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.
\[ \begin{equation}\begin{split}[1/2]Find\ the\ 4th\ derivative\ of\ function\ xlg(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( xlg(x)\right)}{dx}\\=&lg(x) + \frac{x}{ln{10}(x)}\\=&lg(x) + \frac{1}{ln{10}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( lg(x) + \frac{1}{ln{10}}\right)}{dx}\\=&\frac{1}{ln{10}(x)} + \frac{-0}{ln^{2}{10}}\\=&\frac{1}{xln{10}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{1}{xln{10}}\right)}{dx}\\=&\frac{-1}{x^{2}ln{10}} + \frac{-0}{xln^{2}{10}}\\=&\frac{-1}{x^{2}ln{10}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-1}{x^{2}ln{10}}\right)}{dx}\\=&\frac{--2}{x^{3}ln{10}} - \frac{-0}{x^{2}ln^{2}{10}}\\=&\frac{2}{x^{3}ln{10}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/2]Find\ the\ 4th\ derivative\ of\ function\ {x}^{lg(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( {x}^{lg(x)}\right)}{dx}\\=&({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))\\=&\frac{{x}^{lg(x)}ln(x)}{xln{10}} + \frac{{x}^{lg(x)}lg(x)}{x}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{{x}^{lg(x)}ln(x)}{xln{10}} + \frac{{x}^{lg(x)}lg(x)}{x}\right)}{dx}\\=&\frac{-{x}^{lg(x)}ln(x)}{x^{2}ln{10}} + \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)}{xln{10}} + \frac{{x}^{lg(x)}}{x(x)ln{10}} + \frac{{x}^{lg(x)}ln(x)*-0}{xln^{2}{10}} + \frac{-{x}^{lg(x)}lg(x)}{x^{2}} + \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))lg(x)}{x} + \frac{{x}^{lg(x)}}{xln{10}(x)}\\=&\frac{{x}^{lg(x)}ln(x)lg(x)}{x^{2}ln{10}} + \frac{{x}^{lg(x)}ln(x)lg(x)}{x^{2}ln{10}} - \frac{{x}^{lg(x)}ln(x)}{x^{2}ln{10}} + \frac{{x}^{lg(x)}ln^{2}(x)}{x^{2}ln^{2}{10}} - \frac{{x}^{lg(x)}lg(x)}{x^{2}} + \frac{2{x}^{lg(x)}}{x^{2}ln{10}} + \frac{{x}^{lg(x)}lg^{2}(x)}{x^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{{x}^{lg(x)}ln(x)lg(x)}{x^{2}ln{10}} + \frac{{x}^{lg(x)}ln(x)lg(x)}{x^{2}ln{10}} - \frac{{x}^{lg(x)}ln(x)}{x^{2}ln{10}} + \frac{{x}^{lg(x)}ln^{2}(x)}{x^{2}ln^{2}{10}} - \frac{{x}^{lg(x)}lg(x)}{x^{2}} + \frac{2{x}^{lg(x)}}{x^{2}ln{10}} + \frac{{x}^{lg(x)}lg^{2}(x)}{x^{2}}\right)}{dx}\\=&\frac{-2{x}^{lg(x)}ln(x)lg(x)}{x^{3}ln{10}} + \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)lg(x)}{x^{2}ln{10}} + \frac{{x}^{lg(x)}lg(x)}{x^{2}(x)ln{10}} + \frac{{x}^{lg(x)}ln(x)*-0lg(x)}{x^{2}ln^{2}{10}} + \frac{{x}^{lg(x)}ln(x)}{x^{2}ln{10}ln{10}(x)} + \frac{-2{x}^{lg(x)}ln(x