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.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ arctan(cot(sqrt(e^{x})))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( arctan(cot(sqrt(e^{x})))\right)}{dx}\\=&(\frac{(\frac{-csc^{2}(sqrt(e^{x}))e^{x}*\frac{1}{2}}{(e^{x})^{\frac{1}{2}}})}{(1 + (cot(sqrt(e^{x})))^{2})})\\=&\frac{-e^{{x}*{\frac{1}{2}}}csc^{2}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-e^{{x}*{\frac{1}{2}}}csc^{2}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)}\right)}{dx}\\=&\frac{-(\frac{-(\frac{-2cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))e^{x}*\frac{1}{2}}{(e^{x})^{\frac{1}{2}}} + 0)}{(cot^{2}(sqrt(e^{x})) + 1)^{2}})e^{{x}*{\frac{1}{2}}}csc^{2}(sqrt(e^{x}))}{2} - \frac{\frac{1}{2}e^{x}csc^{2}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)e^{{x}*{\frac{1}{2}}}} - \frac{e^{{x}*{\frac{1}{2}}}*-2csc^{2}(sqrt(e^{x}))cot(sqrt(e^{x}))e^{x}*\frac{1}{2}}{2(cot^{2}(sqrt(e^{x})) + 1)(e^{x})^{\frac{1}{2}}}\\=&\frac{-e^{x}cot(sqrt(e^{x}))csc^{4}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)^{2}} - \frac{e^{{x}*{\frac{1}{2}}}csc^{2}(sqrt(e^{x}))}{4(cot^{2}(sqrt(e^{x})) + 1)} + \frac{e^{x}cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-e^{x}cot(sqrt(e^{x}))csc^{4}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)^{2}} - \frac{e^{{x}*{\frac{1}{2}}}csc^{2}(sqrt(e^{x}))}{4(cot^{2}(sqrt(e^{x})) + 1)} + \frac{e^{x}cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)}\right)}{dx}\\=&\frac{-(\frac{-2(\frac{-2cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))e^{x}*\frac{1}{2}}{(e^{x})^{\frac{1}{2}}} + 0)}{(cot^{2}(sqrt(e^{x})) + 1)^{3}})e^{x}cot(sqrt(e^{x}))csc^{4}(sqrt(e^{x}))}{2} - \frac{e^{x}cot(sqrt(e^{x}))csc^{4}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)^{2}} - \frac{e^{x}*-csc^{2}(sqrt(e^{x}))e^{x}*\frac{1}{2}csc^{4}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)^{2}(e^{x})^{\frac{1}{2}}} - \frac{e^{x}cot(sqrt(e^{x}))*-4csc^{4}(sqrt(e^{x}))cot(sqrt(e^{x}))e^{x}*\frac{1}{2}}{2(cot^{2}(sqrt(e^{x})) + 1)^{2}(e^{x})^{\frac{1}{2}}} - \frac{(\frac{-(\frac{-2cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))e^{x}*\frac{1}{2}}{(e^{x})^{\frac{1}{2}}} + 0)}{(cot^{2}(sqrt(e^{x})) + 1)^{2}})e^{{x}*{\frac{1}{2}}}csc^{2}(sqrt(e^{x}))}{4} - \frac{\frac{1}{2}e^{x}csc^{2}(sqrt(e^{x}))}{4(cot^{2}(sqrt(e^{x})) + 1)e^{{x}*{\frac{1}{2}}}} - \frac{e^{{x}*{\frac{1}{2}}}*-2csc^{2}(sqrt(e^{x}))cot(sqrt(e^{x}))e^{x}*\frac{1}{2}}{4(cot^{2}(sqrt(e^{x})) + 1)(e^{x})^{\frac{1}{2}}} + \frac{(\frac{-(\frac{-2cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))e^{x}*\frac{1}{2}}{(e^{x})^{\frac{1}{2}}} + 