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.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ lg(e^{xx} - 2e^{x} - 3)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = lg(e^{x^{2}} - 2e^{x} - 3)\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( lg(e^{x^{2}} - 2e^{x} - 3)\right)}{dx}\\=&\frac{(e^{x^{2}}*2x - 2e^{x} + 0)}{ln{10}(e^{x^{2}} - 2e^{x} - 3)}\\=&\frac{2xe^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} - \frac{2e^{x}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{2xe^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} - \frac{2e^{x}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}}\right)}{dx}\\=&\frac{2(\frac{-(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{2}})xe^{x^{2}}}{ln{10}} + \frac{2e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} + \frac{2xe^{x^{2}}*2x}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} + \frac{2xe^{x^{2}}*-0}{(e^{x^{2}} - 2e^{x} - 3)ln^{2}{10}} - \frac{2(\frac{-(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{2}})e^{x}}{ln{10}} - \frac{2e^{x}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} - \frac{2e^{x}*-0}{(e^{x^{2}} - 2e^{x} - 3)ln^{2}{10}}\\=&\frac{-4x^{2}e^{{x^{2}}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{4xe^{x}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{2e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} + \frac{4x^{2}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} + \frac{4xe^{x^{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{4e^{{x}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{2e^{x}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-4x^{2}e^{{x^{2}}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{4xe^{x}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{2e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} + \frac{4x^{2}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} + \frac{4xe^{x^{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{4e^{{x}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{2e^{x}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}}\right)}{dx}\\=&\frac{-4(\frac{-2(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{3}})x^{2}e^{{x^{2}}*{2}}}{ln{10}} - \frac{4*2xe^{{x^{2}}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{4x^{2}*2e^{x^{2}}e^{x^{2}}*2x}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{4x^{2}e^{{x^{2}}*{2}}*-0}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln^{2}{10}} + \frac{4(\frac{-2(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{3}})xe^{x}e^{x^{2}}}{ln{10}} + \frac{4e^{x}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{4xe^{x}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{4xe^{x}e^{x^{2}}*2x}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{4xe^{x}e^{x^{2}}*-0}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln^{2}{10}} + \frac{2(\frac{-(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{2}})e^{x^{2}}}{ln{10}} + \frac{2e^{x^{2}}*2x}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} + \frac{2e^{x^{2}}*-0}{(e^{x^{2}} - 2e^{x} - 3)ln^{2}{10}} + \frac{4(\frac{-(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{2}})x^{2}e^{x^{2}}}{ln{10}} + \frac{4*2xe^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} + \frac{4x^{2}e^{x^{2}}*2x}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} + \frac{4x^{2}e^{x^{2}}*-0}{(e^{x^{2}} - 2e^{x} - 3)ln^{2}{10}} + \frac{4(\frac{-2(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{3}})xe^{x^{2}}e^{x}}{ln{10}} + \frac{4e^{x^{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{4xe^{x^{2}}*2xe^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{4xe^{x^{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{4xe^{x^{2}}e^{x}*-0}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln^{2}{10}} - \frac{4(\frac{-2(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{3}})e^{{x}*{2}}}{ln{10}} - \frac{4*2e^{x}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{4e^{{x}*{2}}*-0}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln^{2}{10}} - \frac{2(\frac{-(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{2}})e^{x}}{ln{10}} - \frac{2e^{x}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} - \frac{2e^{x}*-0}{(e^{x^{2}} - 2e^{x} - 3)ln^{2}{10}}\\=&\frac{16x^{3}e^{{x^{2}}*{3}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} - \frac{16x^{2}e^{x}e^{{x^{2}}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} - \frac{12xe^{{x^{2}}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{24x^{3}e^{{x^{2}}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{32x^{2}e^{{x^{2}}*{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} + \frac{32xe^{{x}*{2}}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} + \frac{8e^{x}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{8xe^{x}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{16x^{2}e^{x^{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{12xe^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} + \frac{8x^{2}e^{x}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{8x^{3}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} + \frac{4e^{x^{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{16xe^{x^{2}}e^{{x}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} - \frac{16e^{{x}*{3}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} - \frac{12e^{{x}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{4xe^{x^{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