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.
\[ \begin{equation}\begin{split}[1/2]Find\ the\ 4th\ derivative\ of\ function\ (e^{x})(sin(x))(ln(x))(xx)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = x^{2}e^{x}ln(x)sin(x)\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( x^{2}e^{x}ln(x)sin(x)\right)}{dx}\\=&2xe^{x}ln(x)sin(x) + x^{2}e^{x}ln(x)sin(x) + \frac{x^{2}e^{x}sin(x)}{(x)} + x^{2}e^{x}ln(x)cos(x)\\=&2xe^{x}ln(x)sin(x) + x^{2}e^{x}ln(x)sin(x) + xe^{x}sin(x) + x^{2}e^{x}ln(x)cos(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( 2xe^{x}ln(x)sin(x) + x^{2}e^{x}ln(x)sin(x) + xe^{x}sin(x) + x^{2}e^{x}ln(x)cos(x)\right)}{dx}\\=&2e^{x}ln(x)sin(x) + 2xe^{x}ln(x)sin(x) + \frac{2xe^{x}sin(x)}{(x)} + 2xe^{x}ln(x)cos(x) + 2xe^{x}ln(x)sin(x) + x^{2}e^{x}ln(x)sin(x) + \frac{x^{2}e^{x}sin(x)}{(x)} + x^{2}e^{x}ln(x)cos(x) + e^{x}sin(x) + xe^{x}sin(x) + xe^{x}cos(x) + 2xe^{x}ln(x)cos(x) + x^{2}e^{x}ln(x)cos(x) + \frac{x^{2}e^{x}cos(x)}{(x)} + x^{2}e^{x}ln(x)*-sin(x)\\=&2e^{x}ln(x)sin(x) + 4xe^{x}ln(x)sin(x) + 3e^{x}sin(x) + 4xe^{x}ln(x)cos(x) + 2xe^{x}sin(x) + 2x^{2}e^{x}ln(x)cos(x) + 2xe^{x}cos(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 2e^{x}ln(x)sin(x) + 4xe^{x}ln(x)sin(x) + 3e^{x}sin(x) + 4xe^{x}ln(x)cos(x) + 2xe^{x}sin(x) + 2x^{2}e^{x}ln(x)cos(x) + 2xe^{x}cos(x)\right)}{dx}\\=&2e^{x}ln(x)sin(x) + \frac{2e^{x}sin(x)}{(x)} + 2e^{x}ln(x)cos(x) + 4e^{x}ln(x)sin(x) + 4xe^{x}ln(x)sin(x) + \frac{4xe^{x}sin(x)}{(x)} + 4xe^{x}ln(x)cos(x) + 3e^{x}sin(x) + 3e^{x}cos(x) + 4e^{x}ln(x)cos(x) + 4xe^{x}ln(x)cos(x) + \frac{4xe^{x}cos(x)}{(x)} + 4xe^{x}ln(x)*-sin(x) + 2e^{x}sin(x) + 2xe^{x}sin(x) + 2xe^{x}cos(x) + 2*2xe^{x}ln(x)cos(x) + 2x^{2}e^{x}ln(x)cos(x) + \frac{2x^{2}e^{x}cos(x)}{(x)} + 2x^{2}e^{x}ln(x)*-sin(x) + 2e^{x}cos(x) + 2xe^{x}cos(x) + 2xe^{x}*-sin(x)\\=&6e^{x}ln(x)sin(x) + \frac{2e^{x}sin(x)}{x} + 6e^{x}ln(x)cos(x) + 9e^{x}sin(x) + 12xe^{x}ln(x)cos(x) + 9e^{x}cos(x) + 6xe^{x}cos(x) + 2x^{2}e^{x}ln(x)cos(x) - 2x^{2}e^{x}ln(x)sin(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 6e^{x}ln(x)sin(x) + \frac{2e^{x}sin(x)}{x} + 6e^{x}ln(x)cos(x) + 9e^{x}sin(x) + 12xe^{x}ln(x)cos(x) + 9e^{x}cos(x) + 6xe^{x}cos(x) + 2x^{2}e^{x}ln(x)cos(x) - 2x^{2}e^{x}ln(x)sin(x)\right)}{dx}\\=&6e^{x}ln(x)sin(x) + \frac{6e^{x}sin(x)}{(x)} + 