0)}{(cot^{2}(sqrt(e^{x})) + 1)^{2}})e^{x}cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))}{2} + \frac{e^{x}cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)} + \frac{e^{x}*-csc^{2}(sqrt(e^{x}))e^{x}*\frac{1}{2}csc^{2}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)(e^{x})^{\frac{1}{2}}} + \frac{e^{x}cot(sqrt(e^{x}))*-2csc^{2}(sqrt(e^{x}))cot(sqrt(e^{x}))e^{x}*\frac{1}{2}}{2(cot^{2}(sqrt(e^{x})) + 1)(e^{x})^{\frac{1}{2}}}\\=&\frac{-e^{{x}*{\frac{3}{2}}}cot^{2}(sqrt(e^{x}))csc^{6}(sqrt(e^{x}))}{(cot^{2}(sqrt(e^{x})) + 1)^{3}} - \frac{3e^{x}cot(sqrt(e^{x}))csc^{4}(sqrt(e^{x}))}{4(cot^{2}(sqrt(e^{x})) + 1)^{2}} + \frac{e^{{x}*{\frac{3}{2}}}csc^{6}(sqrt(e^{x}))}{4(cot^{2}(sqrt(e^{x})) + 1)^{2}} + \frac{3e^{{x}*{\frac{3}{2}}}cot^{2}(sqrt(e^{x}))csc^{4}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)^{2}} - \frac{e^{{x}*{\frac{1}{2}}}csc^{2}(sqrt(e^{x}))}{8(cot^{2}(sqrt(e^{x})) + 1)} + \frac{3e^{x}cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))}{4(cot^{2}(sqrt(e^{x})) + 1)} - \frac{e^{{x}*{\frac{3}{2}}}csc^{4}(sqrt(e^{x}))}{4(cot^{2}(sqrt(e^{x})) + 1)} - \frac{e^{{x}*{\frac{3}{2}}}cot^{2}(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-e^{{x}*{\frac{3}{2}}}cot^{2}(sqrt(e^{x}))csc^{6}(sqrt(e^{x}))}{(cot^{2}(sqrt(e^{x})) + 1)^{3}} - \frac{3e^{x}cot(sqrt(e^{x}))csc^{4}(sqrt(e^{x}))}{4(cot^{2}(sqrt(e^{x})) + 1)^{2}} + \frac{e^{{x}*{\frac{3}{2}}}csc^{6}(sqrt(e^{x}))}{4(cot^{2}(sqrt(e^{x})) + 1)^{2}} + \frac{3e^{{x}*{\frac{3}{2}}}cot^{2}(sqrt(e^{x}))csc^{4}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)^{2}} - \frac{e^{{x}*{\frac{1}{2}}}csc^{2}(sqrt(e^{x}))}{8(cot^{2}(sqrt(e^{x})) + 1)} + \frac{3e^{x}cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))}{4(cot^{2}(sqrt(e^{x})) + 1)} - \frac{e^{{x}*{\frac{3}{2}}}csc^{4}(sqrt(e^{x}))}{4(cot^{2}(sqrt(e^{x})) + 1)} - \frac{e^{{x}*{\frac{3}{2}}}cot^{2}(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)}\right)}{dx}\\=&-(\frac{-3(\frac{-2cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))e^{x}*\frac{1}{2}}{(e^{x})^{\frac{1}{2}}} + 0)}{(cot^{2}(sqrt(e^{x})) + 1)^{4}})e^{{x}*{\frac{3}{2}}}cot^{2}(sqrt(e^{x}))csc^{6}(sqrt(e^{x})) - \frac{\frac{3}{2}e^{{x}*{\frac{1}{2}}}e^{x}cot^{2}(sqrt(e^{x}))csc^{6}(sqrt(e^{x}))}{(cot^{2}(sqrt(e^{x})) + 1)^{3}} - \frac{e^{{x}*{\frac{3}{2}}}*-2cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))e^{x}*\frac{1}{2}csc^{6}(sqrt(e^{x}))}{(cot^{2}(sqrt(e^{x})) + 1)^{3}(e^{x})^{\frac{1}{2}}} - \frac{e^{{x}*{\frac{3}{2}}}cot^{2}(sqrt(e^{x}))*-6csc^{6}(sqrt(e^{x}))cot(sqrt(e^{x}))e^{x}*\frac{1}{2}}{(cot^{2}(sqrt(e^{x})) + 