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.
\[ \begin{equation}\begin{split}[1/2]Find\ the\ 4th\ derivative\ of\ function\ ln(\frac{(e^{x} + e^{-x})}{2})\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = ln(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( ln(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})\right)}{dx}\\=&\frac{(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x}*-1)}{(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})}\\=&\frac{e^{x}}{2(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})} - \frac{e^{-x}}{2(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{e^{x}}{2(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})} - \frac{e^{-x}}{2(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})}\right)}{dx}\\=&\frac{(\frac{-(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x}*-1)}{(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{2}})e^{x}}{2} + \frac{e^{x}}{2(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})} - \frac{(\frac{-(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x}*-1)}{(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{2}})e^{-x}}{2} - \frac{e^{-x}*-1}{2(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})}\\=&\frac{e^{-x}e^{x}}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{2}} + \frac{e^{x}e^{-x}}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{2}} + \frac{e^{x}}{2(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})} - \frac{e^{{x}*{2}}}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{2}} - \frac{e^{{-x}*{2}}}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{2}} + \frac{e^{-x}}{2(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{e^{-x}e^{x}}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{2}} + \frac{e^{x}e^{-x}}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{2}} + \frac{e^{x}}{2(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})} - \frac{e^{{x}*{2}}}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{2}} - \frac{e^{{-x}*{2}}}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{2}} + \frac{e^{-x}}{2(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})}\right)}{dx}\\=&\frac{(\frac{-2(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x}*-1)}{(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{3}})e^{-x}e^{x}}{4} + \frac{e^{-x}*-e^{x}}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{2}} + \frac{e^{-x}e^{x}}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{2}} + \frac{(\frac{-2(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x}*-1)}{(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{3}})e^{x}e^{-x}}{4} + \frac{e^{x}e^{-x}}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{2}} + \frac{e^{x}e^{-x}*-1}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{2}} + \frac{(\frac{-(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x}*-1)}{(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{2}})e^{x}}{2} + \frac{e^{x}}{2(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})} - \frac{(\frac{-2(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x}*-1)}{(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{3}})e^{{x}*{2}}}{4} - \frac{2e^{x}e^{x}}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{2}} - \frac{(\frac{-2(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x}*-1)}{(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{3}})e^{{-x}*{2}}}{4} - \frac{2e^{-x}e^{-x}*-1}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{2}} + \frac{(\frac{-(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x}*-1)}{(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{2}})e^{-x}}{2} + \frac{e^{-x}*-1}{2(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})}\\=& - \frac{e^{{x}*{2}}e^{-x}}{2(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{3}} + \frac{e^{{-x}*{2}}e^{x}}{2(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{3}} - \frac{e^{-x}e^{x}}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{2}} + \frac{e^{x}e^{-x}}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{2}} - \frac{e^{-x}e^{{x}*{2}}}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{3}} + \frac{e^{x}e^{{-x}*{2}}}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{3}} + \frac{e^{{x}*{3}}}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{3}} - \frac{3e^{{x}*{2}}}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{2}} + \frac{e^{x}}{2(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})} - \frac{e^{{-x}*{3}}}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{3}} + \frac{3e^{{-x}*{2}}}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{2}} - \frac{e^{-x}}{2(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( - \frac{e^{{x}*{2}}e^{-x}}{2(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{3}} + \frac{e^{{-x}*{2}}e^{x}}{2(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{3}} - \frac{e^{-x}e^{x}}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{2}} + \frac{e^{x}e^{-x}}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{2}} - \frac{e^{-x}e^{{x}*{2}}}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{3}} + \frac{e^{x}e^{{-x}*{2}}}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{3}} + \frac{e^{{x}*{3}}}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{3}} - \frac{3e^{{x}*