{2e^{x}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{16x^{3}e^{{x^{2}}*{3}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} - \frac{16x^{2}e^{x}e^{{x^{2}}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} - \frac{12xe^{{x^{2}}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{24x^{3}e^{{x^{2}}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{32x^{2}e^{{x^{2}}*{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} + \frac{32xe^{{x}*{2}}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} + \frac{8e^{x}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{8xe^{x}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{16x^{2}e^{x^{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{12xe^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} + \frac{8x^{2}e^{x}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{8x^{3}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} + \frac{4e^{x^{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{16xe^{x^{2}}e^{{x}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} - \frac{16e^{{x}*{3}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} - \frac{12e^{{x}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{4xe^{x^{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{2e^{x}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}}\right)}{dx}\\=&\frac{16(\frac{-3(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{4}})x^{3}e^{{x^{2}}*{3}}}{ln{10}} + \frac{16*3x^{2}e^{{x^{2}}*{3}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} + \frac{16x^{3}*3e^{{x^{2}}*{2}}e^{x^{2}}*2x}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} + \frac{16x^{3}e^{{x^{2}}*{3}}*-0}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln^{2}{10}} - \frac{16(\frac{-3(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{4}})x^{2}e^{x}e^{{x^{2}}*{2}}}{ln{10}} - \frac{16*2xe^{x}e^{{x^{2}}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} - \frac{16x^{2}e^{x}e^{{x^{2}}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} - \frac{16x^{2}e^{x}*2e^{x^{2}}e^{x^{2}}*2x}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} - \frac{16x^{2}e^{x}e^{{x^{2}}*{2}}*-0}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln^{2}{10}} - \frac{12(\frac{-2(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{3}})xe^{{x^{2}}*{2}}}{ln{10}} - \frac{12e^{{x^{2}}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{12x*2e^{x^{2}}e^{x^{2}}*2x}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{12xe^{{x^{2}}*{2}}*-0}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln^{2}{10}} - \frac{24(\frac{-2(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{3}})x^{3}e^{{x^{2}}*{2}}}{ln{10}} - \frac{24*3x^{2}e^{{x^{2}}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{24x^{3}*2e^{x^{2}}e^{x^{2}}*2x}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{24x^{3}e^{{x^{2}}*{2}}*-0}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln^{2}{10}} - \frac{32(\frac{-3(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{4}})x^{2}e^{{x^{2}}*{2}}e^{x}}{ln{10}} - \frac{32*2xe^{{x^{2}}*{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} - \frac{32x^{2}*2e^{x^{2}}e^{x^{2}}*2xe^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} - \frac{32x^{2}e^{{x^{2}}*{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} - \frac{32x^{2}e^{{x^{2}}*{2}}e^{x}*-0}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln^{2}{10}} + \frac{32(\frac{-3(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{4}})xe^{{x}*{2}}e^{x^{2}}}{ln{10}} + \frac{32e^{{x}*{2}}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} + \frac{32x*2e^{x}e^{x}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} + \frac{32xe^{{x}*{2}}e^{x^{2}}*2x}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} + \frac{32xe^{{x}*{2}}e^{x^{2}}*-0}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln^{2}{10}} + \frac{8(\frac{-2(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{3}})e^{x}e^{x^{2}}}{ln{10}} + \frac{8e^{x}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{8e^{x}e^{x^{2}}*2x}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{8e^{x}e^{x^{2}}*-0}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln^{2}{10}} + \frac{8(\frac{-2(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{3}})xe^{x}e^{x^{2}}}{ln{10}} + \frac{8e^{x}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{8xe^{x}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{8xe^{x}e^{x^{2}}*2x}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{8xe^{x}e^{x^{2}}*-0}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln^{2}{10}} + \frac{16(\frac{-2(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{3}})x^{2}e^{x^{2}}e^{x}}{ln{10}} + \frac{16*2xe^{x^{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{16x^{2}e^{x^{2}}*2xe^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{16x^{2}e^{x^{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{16x^{2}e^{x^{2}}e^{x}*-0}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln^{2}{10}} + \frac{12(\frac{-(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{2}})xe^{x^{2}}}{ln{10}} + \frac{12e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} + \frac{12xe^{x^{2}}*2x}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} + \frac{12xe^{x^{2}}*-0}{(e^{x^{2}} - 2e^{x} - 