1)^{3}(e^{x})^{\frac{1}{2}}} - \frac{3(\frac{-2(\frac{-2cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))e^{x}*\frac{1}{2}}{(e^{x})^{\frac{1}{2}}} + 0)}{(cot^{2}(sqrt(e^{x})) + 1)^{3}})e^{x}cot(sqrt(e^{x}))csc^{4}(sqrt(e^{x}))}{4} - \frac{3e^{x}cot(sqrt(e^{x}))csc^{4}(sqrt(e^{x}))}{4(cot^{2}(sqrt(e^{x})) + 1)^{2}} - \frac{3e^{x}*-csc^{2}(sqrt(e^{x}))e^{x}*\frac{1}{2}csc^{4}(sqrt(e^{x}))}{4(cot^{2}(sqrt(e^{x})) + 1)^{2}(e^{x})^{\frac{1}{2}}} - \frac{3e^{x}cot(sqrt(e^{x}))*-4csc^{4}(sqrt(e^{x}))cot(sqrt(e^{x}))e^{x}*\frac{1}{2}}{4(cot^{2}(sqrt(e^{x})) + 1)^{2}(e^{x})^{\frac{1}{2}}} + \frac{(\frac{-2(\frac{-2cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))e^{x}*\frac{1}{2}}{(e^{x})^{\frac{1}{2}}} + 0)}{(cot^{2}(sqrt(e^{x})) + 1)^{3}})e^{{x}*{\frac{3}{2}}}csc^{6}(sqrt(e^{x}))}{4} + \frac{\frac{3}{2}e^{{x}*{\frac{1}{2}}}e^{x}csc^{6}(sqrt(e^{x}))}{4(cot^{2}(sqrt(e^{x})) + 1)^{2}} + \frac{e^{{x}*{\frac{3}{2}}}*-6csc^{6}(sqrt(e^{x}))cot(sqrt(e^{x}))e^{x}*\frac{1}{2}}{4(cot^{2}(sqrt(e^{x})) + 1)^{2}(e^{x})^{\frac{1}{2}}} + \frac{3(\frac{-2(\frac{-2cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))e^{x}*\frac{1}{2}}{(e^{x})^{\frac{1}{2}}} + 0)}{(cot^{2}(sqrt(e^{x})) + 1)^{3}})e^{{x}*{\frac{3}{2}}}cot^{2}(sqrt(e^{x}))csc^{4}(sqrt(e^{x}))}{2} + \frac{3*\frac{3}{2}e^{{x}*{\frac{1}{2}}}e^{x}cot^{2}(sqrt(e^{x}))csc^{4}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)^{2}} + \frac{3e^{{x}*{\frac{3}{2}}}*-2cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))e^{x}*\frac{1}{2}csc^{4}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)^{2}(e^{x})^{\frac{1}{2}}} + \frac{3e^{{x}*{\frac{3}{2}}}cot^{2}(sqrt(e^{x}))*-4csc^{4}(sqrt(e^{x}))cot(sqrt(e^{x}))e^{x}*\frac{1}{2}}{2(cot^{2}(sqrt(e^{x})) + 1)^{2}(e^{x})^{\frac{1}{2}}} - \frac{(\frac{-(\frac{-2cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))e^{x}*\frac{1}{2}}{(e^{x})^{\frac{1}{2}}} + 0)}{(cot^{2}(sqrt(e^{x})) + 1)^{2}})e^{{x}*{\frac{1}{2}}}csc^{2}(sqrt(e^{x}))}{8} - \frac{\frac{1}{2}e^{x}csc^{2}(sqrt(e^{x}))}{8(cot^{2}(sqrt(e^{x})) + 1)e^{{x}*{\frac{1}{2}}}} - \frac{e^{{x}*{\frac{1}{2}}}*-2csc^{2}(sqrt(e^{x}))cot(sqrt(e^{x}))e^{x}*\frac{1}{2}}{8(cot^{2}(sqrt(e^{x})) + 1)(e^{x})^{\frac{1}{2}}} + \frac{3(\frac{-(\frac{-2cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))e^{x}*\frac{1}{2}}{(e^{x})^{\frac{1}{2}}} + 0)}{(cot^{2}(sqrt(e^{x})) + 1)^{2}})e^{x}cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))}{4} + \frac{3e^{x}cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))}{4(cot^{2}(sqrt(e^{x})) + 1)} + \frac{3e^{x}*-csc^{2}(sqrt(e^{x}))e^{x}*\frac{1}{2}csc^{2}(sqrt(e^{x}))}{4(cot^{2}(sqrt(e^{x})) + 