{2}}}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{2}} + \frac{e^{x}}{2(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})} - \frac{e^{{-x}*{3}}}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{3}} + \frac{3e^{{-x}*{2}}}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{2}} - \frac{e^{-x}}{2(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})}\right)}{dx}\\=& - \frac{(\frac{-3(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x}*-1)}{(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{4}})e^{{x}*{2}}e^{-x}}{2} - \frac{2e^{x}e^{x}e^{-x}}{2(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{3}} - \frac{e^{{x}*{2}}e^{-x}*-1}{2(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{3}} + \frac{(\frac{-3(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x}*-1)}{(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{4}})e^{{-x}*{2}}e^{x}}{2} + \frac{2e^{-x}e^{-x}*-e^{x}}{2(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{3}} + \frac{e^{{-x}*{2}}e^{x}}{2(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{3}} - \frac{(\frac{-2(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x}*-1)}{(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{3}})e^{-x}e^{x}}{4} - \frac{e^{-x}*-e^{x}}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{2}} - \frac{e^{-x}e^{x}}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{2}} + \frac{(\frac{-2(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x}*-1)}{(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{3}})e^{x}e^{-x}}{4} + \frac{e^{x}e^{-x}}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{2}} + \frac{e^{x}e^{-x}*-1}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{2}} - \frac{(\frac{-3(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x}*-1)}{(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{4}})e^{-x}e^{{x}*{2}}}{4} - \frac{e^{-x}*-e^{{x}*{2}}}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{3}} - \frac{e^{-x}*2e^{x}e^{x}}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{3}} + \frac{(\frac{-3(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x}*-1)}{(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{4}})e^{x}e^{{-x}*{2}}}{4} + \frac{e^{x}e^{{-x}*{2}}}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{3}} + \frac{e^{x}*2e^{-x}e^{-x}*-1}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{3}} + \frac{(\frac{-3(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x}*-1)}{(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{4}})e^{{x}*{3}}}{4} + \frac{3e^{{x}*{2}}e^{x}}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{3}} - \frac{3(\frac{-2(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x}*-1)}{(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{3}})e^{{x}*{2}}}{4} - \frac{3*2e^{x}e^{x}}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{2}} + \frac{(\frac{-(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x}*-1)}{(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{2}})e^{x}}{2} + \frac{e^{x}}{2(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})} - \frac{(\frac{-3(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x}*-1)}{(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{4}})e^{{-x}*{3}}}{4} - \frac{3e^{{-x}*{2}}e^{-x}*-1}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{3}} + \frac{3(\frac{-2(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x}*-1)}{(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{3}})e^{{-x}*{2}}}{4} + \frac{3*2e^{-x}e^{-x}*-1}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{2}} - \frac{(\frac{-(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x}*-1)}{(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{2}})e^{-x}}{2} - \frac{e^{-x}*-1}{2(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})}\\=&\frac{9e^{{x}*{3}}e^{-x}}{8(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{4}} - \frac{9e^{{-x}*{2}}e^{{x}*{2}}}{8(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{4}} - \frac{3e^{{x}*{2}}e^{-x}}{2(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{3}} - \frac{9e^{{x}*{2}}e^{{-x}*{2}}}{8(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{4}} + \frac{9e^{{-x}*{3}}e^{x}}{8(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{4}} - \frac{3e^{{-x}*{2}}e^{x}}{2(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{3}} + \frac{3e^{-x}e^{{x}*{3}}}{8(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{4}} + \frac{e^{-x}e^{x}}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{2}} + \frac{3e^{x}e^{{-x}*{3}}}{8(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{4}} + \frac{e^{x}e^{-x}}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{2}} - \frac{3e^{{x}*{4}}}{8(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{4}} + \frac{e^{x}}{2(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})} + \frac{3e^{{x}*{3}}}{2(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{3}} - \frac{3e^{{-x}*{4}}}{8(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{4}} + \frac{3e^{{-x}*{3}}}{2(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{3}} - \frac{7e^{{-x}*{2}}}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{2}} - \frac{7e^{{x}*{2}}}{4(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})^{2}} + \frac{e^{-x}}{2(\frac{1}{2}e^{x} + \frac{1}{2}e^{-x})}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/2]Find\ the\ 4th\ derivative\ of\ function\ ln(\frac{(e^{x} - e^{-x})}{2})\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = ln(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( ln(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})\right)}{dx}\\=&\frac{(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x}*-1)}{(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})}\\=&\frac{e^{x}}{2(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})} + \frac{e^{-x}}{2(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{e^{x}}{2(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})} + \frac{e^{-x}}{2(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})}\right)}{dx}\\=&\frac{(\frac{-(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x}*-1)}{(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{2}})e^{x}}{2} + \frac{e^{x}}{2(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})} + \frac{(\frac{-(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x}*-1)}{(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{2}})e^{-x}}{2} + \frac{e^{-x}*-1}{2(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})}\\=& - \frac{e^{-x}e^{x}}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{2}} - \frac{e^{x}e^{-x}}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{2}} + \frac{e^{x}}{2(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})} - \frac{e^{{x}*{2}}}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{2}} - \frac{e^{{-x}*{2}}}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{2}} - \frac{e^{-x}}{2(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( - \frac{e^{-x}e^{x}}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{2}} - \frac{e^{x}e^{-x}}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{2}} + \frac{e^{x}}{2(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})} - \frac{e^{{x}*{2}}}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{2}} - \frac{e^{{-x}*{2}}}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{2}} - \frac{e^{-x}}{2(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})}\right)}{dx}\\=& - \frac{(\frac{-2(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x}*-1)}{(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{3}})e^{-x}e^{x}}{4} - \frac{e^{-x}*-e^{x}}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{2}} - \frac{e^{-x}e^{x}}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{2}} - \frac{(\frac{-2(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x}*-1)}{(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{3}})e^{x}e^{-x}}{4} - \frac{e^{x}e^{-x}}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{2}} - \frac{e^{x}e^{-x}*-1}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{2}} + \frac{(\frac{-(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x}*-1)}{(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{2}})e^{x}}{2} + \frac{e^{x}}{2(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})} - \frac{(\frac{-2(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x}*-1)}{(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{3}})e^{{x}*{2}}}{4} - \frac{2e^{x}e^{x}}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{2}} - \frac{(\frac{-2(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x}*-1)}{(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{3}})e^{{-x}*{2}}}{4} - \frac{2e^{-x}e^{-x}*-1}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{2}} - \frac{(\frac{-(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x}*-1)}{(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{2}})e^{-x}}{2} - \frac{e^{-x}*-1}{2(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})}\\=&\frac{e^{{x}*{2}}e^{-x}}{2(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{3}} + \frac{e^{{-x}*{2}}e^{x}}{2(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{3}} + \frac{e^{-x}e^{x}}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{2}} - \frac{e^{x}e^{-x}}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{2}} + \frac{e^{-x}e^{{x}*{2}}}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{3}} + \frac{e^{x}e^{{-x}*{2}}}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{3}} + \frac{e^{{x}*{3}}}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{3}} - \frac{3e^{{x}*{2}}}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{2}} + \frac{e^{x}}{2(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})} + \frac{e^{{-x}*{3}}}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{3}} + \frac{3e^{{-x}*{2}}}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{2}} + \frac{e^{-x}}{2(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{e^{{x}*{2}}e^{-x}}{2(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{3}} + \frac{e^{{-x}*{2}}e^{x}}{2(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{3}} + \frac{e^{-x}e^{x}}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{2}} - \frac{e^{x}e^{-x}}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{2}} + \frac{e^{-x}e^{{x}*{2}}}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{3}} + \frac{e^{x}e^{{-x}*{2}}}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{3}} + \frac{e^{{x}*{3}}}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{3}} - \frac{3e^{{x}*{2}}}