3)ln^{2}{10}} + \frac{8(\frac{-2(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{3}})x^{2}e^{x}e^{x^{2}}}{ln{10}} + \frac{8*2xe^{x}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{8x^{2}e^{x}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{8x^{2}e^{x}e^{x^{2}}*2x}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{8x^{2}e^{x}e^{x^{2}}*-0}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln^{2}{10}} + \frac{8(\frac{-(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{2}})x^{3}e^{x^{2}}}{ln{10}} + \frac{8*3x^{2}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} + \frac{8x^{3}e^{x^{2}}*2x}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} + \frac{8x^{3}e^{x^{2}}*-0}{(e^{x^{2}} - 2e^{x} - 3)ln^{2}{10}} + \frac{4(\frac{-2(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{3}})e^{x^{2}}e^{x}}{ln{10}} + \frac{4e^{x^{2}}*2xe^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{4e^{x^{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{4e^{x^{2}}e^{x}*-0}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln^{2}{10}} + \frac{16(\frac{-3(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{4}})xe^{x^{2}}e^{{x}*{2}}}{ln{10}} + \frac{16e^{x^{2}}e^{{x}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} + \frac{16xe^{x^{2}}*2xe^{{x}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} + \frac{16xe^{x^{2}}*2e^{x}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} + \frac{16xe^{x^{2}}e^{{x}*{2}}*-0}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln^{2}{10}} - \frac{16(\frac{-3(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{4}})e^{{x}*{3}}}{ln{10}} - \frac{16*3e^{{x}*{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} - \frac{16e^{{x}*{3}}*-0}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln^{2}{10}} - \frac{12(\frac{-2(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{3}})e^{{x}*{2}}}{ln{10}} - \frac{12*2e^{x}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{12e^{{x}*{2}}*-0}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln^{2}{10}} + \frac{4(\frac{-2(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{3}})xe^{x^{2}}e^{x}}{ln{10}} + \frac{4e^{x^{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{4xe^{x^{2}}*2xe^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{4xe^{x^{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{4xe^{x^{2}}e^{x}*-0}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln^{2}{10}} - \frac{2(\frac{-(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{2}})e^{x}}{ln{10}} - \frac{2e^{x}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} - \frac{2e^{x}*-0}{(e^{x^{2}} - 2e^{x} - 3)ln^{2}{10}}\\=&\frac{-96x^{4}e^{{x^{2}}*{4}}}{(e^{x^{2}} - 2e^{x} - 3)^{4}ln{10}} + \frac{96x^{3}e^{x}e^{{x^{2}}*{3}}}{(e^{x^{2}} - 2e^{x} - 3)^{4}ln{10}} + \frac{96x^{2}e^{{x^{2}}*{3}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} + \frac{192x^{4}e^{{x^{2}}*{3}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} + \frac{288x^{3}e^{{x^{2}}*{3}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{4}ln{10}} - \frac{288x^{2}e^{{x}*{2}}e^{{x^{2}}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{4}ln{10}} - \frac{80xe^{x}e^{{x^{2}}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} - \frac{48x^{2}e^{x}e^{{x^{2}}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} - \frac{160x^{3}e^{x}e^{{x^{2}}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} - \frac{12e^{{x^{2}}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{144x^{2}e^{{x^{2}}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{112x^{4}e^{{x^{2}}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{112xe^{{x^{2}}*{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} - \frac{224x^{3}e^{{x^{2}}*{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} - \frac{288x^{2}e^{{x^{2}}*{2}}e^{{x}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{4}ln{10}} + \frac{288xe^{{x}*{3}}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{4}ln{10}} + \frac{80e^{{x}*{2}}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} + \frac{144xe^{{x}*{2}}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} + \frac{96x^{2}e^{x^{2}}e^{{x}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} + \frac{20e^{x}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{60xe^{x^{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{48x^{2}e^{{x^{2}}*{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} + \frac{52xe^{x}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{24x^{2}e^{x^{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{96x^{2}e^{{x}*{2}}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} + \frac{48x^{3}e^{x^{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{24x^{2}e^{x}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{12e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} + \frac{48x^{2}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} + \frac{16x^{3}e^{x}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{16x^{4}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} + \frac{16e^{x^{2}}e^{{x}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} + \frac{96xe^{x^{2}}e^{{x}*{3}}}{(e^{x^{2}} - 2e^{x} - 3)^{4}ln{10}} - \frac{96e^{{x}*{4}}}{(e^{x^{2}} - 2e^{x} - 3)^{4}ln{10}} - \frac{96e^{{x}*{3}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} + \frac{48xe^{x^{2}}e^{{x}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} - \frac{28e^{{x}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{4e^{x^{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{2e^{x}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ lg(e^{xx} - 2e^{x} - 3)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = lg(e^{x^{2}} - 2e^{x} - 3)\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( lg(e^{x^{2}} - 2e^{x} - 3)\right)}{dx}\\=&\frac{(e^{x^{2}}*2x - 2e^{x} + 0)}{ln{10}(e^{x^{2}} - 2e^{x} - 3)}\\=&\frac{2xe^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} - \frac{2e^{x}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{2xe^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} - \frac{2e^{x}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}}\right)}{dx}\\=&\frac{2(\frac{-(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{2}})xe^{x^{2}}}{ln{10}} + \frac{2e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} + \frac{2xe^{x^{2}}*2x}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} + \frac{2xe^{x^{2}}*-0}{(e^{x^{2}} - 2e^{x} - 3)ln^{2}{10}} - \frac{2(\frac{-(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{2}})e^{x}}{ln{10}} - \frac{2e^{x}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} - \frac{2e^{x}*-0}{(e^{x^{2}} - 2e^{x} - 3)ln^{2}{10}}\\=&\frac{-4x^{2}e^{{x^{2}}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{4xe^{x}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{2e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} + \frac{4x^{2}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} + \frac{4xe^{x^{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{4e^{{x}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{2e^{x}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-4x^{2}e^{{x^{2}}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{4xe^{x}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{2e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} + \frac{4x^{2}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} + \frac{4xe^{x^{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{4e^{{x}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{2e^{x}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}}\right)}{dx}\\=&\frac{-4(\frac{-2(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{3}})x^{2}e^{{x^{2}}*{2}}}{ln{10}} - \frac{4*2xe^{{x^{2}}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{4x^{2}*2e^{x^{2}}e^{x^{2}}*2x}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{4x^{2}e^{{x^{2}}*{2}}*-0}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln^{2}{10}} + \frac{4(\frac{-2(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{3}})xe^{x}e^{x^{2}}}{ln{10}} + \frac{4e^{x}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{4xe^{x}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{4xe^{x}e^{x^{2}}*2x}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{4xe^{x}e^{x^{2}}*-0}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln^{2}{10}} + \frac{2(\frac{-(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{2}})e^{x^{2}}}{ln{10}} + \frac{2e^{x^{2}}*2x}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} + \frac{2e^{x^{2}}*-0}{(e^{x^{2}} - 2e^{x} - 3)ln^{2}{10}} + \frac{4(\frac{-(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{2}})x^{2}e^{x^{2}}}{ln{10}} + \frac{4*2xe^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} + \frac{4x^{2}e^{x^{2}}*2x}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} + \frac{4x^{2}e^{x^{2}}*-0}{(e^{x^{2}} - 2e^{x} - 3)ln^{2}{10}} + \frac{4(\frac{-2(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{3}})xe^{x^{2}}e^{x}}{ln{10}} + \frac{4e^{x^{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{4xe^{x^{2}}*2xe^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{4xe^{x^{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{4xe^{x^{2}}e^{x}*-0}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln^{2}{10}} - \frac{4(\frac{-2(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{3}})e^{{x}*{2}}}{ln{10}} - \frac{4*2e^{x}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{4e^{{x}*{2}}*-0}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln^{2}{10}} - \frac{2(\frac{-(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{2}})e^{x}}{ln{10}} - \frac{2e^{x}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} - \frac{2e^{x}*-0}{(e^{x^{2}} - 2e^{x} - 3)ln^{2}{10}}\\=&\frac{16x^{3}e^{{x^{2}}*{3}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} - \frac{16x^{2}e^{x}e^{{x^{2}}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} - \frac{12xe^{{x^{2}}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{24x^{3}e^{{x^{2}}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{32x^{2}e^{{x^{2}}*{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} + \frac{32xe^{{x}*{2}}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} + \frac{8e^{x}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{8xe^{x}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{16x^{2}e^{x^{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{12xe^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} + \frac{8x^{2}e^{x}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{8x^{3}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} + \frac{4e^{x^{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{16xe^{x^{2}}e^{{x}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} - \frac{16e^{{x}*{3}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} - \frac{12e^{{x}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{4xe^{x^{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{2e^{x}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