1)(e^{x})^{\frac{1}{2}}} + \frac{3e^{x}cot(sqrt(e^{x}))*-2csc^{2}(sqrt(e^{x}))cot(sqrt(e^{x}))e^{x}*\frac{1}{2}}{4(cot^{2}(sqrt(e^{x})) + 1)(e^{x})^{\frac{1}{2}}} - \frac{(\frac{-(\frac{-2cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))e^{x}*\frac{1}{2}}{(e^{x})^{\frac{1}{2}}} + 0)}{(cot^{2}(sqrt(e^{x})) + 1)^{2}})e^{{x}*{\frac{3}{2}}}csc^{4}(sqrt(e^{x}))}{4} - \frac{\frac{3}{2}e^{{x}*{\frac{1}{2}}}e^{x}csc^{4}(sqrt(e^{x}))}{4(cot^{2}(sqrt(e^{x})) + 1)} - \frac{e^{{x}*{\frac{3}{2}}}*-4csc^{4}(sqrt(e^{x}))cot(sqrt(e^{x}))e^{x}*\frac{1}{2}}{4(cot^{2}(sqrt(e^{x})) + 1)(e^{x})^{\frac{1}{2}}} - \frac{(\frac{-(\frac{-2cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))e^{x}*\frac{1}{2}}{(e^{x})^{\frac{1}{2}}} + 0)}{(cot^{2}(sqrt(e^{x})) + 1)^{2}})e^{{x}*{\frac{3}{2}}}cot^{2}(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))}{2} - \frac{\frac{3}{2}e^{{x}*{\frac{1}{2}}}e^{x}cot^{2}(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)} - \frac{e^{{x}*{\frac{3}{2}}}*-2cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))e^{x}*\frac{1}{2}csc^{2}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)(e^{x})^{\frac{1}{2}}} - \frac{e^{{x}*{\frac{3}{2}}}cot^{2}(sqrt(e^{x}))*-2csc^{2}(sqrt(e^{x}))cot(sqrt(e^{x}))e^{x}*\frac{1}{2}}{2(cot^{2}(sqrt(e^{x})) + 1)(e^{x})^{\frac{1}{2}}}\\=&\frac{-3e^{{x}*{2}}cot^{3}(sqrt(e^{x}))csc^{8}(sqrt(e^{x}))}{(cot^{2}(sqrt(e^{x})) + 1)^{4}} - \frac{3e^{{x}*{\frac{3}{2}}}cot^{2}(sqrt(e^{x}))csc^{6}(sqrt(e^{x}))}{(cot^{2}(sqrt(e^{x})) + 1)^{3}} + \frac{3e^{{x}*{2}}cot(sqrt(e^{x}))csc^{8}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)^{3}} + \frac{6e^{{x}*{2}}cot^{3}(sqrt(e^{x}))csc^{6}(sqrt(e^{x}))}{(cot^{2}(sqrt(e^{x})) + 1)^{3}} - \frac{7e^{x}cot(sqrt(e^{x}))csc^{4}(sqrt(e^{x}))}{8(cot^{2}(sqrt(e^{x})) + 1)^{2}} + \frac{3e^{{x}*{\frac{3}{2}}}csc^{6}(sqrt(e^{x}))}{4(cot^{2}(sqrt(e^{x})) + 1)^{2}} + \frac{9e^{{x}*{\frac{3}{2}}}cot^{2}(sqrt(e^{x}))csc^{4}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)^{2}} - \frac{5e^{{x}*{2}}cot(sqrt(e^{x}))csc^{6}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)^{2}} + \frac{e^{{x}*{2}}cot(sqrt(e^{x}))csc^{4}(sqrt(e^{x}))}{(cot^{2}(sqrt(e^{x})) + 1)} - \frac{7e^{{x}*{2}}cot^{3}(sqrt(e^{x}))csc^{4}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)^{2}} - \frac{e^{{x}*{\frac{1}{2}}}csc^{2}(sqrt(e^{x}))}{16(cot^{2}(sqrt(e^{x})) + 1)} + \frac{7e^{x}cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))}{8(cot^{2}(sqrt(e^{x})) + 1)} - \frac{3e^{{x}*{\frac{3}{2}}}csc^{4}(sqrt(e^{x}))}{4(cot^{2}(sqrt(e^{x})) + 1)} - \frac{3e^{{x}*{\frac{3}{2}}}cot^{2}(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)} + \frac{e^{{x}*{2}}cot^{3}(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ arctan(cot(sqrt(e^{x})))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( arctan(cot(sqrt(e^{x})))\right)}{dx}\\=&(\frac{(\frac{-csc^{2}(sqrt(e^{x}))e^{x}*\frac{1}{2}}{(e^{x})^{\frac{1}{2}}})}{(1 + (cot(sqrt(e^{x})))^{2})})\\=&\frac{-e^{{x}*{\frac{1}{2}}}csc^{2}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-e^{{x}*{\frac{1}{2}}}csc^{2}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)}\right)}{dx}\\=&\frac{-(\frac{-(\frac{-2cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))e^{x}*\frac{1}{2}}{(e^{x})^{\frac{1}{2}}} + 0)}{(cot^{2}(sqrt(e^{x})) + 1)^{2}})e^{{x}*{\frac{1}{2}}}csc^{2}(sqrt(e^{x}))}{2} - \frac{\frac{1}{2}e^{x}csc^{2}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)e^{{x}*{\frac{1}{2}}}} - \frac{e^{{x}*{\frac{1}{2}}}*-2csc^{2}(sqrt(e^{x}))cot(sqrt(e^{x}))e^{x}*\frac{1}{2}}{2(cot^{2}(sqrt(e^{x})) + 1)(e^{x})^{\frac{1}{2}}}\\=&\frac{-e^{x}cot(sqrt(e^{x}))csc^{4}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)^{2}} - \frac{e^{{x}*{\frac{1}{2}}}csc^{2}(sqrt(e^{x}))}{4(cot^{2}(sqrt(e^{x})) + 1)} + \frac{e^{x}cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-e^{x}cot(sqrt(e^{x}))csc^{4}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)^{2}} - \frac{e^{{x}*{\frac{1}{2}}}csc^{2}(sqrt(e^{x}))}{4(cot^{2}(sqrt(e^{x})) + 1)} + \frac{e^{x}cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)}\right)}{dx}\\=&\frac{-(\frac{-2(\frac{-2cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))e^{x}*\frac{1}{2}}{(e^{x})^{\frac{1}{2}}} + 0)}{(cot^{2}(sqrt(e^{x})) + 1)^{3}})e^{x}cot(sqrt(e^{x}))csc^{4}(sqrt(e^{x}))}{2} - \frac{e^{x}cot(sqrt(e^{x}))csc^{4}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)^{2}} - \frac{e^{x}*-csc^{2}(sqrt(e^{x}))e^{x}*\frac{1}{2}csc^{4}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)^{2}(e^{x})^{\frac{1}{2}}} - \frac{e^{x}cot(sqrt(e^{x}))*-4csc^{4}(sqrt(e^{x}))cot(sqrt(e^{x}))e^{x}*\frac{1}{2}}{2(cot^{2}(sqrt(e^{x})) + 1)^{2}(e^{x})^{\frac{1}{2}}} - \frac{(\frac{-(\frac{-2cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))e^{x}*\frac{1}{2}}{(e^{x})^{\frac{1}{2}}} + 0)}{(cot^{2}(sqrt(e^{x})) + 1)^{2}})e^{{x}*{\frac{1}{2}}}csc^{2}(sqrt(e^{x}))}{4} - \frac{\frac{1}{2}e^{x}csc^{2}(sqrt(e^{x}))}{4(cot^{2}(sqrt(e^{x})) + 1)e^{{x}*{\frac{1}{2}}}} - \frac{e^{{x}*{\frac{1}{2}}}*-2csc^{2}(sqrt(e^{x}))cot(sqrt(e^{x}))e^{x}*\frac{1}{2}}{4(cot^{2}(sqrt(e^{x})) + 1)(e^{x})^{\frac{1}{2}}} + \frac{(\frac{-(\frac{-2cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))e^{x}*\frac{1}{2}}{(e^{x})^{\frac{1}{2}}} + 0)}{(cot^{2}(sqrt(e^{x})) + 