)lg(x)}{x^{3}ln{10}} + \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)lg(x)}{x^{2}ln{10}} + \frac{{x}^{lg(x)}*-0ln(x)lg(x)}{x^{2}ln^{2}{10}} + \frac{{x}^{lg(x)}lg(x)}{x^{2}ln{10}(x)} + \frac{{x}^{lg(x)}ln(x)}{x^{2}ln{10}ln{10}(x)} - \frac{-2{x}^{lg(x)}ln(x)}{x^{3}ln{10}} - \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)}{x^{2}ln{10}} - \frac{{x}^{lg(x)}}{x^{2}(x)ln{10}} - \frac{{x}^{lg(x)}ln(x)*-0}{x^{2}ln^{2}{10}} + \frac{-2{x}^{lg(x)}ln^{2}(x)}{x^{3}ln^{2}{10}} + \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln^{2}(x)}{x^{2}ln^{2}{10}} + \frac{{x}^{lg(x)}*-2*0ln^{2}(x)}{x^{2}ln^{3}{10}} + \frac{{x}^{lg(x)}*2ln(x)}{x^{2}ln^{2}{10}(x)} - \frac{-2{x}^{lg(x)}lg(x)}{x^{3}} - \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))lg(x)}{x^{2}} - \frac{{x}^{lg(x)}}{x^{2}ln{10}(x)} + \frac{2*-2{x}^{lg(x)}}{x^{3}ln{10}} + \frac{2({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))}{x^{2}ln{10}} + \frac{2{x}^{lg(x)}*-0}{x^{2}ln^{2}{10}} + \frac{-2{x}^{lg(x)}lg^{2}(x)}{x^{3}} + \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))lg^{2}(x)}{x^{2}} + \frac{{x}^{lg(x)}*2lg(x)}{x^{2}ln{10}(x)}\\=&\frac{-3{x}^{lg(x)}ln(x)lg(x)}{x^{3}ln{10}} + \frac{3{x}^{lg(x)}ln^{2}(x)lg(x)}{x^{3}ln^{2}{10}} + \frac{{x}^{lg(x)}ln(x)lg^{2}(x)}{x^{3}ln{10}} + \frac{6{x}^{lg(x)}lg(x)}{x^{3}ln{10}} - \frac{3{x}^{lg(x)}ln(x)lg(x)}{x^{3}ln{10}} + \frac{2{x}^{lg(x)}ln(x)lg^{2}(x)}{x^{3}ln{10}} + \frac{3{x}^{lg(x)}ln(x)}{x^{3}ln^{2}{10}} + \frac{3{x}^{lg(x)}ln(x)}{x^{3}ln^{2}{10}} + \frac{2{x}^{lg(x)}ln(x)}{x^{3}ln{10}} - \frac{3{x}^{lg(x)}ln^{2}(x)}{x^{3}ln^{2}{10}} + \frac{{x}^{lg(x)}ln^{3}(x)}{x^{3}ln^{3}{10}} - \frac{6{x}^{lg(x)}}{x^{3}ln{10}} + \frac{2{x}^{lg(x)}lg(x)}{x^{3}} - \frac{3{x}^{lg(x)}lg^{2}(x)}{x^{3}} + \frac{{x}^{lg(x)}lg^{3}(x)}{x^{3}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-3{x}^{lg(x)}ln(x)lg(x)}{x^{3}ln{10}} + \frac{3{x}^{lg(x)}ln^{2}(x)lg(x)}{x^{3}ln^{2}{10}} + \frac{{x}^{lg(x)}ln(x)lg^{2}(x)}{x^{3}ln{10}} + \frac{6{x}^{lg(x)}lg(x)}{x^{3}ln{10}} - \frac{3{x}^{lg(x)}ln(x)lg(x)}{x^{3}ln{10}} + \frac{2{x}^{lg(x)}ln(x)lg^{2}(x)}{x^{3}ln{10}} + \frac{3{x}^{lg(x)}ln(x)}{x^{3}ln^{2}{10}} + \frac{3{x}^{lg(x)}ln(x)}{x^{3}ln^{2}{10}} + \frac{2{x}^{lg(x)}ln(x)}{x^{3}ln{10}} - \frac{3{x}^{lg(x)}ln^{2}(x)}{x^{3}ln^{2}{10}} + \frac{{x}^{lg(x)}ln^{3}(x)}{x^{3}ln^{3}{10}} - \frac{6{x}^{lg(x)}}{x^{3}ln{10}} + \frac{2{x}^{lg(x)}lg(x)}{x^{3