{16x^{3}e^{{x^{2}}*{3}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} - \frac{16x^{2}e^{x}e^{{x^{2}}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} - \frac{12xe^{{x^{2}}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{24x^{3}e^{{x^{2}}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{32x^{2}e^{{x^{2}}*{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} + \frac{32xe^{{x}*{2}}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} + \frac{8e^{x}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{8xe^{x}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{16x^{2}e^{x^{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{12xe^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} + \frac{8x^{2}e^{x}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{8x^{3}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} + \frac{4e^{x^{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{16xe^{x^{2}}e^{{x}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} - \frac{16e^{{x}*{3}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} - \frac{12e^{{x}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{4xe^{x^{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{2e^{x}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}}\right)}{dx}\\=&\frac{16(\frac{-3(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{4}})x^{3}e^{{x^{2}}*{3}}}{ln{10}} + \frac{16*3x^{2}e^{{x^{2}}*{3}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} + \frac{16x^{3}*3e^{{x^{2}}*{2}}e^{x^{2}}*2x}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} + \frac{16x^{3}e^{{x^{2}}*{3}}*-0}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln^{2}{10}} - \frac{16(\frac{-3(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{4}})x^{2}e^{x}e^{{x^{2}}*{2}}}{ln{10}} - \frac{16*2xe^{x}e^{{x^{2}}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} - \frac{16x^{2}e^{x}e^{{x^{2}}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} - \frac{16x^{2}e^{x}*2e^{x^{2}}e^{x^{2}}*2x}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} - \frac{16x^{2}e^{x}e^{{x^{2}}*{2}}*-0}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln^{2}{10}} - \frac{12(\frac{-2(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{3}})xe^{{x^{2}}*{2}}}{ln{10}} - \frac{12e^{{x^{2}}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{12x*2e^{x^{2}}e^{x^{2}}*2x}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{12xe^{{x^{2}}*{2}}*-0}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln^{2}{10}} - \frac{24(\frac{-2(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{3}})x^{3}e^{{x^{2}}*{2}}}{ln{10}} - \frac{24*3x^{2}e^{{x^{2}}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{24x^{3}*2e^{x^{2}}e^{x^{2}}*2x}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{24x^{3}e^{{x^{2}}*{2}}*-0}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln^{2}{10}} - \frac{32(\frac{-3(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{4}})x^{2}e^{{x^{2}}*{2}}e^{x}}{ln{10}} - \frac{32*2xe^{{x^{2}}*{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} - \frac{32x^{2}*2e^{x^{2}}e^{x^{2}}*2xe^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} - \frac{32x^{2}e^{{x^{2}}*{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} - \frac{32x^{2}e^{{x^{2}}*{2}}e^{x}*-0}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln^{2}{10}} + \frac{32(\frac{-3(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{4}})xe^{{x}*{2}}e^{x^{2}}}{ln{10}} + \frac{32e^{{x}*{2}}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} + \frac{32x*2e^{x}e^{x}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} + \frac{32xe^{{x}*{2}}e^{x^{2}}*2x}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} + \frac{32xe^{{x}*{2}}e^{x^{2}}*-0}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln^{2}{10}} + \frac{8(\frac{-2(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{3}})e^{x}e^{x^{2}}}{ln{10}} + \frac{8e^{x}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{8e^{x}e^{x^{2}}*2x}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{8e^{x}e^{x^{2}}*-0}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln^{2}{10}} + \frac{8(\frac{-2(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{3}})xe^{x}e^{x^{2}}}{ln{10}} + \frac{8e^{x}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{8xe^{x}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{8xe^{x}e^{x^{2}}*2x}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{8xe^{x}e^{x^{2}}*-0}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln^{2}{10}} + \frac{16(\frac{-2(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{3}})x^{2}e^{x^{2}}e^{x}}{ln{10}} + \frac{16*2xe^{x^{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{16x^{2}e^{x^{2}}*2xe^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{16x^{2}e^{x^{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{16x^{2}e^{x^{2}}e^{x}*-0}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln^{2}{10}} + \frac{12(\frac{-(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{2}})xe^{x^{2}}}{ln{10}} + \frac{12e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} + \frac{12xe^{x^{2}}*2x}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} + \frac{12xe^{x^{2}}*-0}{(e^{x^{2}} - 2e^{x} - 