6e^{x}ln(x)cos(x) + \frac{2*-e^{x}sin(x)}{x^{2}} + \frac{2e^{x}sin(x)}{x} + \frac{2e^{x}cos(x)}{x} + 6e^{x}ln(x)cos(x) + \frac{6e^{x}cos(x)}{(x)} + 6e^{x}ln(x)*-sin(x) + 9e^{x}sin(x) + 9e^{x}cos(x) + 12e^{x}ln(x)cos(x) + 12xe^{x}ln(x)cos(x) + \frac{12xe^{x}cos(x)}{(x)} + 12xe^{x}ln(x)*-sin(x) + 9e^{x}cos(x) + 9e^{x}*-sin(x) + 6e^{x}cos(x) + 6xe^{x}cos(x) + 6xe^{x}*-sin(x) + 2*2xe^{x}ln(x)cos(x) + 2x^{2}e^{x}ln(x)cos(x) + \frac{2x^{2}e^{x}cos(x)}{(x)} + 2x^{2}e^{x}ln(x)*-sin(x) - 2*2xe^{x}ln(x)sin(x) - 2x^{2}e^{x}ln(x)sin(x) - \frac{2x^{2}e^{x}sin(x)}{(x)} - 2x^{2}e^{x}ln(x)cos(x)\\=&\frac{8e^{x}sin(x)}{x} + 24e^{x}ln(x)cos(x) - \frac{2e^{x}sin(x)}{x^{2}} + \frac{8e^{x}cos(x)}{x} + 36e^{x}cos(x) + 16xe^{x}ln(x)cos(x) - 16xe^{x}ln(x)sin(x) + 8xe^{x}cos(x) - 8xe^{x}sin(x) - 4x^{2}e^{x}ln(x)sin(x)\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/2]Find\ the\ 4th\ derivative\ of\ function\ e^{sin(ln(xx))}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = e^{sin(ln(x^{2}))}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( e^{sin(ln(x^{2}))}\right)}{dx}\\=&\frac{e^{sin(ln(x^{2}))}cos(ln(x^{2}))*2x}{(x^{2})}\\=&\frac{2e^{sin(ln(x^{2}))}cos(ln(x^{2}))}{x}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{2e^{sin(ln(x^{2}))}cos(ln(x^{2}))}{x}\right)}{dx}\\=&\frac{2*-e^{sin(ln(x^{2}))}cos(ln(x^{2}))}{x^{2}} + \frac{2e^{sin(ln(x^{2}))}cos(ln(x^{2}))*2xcos(ln(x^{2}))}{x(x^{2})} + \frac{2e^{sin(ln(x^{2}))}*-sin(ln(x^{2}))*2x}{x(x^{2})}\\=&\frac{-2e^{sin(ln(x^{2}))}cos(ln(x^{2}))}{x^{2}} + \frac{4e^{sin(ln(x^{2}))}cos^{2}(ln(x^{2}))}{x^{2}} - \frac{4e^{sin(ln(x^{2}))}sin(ln(x^{2}))}{x^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2e^{sin(ln(x^{2}))}cos(ln(x^{2}))}{x^{2}} + \frac{4e^{sin(ln(x^{2}))}cos^{2}(ln(x^{2}))}{x^{2}} - \frac{4e^{sin(ln(x^{2}))}sin(ln(x^{2}))}{x^{2}}\right)}{dx}\\=&\frac{-2*-2e^{sin(ln(x^{2}))}cos(ln(x^{2}))}{x^{3}} - \frac{2e^{sin(ln(x^{2}))}cos(ln(x^{2}))*2xcos(ln(x^{2}))}{x^{2}(x^{2})} - \frac{2e^{sin(ln(x^{2}))}*-sin(ln(x^{2}))*2x}{x^{2}(x^{2})} + \frac{4*-2e^{sin(ln(x^{2}))}cos^{2}(ln(x^{2}))}{x^{3}} + \frac{4e^{sin(ln(x^{2}))}cos(ln(x^{2}))*2xcos^{2}(ln(x^{2}))}{x^{2}(x^{2})} + \frac{4e^{sin(ln(x^{2}))}*-2cos(ln(x^{2}))sin(ln(x^{2}))*2x}{x^{2}(x^{2})} - \frac{4*-2e^{sin(ln(x^{2}))}sin(ln(x^{2}))}{x^{3}} - \frac{4e^{sin(ln(x^{2