1)^{2}})e^{x}cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))}{2} + \frac{e^{x}cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)} + \frac{e^{x}*-csc^{2}(sqrt(e^{x}))e^{x}*\frac{1}{2}csc^{2}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)(e^{x})^{\frac{1}{2}}} + \frac{e^{x}cot(sqrt(e^{x}))*-2csc^{2}(sqrt(e^{x}))cot(sqrt(e^{x}))e^{x}*\frac{1}{2}}{2(cot^{2}(sqrt(e^{x})) + 1)(e^{x})^{\frac{1}{2}}}\\=&\frac{-e^{{x}*{\frac{3}{2}}}cot^{2}(sqrt(e^{x}))csc^{6}(sqrt(e^{x}))}{(cot^{2}(sqrt(e^{x})) + 1)^{3}} - \frac{3e^{x}cot(sqrt(e^{x}))csc^{4}(sqrt(e^{x}))}{4(cot^{2}(sqrt(e^{x})) + 1)^{2}} + \frac{e^{{x}*{\frac{3}{2}}}csc^{6}(sqrt(e^{x}))}{4(cot^{2}(sqrt(e^{x})) + 1)^{2}} + \frac{3e^{{x}*{\frac{3}{2}}}cot^{2}(sqrt(e^{x}))csc^{4}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)^{2}} - \frac{e^{{x}*{\frac{1}{2}}}csc^{2}(sqrt(e^{x}))}{8(cot^{2}(sqrt(e^{x})) + 1)} + \frac{3e^{x}cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))}{4(cot^{2}(sqrt(e^{x})) + 1)} - \frac{e^{{x}*{\frac{3}{2}}}csc^{4}(sqrt(e^{x}))}{4(cot^{2}(sqrt(e^{x})) + 1)} - \frac{e^{{x}*{\frac{3}{2}}}cot^{2}(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-e^{{x}*{\frac{3}{2}}}cot^{2}(sqrt(e^{x}))csc^{6}(sqrt(e^{x}))}{(cot^{2}(sqrt(e^{x})) + 1)^{3}} - \frac{3e^{x}cot(sqrt(e^{x}))csc^{4}(sqrt(e^{x}))}{4(cot^{2}(sqrt(e^{x})) + 1)^{2}} + \frac{e^{{x}*{\frac{3}{2}}}csc^{6}(sqrt(e^{x}))}{4(cot^{2}(sqrt(e^{x})) + 1)^{2}} + \frac{3e^{{x}*{\frac{3}{2}}}cot^{2}(sqrt(e^{x}))csc^{4}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)^{2}} - \frac{e^{{x}*{\frac{1}{2}}}csc^{2}(sqrt(e^{x}))}{8(cot^{2}(sqrt(e^{x})) + 1)} + \frac{3e^{x}cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))}{4(cot^{2}(sqrt(e^{x})) + 1)} - \frac{e^{{x}*{\frac{3}{2}}}csc^{4}(sqrt(e^{x}))}{4(cot^{2}(sqrt(e^{x})) + 1)} - \frac{e^{{x}*{\frac{3}{2}}}cot^{2}(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)}\right)}{dx}\\=&-(\frac{-3(\frac{-2cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))e^{x}*\frac{1}{2}}{(e^{x})^{\frac{1}{2}}} + 0)}{(cot^{2}(sqrt(e^{x})) + 1)^{4}})e^{{x}*{\frac{3}{2}}}cot^{2}(sqrt(e^{x}))csc^{6}(sqrt(e^{x})) - \frac{\frac{3}{2}e^{{x}*{\frac{1}{2}}}e^{x}cot^{2}(sqrt(e^{x}))csc^{6}(sqrt(e^{x}))}{(cot^{2}(sqrt(e^{x})) + 1)^{3}} - \frac{e^{{x}*{\frac{3}{2}}}*-2cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))e^{x}*\frac{1}{2}csc^{6}(sqrt(e^{x}))}{(cot^{2}(sqrt(e^{x})) + 1)^{3}(e^{x})^{\frac{1}{2}}} - \frac{e^{{x}*{\frac{3}{2}}}cot^{2}(sqrt(e^{x}))*-6csc^{6}(sqrt(e^{x}))cot(sqrt(e^{x}))e^{x}*\frac{1}{2}}{(cot^{2}(sqrt(e^{x})) + 1)^{3}(e^{x})^{\frac{1}{2}}} - \frac{3(\frac{-2(\frac{-2cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))e^{x}*\frac{1}{2}}{(e^{x})^{\frac{1}{2}}} + 