}} - \frac{3{x}^{lg(x)}lg^{2}(x)}{x^{3}} + \frac{{x}^{lg(x)}lg^{3}(x)}{x^{3}}\right)}{dx}\\=&\frac{-3*-3{x}^{lg(x)}ln(x)lg(x)}{x^{4}ln{10}} - \frac{3({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)lg(x)}{x^{3}ln{10}} - \frac{3{x}^{lg(x)}lg(x)}{x^{3}(x)ln{10}} - \frac{3{x}^{lg(x)}ln(x)*-0lg(x)}{x^{3}ln^{2}{10}} - \frac{3{x}^{lg(x)}ln(x)}{x^{3}ln{10}ln{10}(x)} + \frac{3*-3{x}^{lg(x)}ln^{2}(x)lg(x)}{x^{4}ln^{2}{10}} + \frac{3({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln^{2}(x)lg(x)}{x^{3}ln^{2}{10}} + \frac{3{x}^{lg(x)}*-2*0ln^{2}(x)lg(x)}{x^{3}ln^{3}{10}} + \frac{3{x}^{lg(x)}*2ln(x)lg(x)}{x^{3}ln^{2}{10}(x)} + \frac{3{x}^{lg(x)}ln^{2}(x)}{x^{3}ln^{2}{10}ln{10}(x)} + \frac{-3{x}^{lg(x)}ln(x)lg^{2}(x)}{x^{4}ln{10}} + \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)lg^{2}(x)}{x^{3}ln{10}} + \frac{{x}^{lg(x)}lg^{2}(x)}{x^{3}(x)ln{10}} + \frac{{x}^{lg(x)}ln(x)*-0lg^{2}(x)}{x^{3}ln^{2}{10}} + \frac{{x}^{lg(x)}ln(x)*2lg(x)}{x^{3}ln{10}ln{10}(x)} + \frac{6*-3{x}^{lg(x)}lg(x)}{x^{4}ln{10}} + \frac{6({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))lg(x)}{x^{3}ln{10}} + \frac{6{x}^{lg(x)}*-0lg(x)}{x^{3}ln^{2}{10}} + \frac{6{x}^{lg(x)}}{x^{3}ln{10}ln{10}(x)} - \frac{3*-3{x}^{lg(x)}ln(x)lg(x)}{x^{4}ln{10}} - \frac{3({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)lg(x)}{x^{3}ln{10}} - \frac{3{x}^{lg(x)}*-0ln(x)lg(x)}{x^{3}ln^{2}{10}} - \frac{3{x}^{lg(x)}lg(x)}{x^{3}ln{10}(x)} - \frac{3{x}^{lg(x)}ln(x)}{x^{3}ln{10}ln{10}(x)} + \frac{2*-3{x}^{lg(x)}ln(x)lg^{2}(x)}{x^{4}ln{10}} + \frac{2({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)lg^{2}(x)}{x^{3}ln{10}} + \frac{2{x}^{lg(x)}*-0ln(x)lg^{2}(x)}{x^{3}ln^{2}{10}} + \frac{2{x}^{lg(x)}lg^{2}(x)}{x^{3}ln{10}(x)} + \frac{2{x}^{lg(x)}ln(x)*2lg(x)}{x^{3}ln{10}ln{10}(x)} + \frac{3*-3{x}^{lg(x)}ln(x)}{x^{4}ln^{2}{10}} + \frac{3({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)}{x^{3}ln^{2}{10}} + \frac{3{x}^{lg(x)}}{x^{3}(x)ln^{2}{10}} + \frac{3{x}^{lg(x)}ln(x)*-2*0}{x^{3}ln^{3}{10}} + \frac{3*-3{x}^{lg(x)}ln(x)}{x^{4}ln^{2}{10}} + \frac{3({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)}{x^{3}ln^{2}{10}} + \frac{3{x}^{lg(x)}*-2*0ln(x)}{x^{3}ln^{3}{10}} + \frac{3{x}^{lg(x)}}{x^{3}ln^{2}{10}(x)} + \frac{2*-3{x}^{lg(x)}ln(x)}{x^{4}ln{10}} + \frac{2({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)}{x^{3}ln{10}} + \frac{2{x}^{lg(x)}}{x^{3}(x)ln{10}} + \frac{2{x}^{