3)ln^{2}{10}} + \frac{8(\frac{-2(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{3}})x^{2}e^{x}e^{x^{2}}}{ln{10}} + \frac{8*2xe^{x}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{8x^{2}e^{x}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{8x^{2}e^{x}e^{x^{2}}*2x}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{8x^{2}e^{x}e^{x^{2}}*-0}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln^{2}{10}} + \frac{8(\frac{-(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{2}})x^{3}e^{x^{2}}}{ln{10}} + \frac{8*3x^{2}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} + \frac{8x^{3}e^{x^{2}}*2x}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} + \frac{8x^{3}e^{x^{2}}*-0}{(e^{x^{2}} - 2e^{x} - 3)ln^{2}{10}} + \frac{4(\frac{-2(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{3}})e^{x^{2}}e^{x}}{ln{10}} + \frac{4e^{x^{2}}*2xe^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{4e^{x^{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{4e^{x^{2}}e^{x}*-0}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln^{2}{10}} + \frac{16(\frac{-3(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{4}})xe^{x^{2}}e^{{x}*{2}}}{ln{10}} + \frac{16e^{x^{2}}e^{{x}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} + \frac{16xe^{x^{2}}*2xe^{{x}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} + \frac{16xe^{x^{2}}*2e^{x}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} + \frac{16xe^{x^{2}}e^{{x}*{2}}*-0}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln^{2}{10}} - \frac{16(\frac{-3(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{4}})e^{{x}*{3}}}{ln{10}} - \frac{16*3e^{{x}*{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} - \frac{16e^{{x}*{3}}*-0}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln^{2}{10}} - \frac{12(\frac{-2(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{3}})e^{{x}*{2}}}{ln{10}} - \frac{12*2e^{x}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{12e^{{x}*{2}}*-0}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln^{2}{10}} + \frac{4(\frac{-2(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{3}})xe^{x^{2}}e^{x}}{ln{10}} + \frac{4e^{x^{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{4xe^{x^{2}}*2xe^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{4xe^{x^{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{4xe^{x^{2}}e^{x}*-0}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln^{2}{10}} - \frac{2(\frac{-(e^{x^{2}}*2x - 2e^{x} + 0)}{(e^{x^{2}} - 2e^{x} - 3)^{2}})e^{x}}{ln{10}} - \frac{2e^{x}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} - \frac{2e^{x}*-0}{(e^{x^{2}} - 2e^{x} - 3)ln^{2}{10}}\\=&\frac{-96x^{4}e^{{x^{2}}*{4}}}{(e^{x^{2}} - 2e^{x} - 3)^{4}ln{10}} + \frac{96x^{3}e^{x}e^{{x^{2}}*{3}}}{(e^{x^{2}} - 2e^{x} - 3)^{4}ln{10}} + \frac{96x^{2}e^{{x^{2}}*{3}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} + \frac{192x^{4}e^{{x^{2}}*{3}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} + \frac{288x^{3}e^{{x^{2}}*{3}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{4}ln{10}} - \frac{288x^{2}e^{{x}*{2}}e^{{x^{2}}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{4}ln{10}} - \frac{80xe^{x}e^{{x^{2}}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} - \frac{48x^{2}e^{x}e^{{x^{2}}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} - \frac{160x^{3}e^{x}e^{{x^{2}}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} - \frac{12e^{{x^{2}}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{144x^{2}e^{{x^{2}}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{112x^{4}e^{{x^{2}}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{112xe^{{x^{2}}*{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} - \frac{224x^{3}e^{{x^{2}}*{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} - \frac{288x^{2}e^{{x^{2}}*{2}}e^{{x}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{4}ln{10}} + \frac{288xe^{{x}*{3}}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{4}ln{10}} + \frac{80e^{{x}*{2}}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} + \frac{144xe^{{x}*{2}}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} + \frac{96x^{2}e^{x^{2}}e^{{x}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} + \frac{20e^{x}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{60xe^{x^{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{48x^{2}e^{{x^{2}}*{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} + \frac{52xe^{x}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{24x^{2}e^{x^{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{96x^{2}e^{{x}*{2}}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} + \frac{48x^{3}e^{x^{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{24x^{2}e^{x}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{12e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} + \frac{48x^{2}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} + \frac{16x^{3}e^{x}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{16x^{4}e^{x^{2}}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}} + \frac{16e^{x^{2}}e^{{x}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} + \frac{96xe^{x^{2}}e^{{x}*{3}}}{(e^{x^{2}} - 2e^{x} - 3)^{4}ln{10}} - \frac{96e^{{x}*{4}}}{(e^{x^{2}} - 2e^{x} - 3)^{4}ln{10}} - \frac{96e^{{x}*{3}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} + \frac{48xe^{x^{2}}e^{{x}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{3}ln{10}} - \frac{28e^{{x}*{2}}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} + \frac{4e^{x^{2}}e^{x}}{(e^{x^{2}} - 2e^{x} - 3)^{2}ln{10}} - \frac{2e^{x}}{(e^{x^{2}} - 2e^{x} - 3)ln{10}}\\ \end{split}\end{equation} \]