0)}{(cot^{2}(sqrt(e^{x})) + 1)^{3}})e^{x}cot(sqrt(e^{x}))csc^{4}(sqrt(e^{x}))}{4} - \frac{3e^{x}cot(sqrt(e^{x}))csc^{4}(sqrt(e^{x}))}{4(cot^{2}(sqrt(e^{x})) + 1)^{2}} - \frac{3e^{x}*-csc^{2}(sqrt(e^{x}))e^{x}*\frac{1}{2}csc^{4}(sqrt(e^{x}))}{4(cot^{2}(sqrt(e^{x})) + 1)^{2}(e^{x})^{\frac{1}{2}}} - \frac{3e^{x}cot(sqrt(e^{x}))*-4csc^{4}(sqrt(e^{x}))cot(sqrt(e^{x}))e^{x}*\frac{1}{2}}{4(cot^{2}(sqrt(e^{x})) + 1)^{2}(e^{x})^{\frac{1}{2}}} + \frac{(\frac{-2(\frac{-2cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))e^{x}*\frac{1}{2}}{(e^{x})^{\frac{1}{2}}} + 0)}{(cot^{2}(sqrt(e^{x})) + 1)^{3}})e^{{x}*{\frac{3}{2}}}csc^{6}(sqrt(e^{x}))}{4} + \frac{\frac{3}{2}e^{{x}*{\frac{1}{2}}}e^{x}csc^{6}(sqrt(e^{x}))}{4(cot^{2}(sqrt(e^{x})) + 1)^{2}} + \frac{e^{{x}*{\frac{3}{2}}}*-6csc^{6}(sqrt(e^{x}))cot(sqrt(e^{x}))e^{x}*\frac{1}{2}}{4(cot^{2}(sqrt(e^{x})) + 1)^{2}(e^{x})^{\frac{1}{2}}} + \frac{3(\frac{-2(\frac{-2cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))e^{x}*\frac{1}{2}}{(e^{x})^{\frac{1}{2}}} + 0)}{(cot^{2}(sqrt(e^{x})) + 1)^{3}})e^{{x}*{\frac{3}{2}}}cot^{2}(sqrt(e^{x}))csc^{4}(sqrt(e^{x}))}{2} + \frac{3*\frac{3}{2}e^{{x}*{\frac{1}{2}}}e^{x}cot^{2}(sqrt(e^{x}))csc^{4}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)^{2}} + \frac{3e^{{x}*{\frac{3}{2}}}*-2cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))e^{x}*\frac{1}{2}csc^{4}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)^{2}(e^{x})^{\frac{1}{2}}} + \frac{3e^{{x}*{\frac{3}{2}}}cot^{2}(sqrt(e^{x}))*-4csc^{4}(sqrt(e^{x}))cot(sqrt(e^{x}))e^{x}*\frac{1}{2}}{2(cot^{2}(sqrt(e^{x})) + 1)^{2}(e^{x})^{\frac{1}{2}}} - \frac{(\frac{-(\frac{-2cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))e^{x}*\frac{1}{2}}{(e^{x})^{\frac{1}{2}}} + 0)}{(cot^{2}(sqrt(e^{x})) + 1)^{2}})e^{{x}*{\frac{1}{2}}}csc^{2}(sqrt(e^{x}))}{8} - \frac{\frac{1}{2}e^{x}csc^{2}(sqrt(e^{x}))}{8(cot^{2}(sqrt(e^{x})) + 1)e^{{x}*{\frac{1}{2}}}} - \frac{e^{{x}*{\frac{1}{2}}}*-2csc^{2}(sqrt(e^{x}))cot(sqrt(e^{x}))e^{x}*\frac{1}{2}}{8(cot^{2}(sqrt(e^{x})) + 1)(e^{x})^{\frac{1}{2}}} + \frac{3(\frac{-(\frac{-2cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))e^{x}*\frac{1}{2}}{(e^{x})^{\frac{1}{2}}} + 0)}{(cot^{2}(sqrt(e^{x})) + 1)^{2}})e^{x}cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))}{4} + \frac{3e^{x}cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))}{4(cot^{2}(sqrt(e^{x})) + 1)} + \frac{3e^{x}*-csc^{2}(sqrt(e^{x}))e^{x}*\frac{1}{2}csc^{2}(sqrt(e^{x}))}{4(cot^{2}(sqrt(e^{x})) + 1)(e^{x})^{\frac{1}{2}}} + \frac{3e^{x}cot(sqrt(e^{x}))*-2csc^{2}(sqrt(e^{x}))cot(sqrt(e^{x}))e^{x}*\frac{1}{2}}{4(cot^{2}(sqrt(e^{x})) + 