lg(x)}ln(x)*-0}{x^{3}ln^{2}{10}} - \frac{3*-3{x}^{lg(x)}ln^{2}(x)}{x^{4}ln^{2}{10}} - \frac{3({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln^{2}(x)}{x^{3}ln^{2}{10}} - \frac{3{x}^{lg(x)}*-2*0ln^{2}(x)}{x^{3}ln^{3}{10}} - \frac{3{x}^{lg(x)}*2ln(x)}{x^{3}ln^{2}{10}(x)} + \frac{-3{x}^{lg(x)}ln^{3}(x)}{x^{4}ln^{3}{10}} + \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln^{3}(x)}{x^{3}ln^{3}{10}} + \frac{{x}^{lg(x)}*-3*0ln^{3}(x)}{x^{3}ln^{4}{10}} + \frac{{x}^{lg(x)}*3ln^{2}(x)}{x^{3}ln^{3}{10}(x)} - \frac{6*-3{x}^{lg(x)}}{x^{4}ln{10}} - \frac{6({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))}{x^{3}ln{10}} - \frac{6{x}^{lg(x)}*-0}{x^{3}ln^{2}{10}} + \frac{2*-3{x}^{lg(x)}lg(x)}{x^{4}} + \frac{2({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))lg(x)}{x^{3}} + \frac{2{x}^{lg(x)}}{x^{3}ln{10}(x)} - \frac{3*-3{x}^{lg(x)}lg^{2}(x)}{x^{4}} - \frac{3({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))lg^{2}(x)}{x^{3}} - \frac{3{x}^{lg(x)}*2lg(x)}{x^{3}ln{10}(x)} + \frac{-3{x}^{lg(x)}lg^{3}(x)}{x^{4}} + \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))lg^{3}(x)}{x^{3}} + \frac{{x}^{lg(x)}*3lg^{2}(x)}{x^{3}ln{10}(x)}\\=&\frac{11{x}^{lg(x)}ln(x)lg(x)}{x^{4}ln{10}} - \frac{18{x}^{lg(x)}ln^{2}(x)lg(x)}{x^{4}ln^{2}{10}} - \frac{6{x}^{lg(x)}ln(x)lg^{2}(x)}{x^{4}ln{10}} - \frac{36{x}^{lg(x)}lg(x)}{x^{4}ln{10}} + \frac{13{x}^{lg(x)}ln(x)lg(x)}{x^{4}ln^{2}{10}} + \frac{4{x}^{lg(x)}ln^{3}(x)lg(x)}{x^{4}ln^{3}{10}} + \frac{6{x}^{lg(x)}ln^{2}(x)lg^{2}(x)}{x^{4}ln^{2}{10}} + \frac{11{x}^{lg(x)}ln(x)lg(x)}{x^{4}ln^{2}{10}} + \frac{11{x}^{lg(x)}ln(x)lg(x)}{x^{4}ln{10}} + \frac{{x}^{lg(x)}ln(x)lg^{3}(x)}{x^{4}ln{10}} + \frac{12{x}^{lg(x)}lg^{2}(x)}{x^{4}ln{10}} - \frac{12{x}^{lg(x)}ln(x)lg^{2}(x)}{x^{4}ln{10}} + \frac{3{x}^{lg(x)}ln(x)lg^{3}(x)}{x^{4}ln{10}} - \frac{18{x}^{lg(x)}ln(x)}{x^{4}ln^{2}{10}} + \frac{9{x}^{lg(x)}ln^{2}(x)}{x^{4}ln^{3}{10}} - \frac{6{x}^{lg(x)}ln(x)}{x^{4}ln{10}} - \frac{18{x}^{lg(x)}ln(x)}{x^{4}ln^{2}{10}} + \frac{11{x}^{lg(x)}ln^{2}(x)}{x^{4}ln^{2}{10}} - \frac{6{x}^{lg(x)}ln^{3}(x)}{x^{4}ln^{3}{10}} + \frac{3{x}^{lg(x)}ln^{2}(x)}{x^{4}ln^{3}{10}} + \frac{{x}^{lg(x)}ln^{4}(x)}{x^{4}ln^{4}{10}} + \frac{22{x}^{lg(x)}}{x^{4}ln{10}} + \frac{12{x}^{lg(x)}}{x^{4}ln^{2}{10}} - \frac{6{x}^{lg(x)}lg(x)}{x^{4}} + \frac{11{x}^{lg(x)}lg^{2}(x)}{x^{4}} - \frac{6{x}^{lg(x)}lg^{3}(x)}{x^{4}} + \frac{{x}^{lg(x)}lg^{4}(x)}{x^{4}}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/2]Find\ the\ 4th\ derivative\ of\ function\ xlg(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( xlg(x)\right)}{dx}\\=&lg(x) + \frac{x}{ln{10}(x)}\\=&lg(x) + \frac{1}{ln{10}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( lg(x) + \frac{1}{ln{10}}\right)}{dx}\\=&\frac{1}{ln{10}(x)} + \frac{-0}{ln^{2}{10}}\\=&\frac{1}{xln{10}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{1}{xln{10}}\right)}{dx}\\=&\frac{-1}{x^{2}ln{10}} + \frac{-0}{xln^{2}{10}}\\=&\frac{-1}{x^{2}ln{10}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-1}{x^{2}ln{10}}\right)}{dx}\\=&\frac{--2}{x^{3}ln{10}} - \frac{-0}{x^{2}ln^{2}{10}}\\=&\frac{2}{x^{3}ln{10}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/2]Find\ the\ 4th\ derivative\ of\ function\ {x}^{lg(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( {x}^{lg(x)}\right)}{dx}\\=&({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))\\=&\frac{{x}^{lg(x)}ln(x)}{xln{10}} + \frac{{x}^{lg(x)}lg(x)}{x}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{{x}^{lg(x)}ln(x)}{xln{10}} + \frac{{x}^{lg(x)}lg(x)}{x}\right)}{dx}\\=&\frac{-{x}^{lg(x)}ln(x)}{x^{2}ln{10}} + \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)}{xln{10}} + \frac{{x}^{lg(x)}}{x(x)ln{10}} + \frac{{x}^{lg(x)}ln(x)*-0}{xln^{2}{10}} + \frac{-{x}^{lg(x)}lg(x)}{x^{2}} + \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))lg(x)}{x} + \frac{{x}^{lg(x)}}{xln{10}(x)}\\=&\frac{{x}^{lg(x)}ln(x)lg(x)}{x^{2}ln{10}} + \frac{{x}^{lg(x)}ln(x)lg(x)}{x^{2}ln{10}} - \frac{{x}^{lg(x)}ln(x)}{x^{2}ln{10}} + \frac{{x}^{lg(x)}ln^{2}(x)}{x^{2}ln^{2}{10}} - \frac{{x}^{lg(x)}lg(x)}{x^{2}} + \frac{2{x}^{lg(x)}}{x^{2}ln{10}} + \frac{{x}^{lg(x)}lg^{2}(x)}{x^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{{x}^{lg(x)}ln(x)lg(x)}{x^{2}ln{10}} + \frac{{x}^{lg(x)}ln(x)lg(x)}{x^{2}ln{10}} - \frac{{x}^{lg(x)}ln(x)}{x^{2}ln{10}} + \frac{{x}^{lg(x)}ln^{2}(x)}{x^{2}ln^{2}{10}} - \frac{{x}^{lg(x)}lg(x)}{x^{2}} + \frac{2{x}^{lg(x)}}{x^{2}ln{10}} + \frac{{x}^{lg(x)}lg^{2}(x)}{x^{2}}\right)}{dx}\\=&\frac{-2{x}^{lg(x)}ln(x)lg(x)}{x^{3}ln{10}} + \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)lg(x)}{x^{2}ln{10}} + \frac{{x}^{lg(x)}lg(x)}{x^{2}(x)ln{10}} + \frac{{x}^{lg(x)}ln(x)*-0lg(x)}{x^{2}ln^{2}{10}} + \frac{{x}^{lg(x)}ln(x)}{x^{2}ln{10}ln{10}(x)} + \frac{-2{x}^{lg(x)}ln(x)lg(x)}{x^{3}ln{10}} + \