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{2}} + \frac{e^{x}}{2(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})} + \frac{e^{{-x}*{3}}}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{3}} + \frac{3e^{{-x}*{2}}}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{2}} + \frac{e^{-x}}{2(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})}\right)}{dx}\\=&\frac{(\frac{-3(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x}*-1)}{(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{4}})e^{{x}*{2}}e^{-x}}{2} + \frac{2e^{x}e^{x}e^{-x}}{2(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{3}} + \frac{e^{{x}*{2}}e^{-x}*-1}{2(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{3}} + \frac{(\frac{-3(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x}*-1)}{(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{4}})e^{{-x}*{2}}e^{x}}{2} + \frac{2e^{-x}e^{-x}*-e^{x}}{2(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{3}} + \frac{e^{{-x}*{2}}e^{x}}{2(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{3}} + \frac{(\frac{-2(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x}*-1)}{(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{3}})e^{-x}e^{x}}{4} + \frac{e^{-x}*-e^{x}}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{2}} + \frac{e^{-x}e^{x}}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{2}} - \frac{(\frac{-2(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x}*-1)}{(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{3}})e^{x}e^{-x}}{4} - \frac{e^{x}e^{-x}}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{2}} - \frac{e^{x}e^{-x}*-1}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{2}} + \frac{(\frac{-3(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x}*-1)}{(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{4}})e^{-x}e^{{x}*{2}}}{4} + \frac{e^{-x}*-e^{{x}*{2}}}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{3}} + \frac{e^{-x}*2e^{x}e^{x}}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{3}} + \frac{(\frac{-3(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x}*-1)}{(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{4}})e^{x}e^{{-x}*{2}}}{4} + \frac{e^{x}e^{{-x}*{2}}}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{3}} + \frac{e^{x}*2e^{-x}e^{-x}*-1}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{3}} + \frac{(\frac{-3(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x}*-1)}{(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{4}})e^{{x}*{3}}}{4} + \frac{3e^{{x}*{2}}e^{x}}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{3}} - \frac{3(\frac{-2(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x}*-1)}{(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{3}})e^{{x}*{2}}}{4} - \frac{3*2e^{x}e^{x}}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{2}} + \frac{(\frac{-(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x}*-1)}{(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{2}})e^{x}}{2} + \frac{e^{x}}{2(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})} + \frac{(\frac{-3(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x}*-1)}{(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{4}})e^{{-x}*{3}}}{4} + \frac{3e^{{-x}*{2}}e^{-x}*-1}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{3}} + \frac{3(\frac{-2(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x}*-1)}{(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{3}})e^{{-x}*{2}}}{4} + \frac{3*2e^{-x}e^{-x}*-1}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{2}} + \frac{(\frac{-(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x}*-1)}{(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{2}})e^{-x}}{2} + \frac{e^{-x}*-1}{2(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})}\\=&\frac{-9e^{{x}*{3}}e^{-x}}{8(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{4}} - \frac{9e^{{-x}*{2}}e^{{x}*{2}}}{8(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{4}} + \frac{3e^{{x}*{2}}e^{-x}}{2(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{3}} - \frac{9e^{{x}*{2}}e^{{-x}*{2}}}{8(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{4}} - \frac{9e^{{-x}*{3}}e^{x}}{8(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{4}} - \frac{3e^{{-x}*{2}}e^{x}}{2(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{3}} - \frac{3e^{-x}e^{{x}*{3}}}{8(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{4}} - \frac{e^{-x}e^{x}}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{2}} - \frac{3e^{x}e^{{-x}*{3}}}{8(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{4}} - \frac{e^{x}e^{-x}}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{2}} - \frac{3e^{{x}*{4}}}{8(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{4}} + \frac{e^{x}}{2(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})} + \frac{3e^{{x}*{3}}}{2(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{3}} - \frac{3e^{{-x}*{4}}}{8(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{4}} - \frac{3e^{{-x}*{3}}}{2(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{3}} - \frac{7e^{{-x}*{2}}}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{2}} - \frac{7e^{{x}*{2}}}{4(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})^{2}} - \frac{e^{-x}}{2(\frac{1}{2}e^{x} - \frac{1}{2}e^{-x})}\\ \end{split}\end{equation} \]