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 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ e^{x(e^{x} - 1) - \frac{1}{(2{x}^{2})}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}\right)}{dx}\\=&e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})\\=&e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x} + xe^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} - e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + \frac{e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{3}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x} + xe^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} - e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + \frac{e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{3}}\right)}{dx}\\=&e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{x} + e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x} + e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + xe^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + xe^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}}) - e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}}) + \frac{-3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{4}} + \frac{e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})}{x^{3}}\\=&e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}} + xe^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} - 2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x} + \frac{2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{3}} + 2e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}} + x^{2}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}} - xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x} + \frac{e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{2}} + e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + \frac{e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{2}} - \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{4}} - \frac{2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{3}} + \frac{e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{6}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}} + xe^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} - 2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x} + \frac{2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{3}} + 2e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}} + x^{2}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}} - xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x} + \frac{e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{2}} + e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + \frac{e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{2}} - \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{4}} - \frac{2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{3}} + \frac{e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{6}}\right)}{dx}\\=&e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{{x}*{2}} + e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}*2e^{x}e^{x} + e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + x*2e^{x}e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + xe^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}}) - 2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{x} - 2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x} + \frac{2*-3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{4}} + \frac{2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{x}}{x^{3}} + \frac{2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{3}} + 2e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + 2e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}}) + e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}} + xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{{x}*{2}} + xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}*2e^{x}e^{x} + 2xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}} + x^{2}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{{x}*{2}} + x^{2}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}*2e^{x}e^{x} - e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x} - xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{x} - xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x} + \frac{-2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{3}} + \frac{e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{x}}{x^{2}} + \frac{e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{2}} + e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}}) + \frac{-2e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{3}} + \frac{e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{2}} + \frac{e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})}{x^{2}} - \frac{3*-4e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{5}} - \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})}{x^{4}} - \frac{2*-3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{4}} - \frac{2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})}{x^{3}} + \frac{-6e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{7}} + \frac{e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})}{x^{6}}\\=&e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{3}} + xe^{{x}*{3}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + 2xe^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + 3e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + 2xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{3}} + 2x^{2}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{3}} + \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}}}{x^{3}} + \frac{2e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{2}} + xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}} + x^{2}e^{{x}*{3}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + x^{3}e^{{x}*{3}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + \frac{4e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}}}{x^{2}} + x^{2}e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} - \frac{6e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{3}} + \frac{e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x} + \frac{2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}}}{x} - x^{2}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}} + xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x} - \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{2}} - \frac{3e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{3}} + \frac{2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{5}} - e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} - \frac{9e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{4}} + \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{6}} + \frac{e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{5}} + \frac{9e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{4}} + \frac{12e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{5}} + \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{3}} - \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{6}} - \frac{9e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{7}} + \frac{e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{9}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{3}} + xe^{{x}*{3}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + 2xe^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + 3e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + 2xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{3}} + 2x^{2}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{3}} + \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}}}{x^{3}} + \frac{2e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{2}} + xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}} + x^{2}e^{{x}*{3}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + x^{3}e^{{x}*{3}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + \frac{4e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}}}{x^{2}} + x^{2}e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} - \frac{6e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{3}} + \frac{e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x} + \frac{2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}}}{x} - x^{2}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}} + xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x} - \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{2}} - \frac{3e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{3}} + \frac{2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{5}} - e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} - \frac{9e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{4}} + \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{6}} + \frac{e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{5}} + \frac{9e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{4}} + \frac{12e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{5}} + \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{3}} - \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{6}} - \frac{9e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{7}} + \frac{e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{9}}\right)}{dx}\\=&e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{{x}*{3}} + e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}*3e^{{x}*{2}}e^{x} + e^{{x}*{3}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + x*3e^{{x}*{2}}e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + xe^{{x}*{3}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}}) + 2e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + 2x*2e^{x}e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + 2xe^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}}) + 3*2e^{x}e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + 3e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}}) + 2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{3}} + 2xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{{x}*{3}} + 2xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}*3e^{{x}*{2}}e^{x} + 2*2xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{3}} + 2x^{2}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{{x}*{3}} + 2x^{2}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}*3e^{{x}*{2}}e^{x} + \frac{3*-3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}}}{x^{4}} + \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{{x}*{2}}}{x^{3}} + \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}*2e^{x}e^{x}}{x^{3}} + \frac{2*-2e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{3}} + \frac{2*2e^{x}e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{2}} + \frac{2e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})}{x^{2}} + e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}} + xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{{x}*{2}} + xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}*2e^{x}e^{x} + 2xe^{{x}*{3}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + x^{2}*3e^{{x}*{2}}e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + x^{2}e^{{x}*{3}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}}) + 3x^{2}e^{{x}*{3}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + x^{3}*3e^{{x}*{2}}e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + x^{3}e^{{x}*{3}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}}) + \frac{4*-2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}}}{x^{3}} + \frac{4e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{{x}*{2}}}{x^{2}} + \frac{4e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}*2e^{x}e^{x}}{x^{2}} + 2xe^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + x^{2}*2e^{x}e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + x^{2}e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}}) - \frac{6*-3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{4}} - \frac{6e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{x}}{x^{3}} - \frac{6e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{3}} + \frac{-e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{2}} + \frac{2e^{x}e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x} + \frac{e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})}{x} + \frac{2*-e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}}}{x^{2}} + \frac{2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{{x}*{2}}}{x} + \frac{2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}*2e^{x}e^{x}}{x} - 2xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}} - x^{2}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{{x}*{2}} - x^{2}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}*2e^{x}e^{x} + e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x} + xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{x} + xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x} - \frac{3*-2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{3}} - \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{x}}{x^{2}} - \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{2}} - \frac{3*-3e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{4}} - \frac{3e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{3}} - \frac{3e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})}{x^{3}} + \frac{2*-5e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{6}} + \frac{2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{x}}{x^{5}} + \frac{2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{5}} - e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}}) - \frac{9*-4e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{5}} - \frac{9e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{x}}{x^{4}} - \frac{9e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{4}} + \frac{3*-6e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{7}} + \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{x}}{x^{6}} + \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{6}} + \frac{-5e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{6}} + \frac{e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{5}} + \frac{e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})}{x^{5}} + \frac{9*-4e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{5}} + \frac{9e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})}{x^{4}} + \frac{12*-5e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{6}} + \frac{12e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})}{x^{5}} + \frac{3*-3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{4}} + \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})}{x^{3}} - \frac{3*-6e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{7}} - \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})}{x^{6}} - \frac{9*-7e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{8}} - \frac{9e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})}{x^{7}} + \frac{-9e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{10}} + \frac{e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})}{x^{9}}\\=&e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{4}} + xe^{{x}*{4}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + 6e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{3}} + 11xe^{{x}*{3}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + 6e^{{x}*{3}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + 8xe^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + 3xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{4}} + 4x^{2}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{4}} + 7xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{3}} - x^{2}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{3}} + 8e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + 2x^{2}e^{{x}*{4}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + 2x^{3}e^{{x}*{4}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} - 2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}} + 13x^{2}e^{{x}*{3}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + \frac{4e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{3}}}{x^{3}} + \frac{3e^{{x}*{3}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{2}} + \frac{9e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{3}}}{x^{2}} + \frac{8e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{3}}}{x} - \frac{7e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{3}} + 2x^{3}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{4}} + \frac{5e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{2}} + x^{4}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{4}} - \frac{17e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}}}{x^{3}} + \frac{4e^{{x}*{3}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x} - 4xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}} - \frac{11e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}}}{x^{2}} + 2x^{3}e^{{x}*{3}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + \frac{9e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}}}{x^{5}} - \frac{14e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}}}{x^{4}} + x^{2}e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + \frac{6e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}}}{x^{6}} + \frac{3e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x} + \frac{2e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{4}} + \frac{3e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{5}} + \frac{36e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{4}} + \frac{18e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{3}} - \frac{25e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{6}} - \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}}}{x} - xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x} + \frac{4e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{2}} + \frac{42e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{5}} + \frac{12e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{4}} + \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{8}} + e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} - \frac{36e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{7}} + \frac{4e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{9}} - \frac{11e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{6}} + \frac{e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{8}} - \frac{18e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{4}} - \frac{4e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{3}} + \frac{36e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{7}} - \frac{54e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{6}} + \frac{75e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{8}} - \frac{4e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{9}} - \frac{48e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{5}} - \frac{18e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{10}} + \frac{e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{12}}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ e^{x(e^{x} - 1) - \frac{1}{(2{x}^{2})}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}\right)}{dx}\\=&e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})\\=&e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x} + xe^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} - e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + \frac{e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{3}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x} + xe^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} - e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + \frac{e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{3}}\right)}{dx}\\=&e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{x} + e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x} + e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + xe^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + xe^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}}) - e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}}) + \frac{-3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{4}} + \frac{e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})}{x^{3}}\\=&e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}} + xe^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} - 2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x} + \frac{2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{3}} + 2e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}} + x^{2}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}} - xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x} + \frac{e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{2}} + e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + \frac{e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{2}} - \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{4}} - \frac{2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{3}} + \frac{e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{6}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}} + xe^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} - 2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x} + \frac{2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{3}} + 2e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}} + x^{2}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}} - xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x} + \frac{e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{2}} + e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + \frac{e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{2}} - \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{4}} - \frac{2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{3}} + \frac{e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{6}}\right)}{dx}\\=&e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{{x}*{2}} + e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}*2e^{x}e^{x} + e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + x*2e^{x}e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + xe^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}}) - 2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{x} - 2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x} + \frac{2*-3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{4}} + \frac{2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{x}}{x^{3}} + \frac{2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{3}} + 2e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + 2e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}}) + e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}} + xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{{x}*{2}} + xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}*2e^{x}e^{x} + 2xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}} + x^{2}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{{x}*{2}} + x^{2}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}*2e^{x}e^{x} - e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x} - xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{x} - xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x} + \frac{-2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{3}} + \frac{e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{x}}{x^{2}} + \frac{e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{2}} + e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}}) + \frac{-2e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{3}} + \frac{e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{2}} + \frac{e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})}{x^{2}} - \frac{3*-4e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{5}} - \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})}{x^{4}} - \frac{2*-3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{4}} - \frac{2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})}{x^{3}} + \frac{-6e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{7}} + \frac{e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})}{x^{6}}\\=&e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{3}} + xe^{{x}*{3}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + 2xe^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + 3e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + 2xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{3}} + 2x^{2}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{3}} + \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}}}{x^{3}} + \frac{2e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{2}} + xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}} + x^{2}e^{{x}*{3}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + x^{3}e^{{x}*{3}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + \frac{4e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}}}{x^{2}} + x^{2}e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} - \frac{6e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{3}} + \frac{e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x} + \frac{2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}}}{x} - x^{2}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}} + xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x} - \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{2}} - \frac{3e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{3}} + \frac{2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{5}} - e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} - \frac{9e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{4}} + \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{6}} + \frac{e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{5}} + \frac{9e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{4}} + \frac{12e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{5}} + \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{3}} - \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{6}} - \frac{9e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{7}} + \frac{e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{9}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{3}} + xe^{{x}*{3}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + 2xe^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + 3e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + 2xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{3}} + 2x^{2}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{3}} + \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}}}{x^{3}} + \frac{2e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{2}} + xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}} + x^{2}e^{{x}*{3}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + x^{3}e^{{x}*{3}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + \frac{4e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}}}{x^{2}} + x^{2}e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} - \frac{6e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{3}} + \frac{e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x} + \frac{2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}}}{x} - x^{2}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}} + xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x} - \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{2}} - \frac{3e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{3}} + \frac{2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{5}} - e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} - \frac{9e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{4}} + \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{6}} + \frac{e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{5}} + \frac{9e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{4}} + \frac{12e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{5}} + \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{3}} - \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{6}} - \frac{9e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{7}} + \frac{e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{9}}\right)}{dx}\\=&e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{{x}*{3}} + e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}*3e^{{x}*{2}}e^{x} + e^{{x}*{3}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + x*3e^{{x}*{2}}e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + xe^{{x}*{3}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}}) + 2e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + 2x*2e^{x}e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + 2xe^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}}) + 3*2e^{x}e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + 3e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}}) + 2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{3}} + 2xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{{x}*{3}} + 2xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}*3e^{{x}*{2}}e^{x} + 2*2xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{3}} + 2x^{2}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{{x}*{3}} + 2x^{2}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}*3e^{{x}*{2}}e^{x} + \frac{3*-3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}}}{x^{4}} + \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{{x}*{2}}}{x^{3}} + \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}*2e^{x}e^{x}}{x^{3}} + \frac{2*-2e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{3}} + \frac{2*2e^{x}e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{2}} + \frac{2e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})}{x^{2}} + e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}} + xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{{x}*{2}} + xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}*2e^{x}e^{x} + 2xe^{{x}*{3}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + x^{2}*3e^{{x}*{2}}e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + x^{2}e^{{x}*{3}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}}) + 3x^{2}e^{{x}*{3}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + x^{3}*3e^{{x}*{2}}e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + x^{3}e^{{x}*{3}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}}) + \frac{4*-2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}}}{x^{3}} + \frac{4e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{{x}*{2}}}{x^{2}} + \frac{4e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}*2e^{x}e^{x}}{x^{2}} + 2xe^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + x^{2}*2e^{x}e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + x^{2}e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}}) - \frac{6*-3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{4}} - \frac{6e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{x}}{x^{3}} - \frac{6e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{3}} + \frac{-e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{2}} + \frac{2e^{x}e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x} + \frac{e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})}{x} + \frac{2*-e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}}}{x^{2}} + \frac{2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{{x}*{2}}}{x} + \frac{2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}*2e^{x}e^{x}}{x} - 2xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}} - x^{2}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{{x}*{2}} - x^{2}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}*2e^{x}e^{x} + e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x} + xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{x} + xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x} - \frac{3*-2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{3}} - \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{x}}{x^{2}} - \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{2}} - \frac{3*-3e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{4}} - \frac{3e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{3}} - \frac{3e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})}{x^{3}} + \frac{2*-5e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{6}} + \frac{2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{x}}{x^{5}} + \frac{2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{5}} - e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}}) - \frac{9*-4e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{5}} - \frac{9e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{x}}{x^{4}} - \frac{9e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{4}} + \frac{3*-6e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{7}} + \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})e^{x}}{x^{6}} + \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{6}} + \frac{-5e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{6}} + \frac{e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{5}} + \frac{e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})}{x^{5}} + \frac{9*-4e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{5}} + \frac{9e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})}{x^{4}} + \frac{12*-5e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{6}} + \frac{12e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})}{x^{5}} + \frac{3*-3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{4}} + \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})}{x^{3}} - \frac{3*-6e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{7}} - \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})}{x^{6}} - \frac{9*-7e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{8}} - \frac{9e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})}{x^{7}} + \frac{-9e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{10}} + \frac{e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}(e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}})}{x^{9}}\\=&e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{4}} + xe^{{x}*{4}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + 6e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{3}} + 11xe^{{x}*{3}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + 6e^{{x}*{3}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + 8xe^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + 3xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{4}} + 4x^{2}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{4}} + 7xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{3}} - x^{2}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{3}} + 8e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + 2x^{2}e^{{x}*{4}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + 2x^{3}e^{{x}*{4}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} - 2e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}} + 13x^{2}e^{{x}*{3}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + \frac{4e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{3}}}{x^{3}} + \frac{3e^{{x}*{3}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{2}} + \frac{9e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{3}}}{x^{2}} + \frac{8e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{3}}}{x} - \frac{7e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{3}} + 2x^{3}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{4}} + \frac{5e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{2}} + x^{4}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{4}} - \frac{17e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}}}{x^{3}} + \frac{4e^{{x}*{3}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x} - 4xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}} - \frac{11e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}}}{x^{2}} + 2x^{3}e^{{x}*{3}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + \frac{9e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}}}{x^{5}} - \frac{14e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}}}{x^{4}} + x^{2}e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} + \frac{6e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}}}{x^{6}} + \frac{3e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x} + \frac{2e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{4}} + \frac{3e^{{x}*{2}}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{5}} + \frac{36e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{4}} + \frac{18e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{3}} - \frac{25e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{6}} - \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{{x}*{2}}}{x} - xe^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x} + \frac{4e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{2}} + \frac{42e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{5}} + \frac{12e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{4}} + \frac{3e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{8}} + e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}} - \frac{36e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{7}} + \frac{4e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}e^{x}}{x^{9}} - \frac{11e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{6}} + \frac{e^{x}e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{8}} - \frac{18e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{4}} - \frac{4e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{3}} + \frac{36e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{7}} - \frac{54e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{6}} + \frac{75e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{8}} - \frac{4e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{9}} - \frac{48e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{5}} - \frac{18e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{10}} + \frac{e^{xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}}}{x^{12}}\\ \end{split}\end{equation} \]