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)lg(x)}{x^{2}ln{10}} + \frac{{x}^{lg(x)}*-0ln(x)lg(x)}{x^{2}ln^{2}{10}} + \frac{{x}^{lg(x)}lg(x)}{x^{2}ln{10}(x)} + \frac{{x}^{lg(x)}ln(x)}{x^{2}ln{10}ln{10}(x)} - \frac{-2{x}^{lg(x)}ln(x)}{x^{3}ln{10}} - \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)}{x^{2}ln{10}} - \frac{{x}^{lg(x)}}{x^{2}(x)ln{10}} - \frac{{x}^{lg(x)}ln(x)*-0}{x^{2}ln^{2}{10}} + \frac{-2{x}^{lg(x)}ln^{2}(x)}{x^{3}ln^{2}{10}} + \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln^{2}(x)}{x^{2}ln^{2}{10}} + \frac{{x}^{lg(x)}*-2*0ln^{2}(x)}{x^{2}ln^{3}{10}} + \frac{{x}^{lg(x)}*2ln(x)}{x^{2}ln^{2}{10}(x)} - \frac{-2{x}^{lg(x)}lg(x)}{x^{3}} - \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))lg(x)}{x^{2}} - \frac{{x}^{lg(x)}}{x^{2}ln{10}(x)} + \frac{2*-2{x}^{lg(x)}}{x^{3}ln{10}} + \frac{2({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))}{x^{2}ln{10}} + \frac{2{x}^{lg(x)}*-0}{x^{2}ln^{2}{10}} + \frac{-2{x}^{lg(x)}lg^{2}(x)}{x^{3}} + \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))lg^{2}(x)}{x^{2}} + \frac{{x}^{lg(x)}*2lg(x)}{x^{2}ln{10}(x)}\\=&\frac{-3{x}^{lg(x)}ln(x)lg(x)}{x^{3}ln{10}} + \frac{3{x}^{lg(x)}ln^{2}(x)lg(x)}{x^{3}ln^{2}{10}} + \frac{{x}^{lg(x)}ln(x)lg^{2}(x)}{x^{3}ln{10}} + \frac{6{x}^{lg(x)}lg(x)}{x^{3}ln{10}} - \frac{3{x}^{lg(x)}ln(x)lg(x)}{x^{3}ln{10}} + \frac{2{x}^{lg(x)}ln(x)lg^{2}(x)}{x^{3}ln{10}} + \frac{3{x}^{lg(x)}ln(x)}{x^{3}ln^{2}{10}} + \frac{3{x}^{lg(x)}ln(x)}{x^{3}ln^{2}{10}} + \frac{2{x}^{lg(x)}ln(x)}{x^{3}ln{10}} - \frac{3{x}^{lg(x)}ln^{2}(x)}{x^{3}ln^{2}{10}} + \frac{{x}^{lg(x)}ln^{3}(x)}{x^{3}ln^{3}{10}} - \frac{6{x}^{lg(x)}}{x^{3}ln{10}} + \frac{2{x}^{lg(x)}lg(x)}{x^{3}} - \frac{3{x}^{lg(x)}lg^{2}(x)}{x^{3}} + \frac{{x}^{lg(x)}lg^{3}(x)}{x^{3}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-3{x}^{lg(x)}ln(x)lg(x)}{x^{3}ln{10}} + \frac{3{x}^{lg(x)}ln^{2}(x)lg(x)}{x^{3}ln^{2}{10}} + \frac{{x}^{lg(x)}ln(x)lg^{2}(x)}{x^{3}ln{10}} + \frac{6{x}^{lg(x)}lg(x)}{x^{3}ln{10}} - \frac{3{x}^{lg(x)}ln(x)lg(x)}{x^{3}ln{10}} + \frac{2{x}^{lg(x)}ln(x)lg^{2}(x)}{x^{3}ln{10}} + \frac{3{x}^{lg(x)}ln(x)}{x^{3}ln^{2}{10}} + \frac{3{x}^{lg(x)}ln(x)}{x^{3}ln^{2}{10}} + \frac{2{x}^{lg(x)}ln(x)}{x^{3}ln{10}} - \frac{3{x}^{lg(x)}ln^{2}(x)}{x^{3}ln^{2}{10}} + \frac{{x}^{lg(x)}ln^{3}(x)}{x^{3}ln^{3}{10}} - \frac{6{x}^{lg(x)}}{x^{3}ln{10}} + \frac{2{x}^{lg(x)}lg(x)}{x^{3}} - \frac{3{x}^{lg(x)}lg^{2}(x)}