frac{{10}^{(xlg(x))}ln^{2}(10)}{xln^{2}{10}} + \frac{{10}^{(xlg(x))}ln^{3}(10)}{ln^{3}{10}}\right)}{dx}\\=&({10}^{(xlg(x))}((lg(x) + \frac{x}{ln{10}(x)})ln(10) + \frac{(xlg(x))(0)}{(10)}))ln^{3}(10)lg^{3}(x) + \frac{{10}^{(xlg(x))}*3ln^{2}(10)*0lg^{3}(x)}{(10)} + \frac{{10}^{(xlg(x))}ln^{3}(10)*3lg^{2}(x)}{ln{10}(x)} + \frac{3({10}^{(xlg(x))}((lg(x) + \frac{x}{ln{10}(x)})ln(10) + \frac{(xlg(x))(0)}{(10)}))ln^{3}(10)lg^{2}(x)}{ln{10}} + \frac{3 * {10}^{(xlg(x))}*3ln^{2}(10)*0lg^{2}(x)}{(10)ln{10}} + \frac{3 * {10}^{(xlg(x))}ln^{3}(10)*-0lg^{2}(x)}{ln^{2}{10}} + \frac{3 * {10}^{(xlg(x))}ln^{3}(10)*2lg(x)}{ln{10}ln{10}(x)} + \frac{2*-{10}^{(xlg(x))}ln^{2}(10)lg(x)}{x^{2}ln{10}} + \frac{2({10}^{(xlg(x))}((lg(x) + \frac{x}{ln{10}(x)})ln(10) + \frac{(xlg(x))(0)}{(10)}))ln^{2}(10)lg(x)}{xln{10}} + \frac{2 * {10}^{(xlg(x))}*-0ln^{2}(10)lg(x)}{xln^{2}{10}} + \frac{2 * {10}^{(xlg(x))}*2ln(10)*0lg(x)}{xln{10}(10)} + \frac{2 * {10}^{(xlg(x))}ln^{2}(10)}{xln{10}ln{10}(x)} + \frac{3({10}^{(xlg(x))}((lg(x) + \frac{x}{ln{10}(x)})ln(10) + \frac{(xlg(x))(0)}{(10)}))ln^{3}(10)lg(x)}{ln^{2}{10}} + \frac{3 * {10}^{(xlg(x))}*3ln^{2}(10)*0lg(x)}{(10)ln^{2}{10}} + \frac{3 * {10}^{(xlg(x))}ln^{3}(10)*-2*0lg(x)}{ln^{3}{10}} + \frac{3 * {10}^{(xlg(x))}ln^{3}(10)}{ln^{2}{10}ln{10}(x)} + \frac{-{10}^{(xlg(x))}ln^{2}(10)lg(x)}{x^{2}ln{10}} + \frac{({10}^{(xlg(x))}((lg(x) + \frac{x}{ln{10}(x)})ln(10) + \frac{(xlg(x))(0)}{(10)}))ln^{2}(10)lg(x)}{xln{10}} + \frac{{10}^{(xlg(x))}*2ln(10)*0lg(x)}{x(10)ln{10}} + \frac{{10}^{(xlg(x))}ln^{2}(10)*-0lg(x)}{xln^{2}{10}} + \frac{{10}^{(xlg(x))}ln^{2}(10)}{xln{10}ln{10}(x)} - \frac{-2 * {10}^{(xlg(x))}ln(10)}{x^{3}ln{10}} - \frac{({10}^{(xlg(x))}((lg(x) + \frac{x}{ln{10}(x)})ln(10) + \frac{(xlg(x))(0)}{(10)}))ln(10)}{x^{2}ln{10}} - \frac{{10}^{(xlg(x))}*-0ln(10)}{x^{2}ln^{2}{10}} - \frac{{10}^{(xlg(x))}*0}{x^{2}ln{10}(10)} + \frac{2*-{10}^{(xlg(x))}ln^{2}(10)}{x^{2}ln^{2}{10}} + \frac{2({10}^{(xlg(x))}((lg(x) + \frac{x}{ln{10}(x)})ln(10) + \frac{(xlg(x))(0)}{(10)}))ln^{2}(10)}{xln^{2}{10}} + \frac{2 * {10}^{(xlg(x))}*2ln(10)*0}{x(10)ln^{2}{10}} + \frac{2 * {10}^{(xlg(x))}ln^{2}(10)*-2*0}{xln^{3}{10}} + \frac{-{10}^{(xlg(x))}ln^{2}(10)}{x^{2}ln^{2}{10}} + \frac{({10}^{(xlg(x))}((lg(x) + \frac{x}{ln{10}(x)})ln(10) + \frac{(xlg(x))(0)}{(10)}))ln^{2}(10)}{xln^{2}{10}} + \frac{{10}^{(xlg(x))}*-2*0ln^{2}(10)}{xln^{3}{10}} + \frac{{10}^{(xlg(x))}*2ln(10)*0}{xln^{2}{10}(10)} + \frac{({10}^{(xlg(x))}((lg(x) + \frac{x}{ln{10}(x)})ln(10) + \frac{(xlg(x))(0)}{(10)}))ln^{3}(10)}{ln^{3}{10}} + \frac{{10}^{(xlg(x))}*3ln^{2}(10)*0}{(10)ln^{3}{10}} + \frac{{10}^{(xlg(x))}ln^{3}(10)*-3*0}{ln^{4}{10}}\\=&{10}^{(xlg(x))}ln^{4}(10)lg^{4}(x) + \frac{4 * {10}^{(xlg(x))}ln^{4}(10)lg^{3}(x)}{ln{10}} + \frac{3 * {10}^{(xlg(x))}ln^{3}(10)lg^{2}(x)}{xln{10}} + \frac{6 * {10}^{(xlg(x))}ln^{4}(10)lg^{2}(x)}{ln^{2}{10}} + \frac{9 * {10}^{(xlg(x))}ln^{3}(10)lg(x)}{xln^{2}{10}} - \frac{2 * {10}^{(xlg(x))}ln^{2}(10)lg(x)}{x^{2}ln{10}} + \frac{3 * {10}^{(xlg(x))}ln^{3}(10)lg^{2}(x)}{xln{10}} - \frac{2 * {10}^{(xlg(x))}ln^{2}(10)lg(x)}{x^{2}ln{10}} + \frac{4 * {10}^{(xlg(x))}ln^{4}(10)lg(x)}{ln^{3}{10}} + \frac{3 * {10}^{(xlg(x))}ln^{3}(10)lg(x)}{xln^{2}{10}} - \frac{{10}^{(xlg(x))}ln^{2}(10)}{x^{2}ln^{2}{10}} + \frac{2 * {10}^{(xlg(x))}ln(10)}{x^{3}ln{10}} + \frac{3 * {10}^{(xlg(x))}ln^{3}(10)}{xln^{3}{10}} + \frac{3 * {10}^{(xlg(x))}ln^{3}(10)}{xln^{3}{10}} + \frac{{10}^{(xlg(x))}ln^{4}(10)}{ln^{4}{10}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/2]Find\ the\ 4th\ derivative\ of\ function\ {x}^{x}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \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} \]