}))}cos(ln(x^{2}))*2xsin(ln(x^{2}))}{x^{2}(x^{2})} - \frac{4e^{sin(ln(x^{2}))}cos(ln(x^{2}))*2x}{x^{2}(x^{2})}\\=&\frac{-4e^{sin(ln(x^{2}))}cos(ln(x^{2}))}{x^{3}} - \frac{12e^{sin(ln(x^{2}))}cos^{2}(ln(x^{2}))}{x^{3}} - \frac{24e^{sin(ln(x^{2}))}sin(ln(x^{2}))cos(ln(x^{2}))}{x^{3}} + \frac{8e^{sin(ln(x^{2}))}cos^{3}(ln(x^{2}))}{x^{3}} + \frac{12e^{sin(ln(x^{2}))}sin(ln(x^{2}))}{x^{3}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-4e^{sin(ln(x^{2}))}cos(ln(x^{2}))}{x^{3}} - \frac{12e^{sin(ln(x^{2}))}cos^{2}(ln(x^{2}))}{x^{3}} - \frac{24e^{sin(ln(x^{2}))}sin(ln(x^{2}))cos(ln(x^{2}))}{x^{3}} + \frac{8e^{sin(ln(x^{2}))}cos^{3}(ln(x^{2}))}{x^{3}} + \frac{12e^{sin(ln(x^{2}))}sin(ln(x^{2}))}{x^{3}}\right)}{dx}\\=&\frac{-4*-3e^{sin(ln(x^{2}))}cos(ln(x^{2}))}{x^{4}} - \frac{4e^{sin(ln(x^{2}))}cos(ln(x^{2}))*2xcos(ln(x^{2}))}{x^{3}(x^{2})} - \frac{4e^{sin(ln(x^{2}))}*-sin(ln(x^{2}))*2x}{x^{3}(x^{2})} - \frac{12*-3e^{sin(ln(x^{2}))}cos^{2}(ln(x^{2}))}{x^{4}} - \frac{12e^{sin(ln(x^{2}))}cos(ln(x^{2}))*2xcos^{2}(ln(x^{2}))}{x^{3}(x^{2})} - \frac{12e^{sin(ln(x^{2}))}*-2cos(ln(x^{2}))sin(ln(x^{2}))*2x}{x^{3}(x^{2})} - \frac{24*-3e^{sin(ln(x^{2}))}sin(ln(x^{2}))cos(ln(x^{2}))}{x^{4}} - \frac{24e^{sin(ln(x^{2}))}cos(ln(x^{2}))*2xsin(ln(x^{2}))cos(ln(x^{2}))}{x^{3}(x^{2})} - \frac{24e^{sin(ln(x^{2}))}cos(ln(x^{2}))*2xcos(ln(x^{2}))}{x^{3}(x^{2})} - \frac{24e^{sin(ln(x^{2}))}sin(ln(x^{2}))*-sin(ln(x^{2}))*2x}{x^{3}(x^{2})} + \frac{8*-3e^{sin(ln(x^{2}))}cos^{3}(ln(x^{2}))}{x^{4}} + \frac{8e^{sin(ln(x^{2}))}cos(ln(x^{2}))*2xcos^{3}(ln(x^{2}))}{x^{3}(x^{2})} + \frac{8e^{sin(ln(x^{2}))}*-3cos^{2}(ln(x^{2}))sin(ln(x^{2}))*2x}{x^{3}(x^{2})} + \frac{12*-3e^{sin(ln(x^{2}))}sin(ln(x^{2}))}{x^{4}} + \frac{12e^{sin(ln(x^{2}))}cos(ln(x^{2}))*2xsin(ln(x^{2}))}{x^{3}(x^{2})} + \frac{12e^{sin(ln(x^{2}))}cos(ln(x^{2}))*2x}{x^{3}(x^{2})}\\=&\frac{36e^{sin(ln(x^{2}))}cos(ln(x^{2}))}{x^{4}} - \frac{20e^{sin(ln(x^{2}))}cos^{2}(ln(x^{2}))}{x^{4}} - \frac{96e^{sin(ln(x^{2}))}sin(ln(x^{2}))cos^{2}(ln(x^{2}))}{x^{4}} - \frac{48e^{sin(ln(x^{2}))}cos^{3}(ln(x^{2}))}{x^{4}} + \frac{144e^{sin(ln(x^{2}))}sin(ln(x^{2}))cos(ln(x^{2}))}{x^{4}} + \frac{48e^{sin(ln(x^{2}))}sin^{2}(ln(x^{2}))}{x^{4}} + \frac{16e^{sin(ln(x^{2}))}cos^{4}(ln(x^{2}))}{x^{4}} - \frac{28e^{sin(ln(x^{2}))}sin(ln(x^{2}))}{x^{4}}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/2]Find\ the\ 4th\ derivative\ of\ function\ (e^{x})(sin(x))(ln(x))(xx)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = x^{2}e^{x}ln(x)sin(x)\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( x^{2}e^{x}ln(x)sin(x)\right)}{dx}\\=&2xe^{x}ln(x)sin(x) + x^{2}e^{x}ln(x)sin(x) + \frac{x^{2}e^{x}sin(x)}{(x)} + x^{2}e^{x}ln(x)cos(x)\\=&2xe^{x}ln(x)sin(x) + x^{2}e^{x}ln(x)sin(x) + xe^{x}sin(x) + x^{2}e^{x}ln(x)cos(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( 2xe^{x}ln(x)sin(x) + x^{2}e^{x}ln(x)sin(x) + xe^{x}sin(x) + x^{2}e^{x}ln(x)cos(x)\right)}{dx}\\=&2e^{x}ln(x)sin(x) + 2xe^{x}ln(x)sin(x) + \frac{2xe^{x}sin(x)}{(x)} + 2xe^{x}ln(x)cos(x) + 2xe^{x}ln(x)sin(x) + x^{2}e^{x}ln(x)sin(x) + \frac{x^{2}e^{x}sin(x)}{(x)} + x^{2}e^{x}ln(x)cos(x) + e^{x}sin(x) + xe^{x}sin(x) + xe^{x}cos(x) + 2xe^{x}ln(x)cos(x) + x^{2}e^{x}ln(x)cos(x) + \frac{x^{2}e^{x}cos(x)}{(x)} + x^{2}e^{x}ln(x)*-sin(x)\\=&2e^{x}ln(x)sin(x) + 4xe^{x}ln(x)sin(x) + 3e^{x}sin(x) + 4xe^{x}ln(x)cos(x) + 2xe^{x}sin(x) + 2x^{2}e^{x}ln(x)cos(x) + 2xe^{x}cos(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 2e^{x}ln(x)sin(x) + 4xe^{x}ln(x)sin(x) + 3e^{x}sin(x) + 4xe^{x}ln(x)cos(x) + 2xe^{x}sin(x) + 2x^{2}e^{x}ln(x)cos(x) + 2xe^{x}cos(x)\right)}{dx}\\=&2e^{x}ln(x)sin(x) + \frac{2e^{x}sin(x)}{(x)} + 2e^{x}ln(x)cos(x) + 4e^{x}ln(x)sin(x) + 4xe^{x}ln(x)sin(x) + \frac{4xe^{x}sin(x)}{(x)} + 4xe^{x}ln(x)cos(x) + 3e^{x}sin(x) + 3e^{x}cos(x) + 4e^{x}ln(x)cos(x) + 4xe^{x}ln(x)cos(x) + \frac{4xe^{x}cos(x)}{(x)} + 4xe^{x}ln(x)*-sin(x) + 2e^{x}sin(x) + 2xe^{x}sin(x) + 2xe^{x}cos(x) + 2*2xe^{x}ln(x)cos(x) + 2x^{2}e^{x}ln(x)cos(x) + \frac{2x^{2}e^{x}cos(x)}{(x)} + 2x^{2}e^{x}ln(x)*-sin(x) + 2e^{x}cos(x) + 2xe^{x}cos(x) + 2xe^{x}*-sin(x)\\=&6e^{x}ln(x)sin(x) + \frac{2e^{x}sin(x)}{x} + 6e^{x}ln(x)cos(x) + 9e^{x}sin(x) + 12xe^{x}ln(x)cos(x) + 9e^{x}cos(x) + 6xe^{x}cos(x) + 2x^{2}e^{x}ln(x)cos(x) - 2x^{2}e^{x}ln(x)sin(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 6e^{x}ln(x)sin(x) + \frac{2e^{x}sin(x)}{x} + 6e^{x}ln(x)cos(x) + 9e^{x}sin(x) + 12xe^{x}ln(x)cos(x) + 9e^{x}cos(x) + 6xe^{x}cos(x) + 2x^{2}e^{x}ln(x)cos(x) - 2x^{2}e^{x}ln(x)sin(x)\right)}{dx}\\=&6e^{x}ln(x)sin(x) + \frac{6e^{x}sin(x)}{(x)} + 