{x^{3}} + \frac{{x}^{lg(x)}lg^{3}(x)}{x^{3}}\right)}{dx}\\=&\frac{-3*-3{x}^{lg(x)}ln(x)lg(x)}{x^{4}ln{10}} - \frac{3({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)lg(x)}{x^{3}ln{10}} - \frac{3{x}^{lg(x)}lg(x)}{x^{3}(x)ln{10}} - \frac{3{x}^{lg(x)}ln(x)*-0lg(x)}{x^{3}ln^{2}{10}} - \frac{3{x}^{lg(x)}ln(x)}{x^{3}ln{10}ln{10}(x)} + \frac{3*-3{x}^{lg(x)}ln^{2}(x)lg(x)}{x^{4}ln^{2}{10}} + \frac{3({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln^{2}(x)lg(x)}{x^{3}ln^{2}{10}} + \frac{3{x}^{lg(x)}*-2*0ln^{2}(x)lg(x)}{x^{3}ln^{3}{10}} + \frac{3{x}^{lg(x)}*2ln(x)lg(x)}{x^{3}ln^{2}{10}(x)} + \frac{3{x}^{lg(x)}ln^{2}(x)}{x^{3}ln^{2}{10}ln{10}(x)} + \frac{-3{x}^{lg(x)}ln(x)lg^{2}(x)}{x^{4}ln{10}} + \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)lg^{2}(x)}{x^{3}ln{10}} + \frac{{x}^{lg(x)}lg^{2}(x)}{x^{3}(x)ln{10}} + \frac{{x}^{lg(x)}ln(x)*-0lg^{2}(x)}{x^{3}ln^{2}{10}} + \frac{{x}^{lg(x)}ln(x)*2lg(x)}{x^{3}ln{10}ln{10}(x)} + \frac{6*-3{x}^{lg(x)}lg(x)}{x^{4}ln{10}} + \frac{6({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))lg(x)}{x^{3}ln{10}} + \frac{6{x}^{lg(x)}*-0lg(x)}{x^{3}ln^{2}{10}} + \frac{6{x}^{lg(x)}}{x^{3}ln{10}ln{10}(x)} - \frac{3*-3{x}^{lg(x)}ln(x)lg(x)}{x^{4}ln{10}} - \frac{3({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)lg(x)}{x^{3}ln{10}} - \frac{3{x}^{lg(x)}*-0ln(x)lg(x)}{x^{3}ln^{2}{10}} - \frac{3{x}^{lg(x)}lg(x)}{x^{3}ln{10}(x)} - \frac{3{x}^{lg(x)}ln(x)}{x^{3}ln{10}ln{10}(x)} + \frac{2*-3{x}^{lg(x)}ln(x)lg^{2}(x)}{x^{4}ln{10}} + \frac{2({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)lg^{2}(x)}{x^{3}ln{10}} + \frac{2{x}^{lg(x)}*-0ln(x)lg^{2}(x)}{x^{3}ln^{2}{10}} + \frac{2{x}^{lg(x)}lg^{2}(x)}{x^{3}ln{10}(x)} + \frac{2{x}^{lg(x)}ln(x)*2lg(x)}{x^{3}ln{10}ln{10}(x)} + \frac{3*-3{x}^{lg(x)}ln(x)}{x^{4}ln^{2}{10}} + \frac{3({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)}{x^{3}ln^{2}{10}} + \frac{3{x}^{lg(x)}}{x^{3}(x)ln^{2}{10}} + \frac{3{x}^{lg(x)}ln(x)*-2*0}{x^{3}ln^{3}{10}} + \frac{3*-3{x}^{lg(x)}ln(x)}{x^{4}ln^{2}{10}} + \frac{3({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)}{x^{3}ln^{2}{10}} + \frac{3{x}^{lg(x)}*-2*0ln(x)}{x^{3}ln^{3}{10}} + \frac{3{x}^{lg(x)}}{x^{3}ln^{2}{10}(x)} + \frac{2*-3{x}^{lg(x)}ln(x)}{x^{4}ln{10}} + \frac{2({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)}{x^{3}ln{10}} + \frac{2{x}^{lg(x)}}{x^{3}(x)ln{10}} + \frac{2{x}^{lg(x)}ln(x)*-0}{x^{3}ln^{2}{10}} - \frac{3*-3{x}^{lg(x)}ln^{2}(x)}{x^{4}ln^{2}{10}} - \frac{3({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln^{2}(x)}{x^{3}ln^{2}{10}} - \frac{3{x}^{lg(x)}*-2*0ln^{2}(x)}{x^{3}ln^{3}{10}} - \frac{3{x}^{lg(x)}*2ln(x)}{x^{3}ln^{2}{10}(x)} + \frac{-3{x}^{lg(x)}ln^{3}(x)}{x^{4}ln^{3}{10}} + \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln^{3}(x)}{x^{3}ln^{3}{10}} + \frac{{x}^{lg(x)}*-3*0ln^{3}(x)}{x^{3}ln^{4}{10}} + \frac{{x}^{lg(x)}*3ln^{2}(x)}{x^{3}ln^{3}{10}(x)} - \frac{6*-3{x}^{lg(x)}}{x^{4}ln{10}} - \frac{6({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))}{x^{3}ln{10}} - \frac{6{x}^{lg(x)}*-0}{x^{3}ln^{2}{10}} + \frac{2*-3{x}^{lg(x)}lg(x)}{x^{4}} + \frac{2({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))lg(x)}{x^{3}} + \frac{2{x}^{lg(x)}}{x^{3}ln{10}(x)} - \frac{3*-3{x}^{lg(x)}lg^{2}(x)}{x^{4}} - \frac{3({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))lg^{2}(x)}{x^{3}} - \frac{3{x}^{lg(x)}*2lg(x)}{x^{3}ln{10}(x)} + \frac{-3{x}^{lg(x)}lg^{3}(x)}{x^{4}} + \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))lg^{3}(x)}{x^{3}} + \frac{{x}^{lg(x)}*3lg^{2}(x)}{x^{3}ln{10}(x)}\\=&\frac{11{x}^{lg(x)}ln(x)lg(x)}{x^{4}ln{10}} - \frac{18{x}^{lg(x)}ln^{2}(x)lg(x)}{x^{4}ln^{2}{10}} - \frac{6{x}^{lg(x)}ln(x)lg^{2}(x)}{x^{4}ln{10}} - \frac{36{x}^{lg(x)}lg(x)}{x^{4}ln{10}} + \frac{13{x}^{lg(x)}ln(x)lg(x)}{x^{4}ln^{2}{10}} + \frac{4{x}^{lg(x)}ln^{3}(x)lg(x)}{x^{4}ln^{3}{10}} + \frac{6{x}^{lg(x)}ln^{2}(x)lg^{2}(x)}{x^{4}ln^{2}{10}} + \frac{11{x}^{lg(x)}ln(x)lg(x)}{x^{4}ln^{2}{10}} + \frac{11{x}^{lg(x)}ln(x)lg(x)}{x^{4}ln{10}} + \frac{{x}^{lg(x)}ln(x)lg^{3}(x)}{x^{4}ln{10}} + \frac{12{x}^{lg(x)}lg^{2}(x)}{x^{4}ln{10}} - \frac{12{x}^{lg(x)}ln(x)lg^{2}(x)}{x^{4}ln{10}} + \frac{3{x}^{lg(x)}ln(x)lg^{3}(x)}{x^{4}ln{10}} - \frac{18{x}^{lg(x)}ln(x)}{x^{4}ln^{2}{10}} + \frac{9{x}^{lg(x)}ln^{2}(x)}{x^{4}ln^{3}{10}} - \frac{6{x}^{lg(x)}ln(x)}{x^{4}ln{10}} - \frac{18{x}^{lg(x)}ln(x)}{x^{4}ln^{2}{10}} + \frac{11{x}^{lg(x)}ln^{2}(x)}{x^{4}ln^{2}{10}} - \frac{6{x}^{lg(x)}ln^{3}(x)}{x^{4}ln^{3}{10}} + \frac{3{x}^{lg(x)}ln^{2}(x)}{x^{4}ln^{3}{10}} + \frac{{x}^{lg(x)}ln^{4}(x)}{x^{4}ln^{4}{10}} + \frac{22{x}^{lg(x)}}{x^{4}ln{10}} + \frac{12{x}^{lg(x)}}{x^{4}ln^{2}{10}} - \frac{6{x}^{lg(x)}lg(x)}{x^{4}} + \frac{11{x}^{lg(x)}lg^{2}(x)}{x^{4}} - \frac{6{x}^{lg(x)}lg^{3}(x)}{x^{4}} + \frac{{x}^{lg(x)}lg^{4}(x)}{x^{4}}\\ \end{split}\end{equation} \]