1)(e^{x})^{\frac{1}{2}}} - \frac{(\frac{-(\frac{-2cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))e^{x}*\frac{1}{2}}{(e^{x})^{\frac{1}{2}}} + 0)}{(cot^{2}(sqrt(e^{x})) + 1)^{2}})e^{{x}*{\frac{3}{2}}}csc^{4}(sqrt(e^{x}))}{4} - \frac{\frac{3}{2}e^{{x}*{\frac{1}{2}}}e^{x}csc^{4}(sqrt(e^{x}))}{4(cot^{2}(sqrt(e^{x})) + 1)} - \frac{e^{{x}*{\frac{3}{2}}}*-4csc^{4}(sqrt(e^{x}))cot(sqrt(e^{x}))e^{x}*\frac{1}{2}}{4(cot^{2}(sqrt(e^{x})) + 1)(e^{x})^{\frac{1}{2}}} - \frac{(\frac{-(\frac{-2cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))e^{x}*\frac{1}{2}}{(e^{x})^{\frac{1}{2}}} + 0)}{(cot^{2}(sqrt(e^{x})) + 1)^{2}})e^{{x}*{\frac{3}{2}}}cot^{2}(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))}{2} - \frac{\frac{3}{2}e^{{x}*{\frac{1}{2}}}e^{x}cot^{2}(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)} - \frac{e^{{x}*{\frac{3}{2}}}*-2cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))e^{x}*\frac{1}{2}csc^{2}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)(e^{x})^{\frac{1}{2}}} - \frac{e^{{x}*{\frac{3}{2}}}cot^{2}(sqrt(e^{x}))*-2csc^{2}(sqrt(e^{x}))cot(sqrt(e^{x}))e^{x}*\frac{1}{2}}{2(cot^{2}(sqrt(e^{x})) + 1)(e^{x})^{\frac{1}{2}}}\\=&\frac{-3e^{{x}*{2}}cot^{3}(sqrt(e^{x}))csc^{8}(sqrt(e^{x}))}{(cot^{2}(sqrt(e^{x})) + 1)^{4}} - \frac{3e^{{x}*{\frac{3}{2}}}cot^{2}(sqrt(e^{x}))csc^{6}(sqrt(e^{x}))}{(cot^{2}(sqrt(e^{x})) + 1)^{3}} + \frac{3e^{{x}*{2}}cot(sqrt(e^{x}))csc^{8}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)^{3}} + \frac{6e^{{x}*{2}}cot^{3}(sqrt(e^{x}))csc^{6}(sqrt(e^{x}))}{(cot^{2}(sqrt(e^{x})) + 1)^{3}} - \frac{7e^{x}cot(sqrt(e^{x}))csc^{4}(sqrt(e^{x}))}{8(cot^{2}(sqrt(e^{x})) + 1)^{2}} + \frac{3e^{{x}*{\frac{3}{2}}}csc^{6}(sqrt(e^{x}))}{4(cot^{2}(sqrt(e^{x})) + 1)^{2}} + \frac{9e^{{x}*{\frac{3}{2}}}cot^{2}(sqrt(e^{x}))csc^{4}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)^{2}} - \frac{5e^{{x}*{2}}cot(sqrt(e^{x}))csc^{6}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)^{2}} + \frac{e^{{x}*{2}}cot(sqrt(e^{x}))csc^{4}(sqrt(e^{x}))}{(cot^{2}(sqrt(e^{x})) + 1)} - \frac{7e^{{x}*{2}}cot^{3}(sqrt(e^{x}))csc^{4}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)^{2}} - \frac{e^{{x}*{\frac{1}{2}}}csc^{2}(sqrt(e^{x}))}{16(cot^{2}(sqrt(e^{x})) + 1)} + \frac{7e^{x}cot(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))}{8(cot^{2}(sqrt(e^{x})) + 1)} - \frac{3e^{{x}*{\frac{3}{2}}}csc^{4}(sqrt(e^{x}))}{4(cot^{2}(sqrt(e^{x})) + 1)} - \frac{3e^{{x}*{\frac{3}{2}}}cot^{2}(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)} + \frac{e^{{x}*{2}}cot^{3}(sqrt(e^{x}))csc^{2}(sqrt(e^{x}))}{2(cot^{2}(sqrt(e^{x})) + 1)}\\ \end{split}\end{equation} \]