6e^{x}ln(x)cos(x) + \frac{2*-e^{x}sin(x)}{x^{2}} + \frac{2e^{x}sin(x)}{x} + \frac{2e^{x}cos(x)}{x} + 6e^{x}ln(x)cos(x) + \frac{6e^{x}cos(x)}{(x)} + 6e^{x}ln(x)*-sin(x) + 9e^{x}sin(x) + 9e^{x}cos(x) + 12e^{x}ln(x)cos(x) + 12xe^{x}ln(x)cos(x) + \frac{12xe^{x}cos(x)}{(x)} + 12xe^{x}ln(x)*-sin(x) + 9e^{x}cos(x) + 9e^{x}*-sin(x) + 6e^{x}cos(x) + 6xe^{x}cos(x) + 6xe^{x}*-sin(x) + 2*2xe^{x}ln(x)cos(x) + 2x^{2}e^{x}ln(x)cos(x) + \frac{2x^{2}e^{x}cos(x)}{(x)} + 2x^{2}e^{x}ln(x)*-sin(x) - 2*2xe^{x}ln(x)sin(x) - 2x^{2}e^{x}ln(x)sin(x) - \frac{2x^{2}e^{x}sin(x)}{(x)} - 2x^{2}e^{x}ln(x)cos(x)\\=&\frac{8e^{x}sin(x)}{x} + 24e^{x}ln(x)cos(x) - \frac{2e^{x}sin(x)}{x^{2}} + \frac{8e^{x}cos(x)}{x} + 36e^{x}cos(x) + 16xe^{x}ln(x)cos(x) - 16xe^{x}ln(x)sin(x) + 8xe^{x}cos(x) - 8xe^{x}sin(x) - 4x^{2}e^{x}ln(x)sin(x)\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/2]Find\ the\ 4th\ derivative\ of\ function\ e^{sin(ln(xx))}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = e^{sin(ln(x^{2}))}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( e^{sin(ln(x^{2}))}\right)}{dx}\\=&\frac{e^{sin(ln(x^{2}))}cos(ln(x^{2}))*2x}{(x^{2})}\\=&\frac{2e^{sin(ln(x^{2}))}cos(ln(x^{2}))}{x}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{2e^{sin(ln(x^{2}))}cos(ln(x^{2}))}{x}\right)}{dx}\\=&\frac{2*-e^{sin(ln(x^{2}))}cos(ln(x^{2}))}{x^{2}} + \frac{2e^{sin(ln(x^{2}))}cos(ln(x^{2}))*2xcos(ln(x^{2}))}{x(x^{2})} + \frac{2e^{sin(ln(x^{2}))}*-sin(ln(x^{2}))*2x}{x(x^{2})}\\=&\frac{-2e^{sin(ln(x^{2}))}cos(ln(x^{2}))}{x^{2}} + \frac{4e^{sin(ln(x^{2}))}cos^{2}(ln(x^{2}))}{x^{2}} - \frac{4e^{sin(ln(x^{2}))}sin(ln(x^{2}))}{x^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2e^{sin(ln(x^{2}))}cos(ln(x^{2}))}{x^{2}} + \frac{4e^{sin(ln(x^{2}))}cos^{2}(ln(x^{2}))}{x^{2}} - \frac{4e^{sin(ln(x^{2}))}sin(ln(x^{2}))}{x^{2}}\right)}{dx}\\=&\frac{-2*-2e^{sin(ln(x^{2}))}cos(ln(x^{2}))}{x^{3}} - \frac{2e^{sin(ln(x^{2}))}cos(ln(x^{2}))*2xcos(ln(x^{2}))}{x^{2}(x^{2})} - \frac{2e^{sin(ln(x^{2}))}*-sin(ln(x^{2}))*2x}{x^{2}(x^{2})} + \frac{4*-2e^{sin(ln(x^{2}))}cos^{2}(ln(x^{2}))}{x^{3}} + \frac{4e^{sin(ln(x^{2}))}cos(ln(x^{2}))*2xcos^{2}(ln(x^{2}))}{x^{2}(x^{2})} + \frac{4e^{sin(ln(x^{2}))}*-2cos(ln(x^{2}))sin(ln(x^{2}))*2x}{x^{2}(x^{2})} - \frac{4*-2e^{sin(ln(x^{2}))}sin(ln(x^{2}))}{x^{3}} - \frac{4e^{sin(ln(x^{2}))}cos(ln(x^{2}))*2xsin(ln(x^{2}))}{x^{2}(x^{2})} - \frac{4e^{sin(ln(x^{2}))}cos(ln(x^{2}))*2x}{x^{2}(x^{2})}\\=&\frac{-4e^{sin(ln(x^{2}))}cos(ln(x^{2}))}{x^{3}} - \frac{12e^{sin(ln(x^{2}))}cos^{2}(ln(x^{2}))}{x^{3}} - \frac{24e^{sin(ln(x^{2}))}sin(ln(x^{2}))cos(ln(x^{2}))}{x^{3}} + \frac{8e^{sin(ln(x^{2}))}cos^{3}(ln(x^{2}))}{x^{3}} + \frac{12e^{sin(ln(x^{2}))}sin(ln(x^{2}))}{x^{3}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-4e^{sin(ln(x^{2}))}cos(ln(x^{2}))}{x^{3}} - \frac{12e^{sin(ln(x^{2}))}cos^{2}(ln(x^{2}))}{x^{3}} - \frac{24e^{sin(ln(x^{2}))}sin(ln(x^{2}))cos(ln(x^{2}))}{x^{3}} + \frac{8e^{sin(ln(x^{2}))}cos^{3}(ln(x^{2}))}{x^{3}} + \frac{12e^{sin(ln(x^{2}))}sin(ln(x^{2}))}{x^{3}}\right)}{dx}\\=&\frac{-4*-3e^{sin(ln(x^{2}))}cos(ln(x^{2}))}{x^{4}} - \frac{4e^{sin(ln(x^{2}))}cos(ln(x^{2}))*2xcos(ln(x^{2}))}{x^{3}(x^{2})} - \frac{4e^{sin(ln(x^{2}))}*-sin(ln(x^{2}))*2x}{x^{3}(x^{2})} - \frac{12*-3e^{sin(ln(x^{2}))}cos^{2}(ln(x^{2}))}{x^{4}} - \frac{12e^{sin(ln(x^{2}))}cos(ln(x^{2}))*2xcos^{2}(ln(x^{2}))}{x^{3}(x^{2})} - \frac{12e^{sin(ln(x^{2}))}*-2cos(ln(x^{2}))sin(ln(x^{2}))*2x}{x^{3}(x^{2})} - \frac{24*-3e^{sin(ln(x^{2}))}sin(ln(x^{2}))cos(ln(x^{2}))}{x^{4}} - \frac{24e^{sin(ln(x^{2}))}cos(ln(x^{2}))*2xsin(ln(x^{2}))cos(ln(x^{2}))}{x^{3}(x^{2})} - \frac{24e^{sin(ln(x^{2}))}cos(ln(x^{2}))*2xcos(ln(x^{2}))}{x^{3}(x^{2})} - \frac{24e^{sin(ln(x^{2}))}sin(ln(x^{2}))*-sin(ln(x^{2}))*2x}{x^{3}(x^{2})} + \frac{8*-3e^{sin(ln(x^{2}))}cos^{3}(ln(x^{2}))}{x^{4}} + \frac{8e^{sin(ln(x^{2}))}cos(ln(x^{2}))*2xcos^{3}(ln(x^{2}))}{x^{3}(x^{2})} + \frac{8e^{sin(ln(x^{2}))}*-3cos^{2}(ln(x^{2}))sin(ln(x^{2}))*2x}{x^{3}(x^{2})} + \frac{12*-3e^{sin(ln(x^{2}))}sin(ln(x^{2}))}{x^{4}} + \frac{12e^{sin(ln(x^{2}))}cos(ln(x^{2}))*2xsin(ln(x^{2}))}{x^{3}(x^{2})} + \frac{12e^{sin(ln(x^{2}))}cos(ln(x^{2}))*2x}{x^{3}(x^{2})}\\=&\frac{36e^{sin(ln(x^{2}))}cos(ln(x^{2}))}{x^{4}} - \frac{20e^{sin(ln(x^{2}))}cos^{2}(ln(x^{2}))}{x^{4}} - \frac{96e^{sin(ln(x^{2}))}sin(ln(x^{2}))cos^{2}(ln(x^{2}))}{x^{4}} - \frac{48e^{sin(ln(x^{2}))}cos^{3}(ln(x^{2}))}{x^{4}} + \frac{144e^{sin(ln(x^{2}))}sin(ln(x^{2}))cos(ln(x^{2}))}{x^{4}} + \frac{48e^{sin(ln(x^{2}))}sin^{2}(ln(x^{2}))}{x^{4}} + \frac{16e^{sin(ln(x^{2}))}cos^{4}(ln(x^{2}))}{x^{4}} - \frac{28e^{sin(ln(x^{2}))}sin(ln(x^{2}))}{x^{4}}\\ \end{split}\end{equation} \]