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 3 derivative of z is calculated.
Note that variables are case sensitive.
\begin{equation}\begin{split}[1/1]Find\ the\ third\ derivative\ of\ function\ ({(z - a{e}^{(\frac{ip}{4})})}^{4}){({z}^{4} + {a}^{4})}^{-1}\ with\ respect\ to\ z：\\\end{split}\end{equation}
\begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{z^{4}}{(z^{4} + a^{4})} - \frac{4az^{3}{e}^{(\frac{1}{4}ip)}}{(z^{4} + a^{4})} + \frac{6a^{2}z^{2}{e}^{(2(\frac{1}{4}ip))}}{(z^{4} + a^{4})} - \frac{4a^{3}z{e}^{(3(\frac{1}{4}ip))}}{(z^{4} + a^{4})} + \frac{a^{4}{e}^{(4(\frac{1}{4}ip))}}{(z^{4} + a^{4})}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{z^{4}}{(z^{4} + a^{4})} - \frac{4az^{3}{e}^{(\frac{1}{4}ip)}}{(z^{4} + a^{4})} + \frac{6a^{2}z^{2}{e}^{(2(\frac{1}{4}ip))}}{(z^{4} + a^{4})} - \frac{4a^{3}z{e}^{(3(\frac{1}{4}ip))}}{(z^{4} + a^{4})} + \frac{a^{4}{e}^{(4(\frac{1}{4}ip))}}{(z^{4} + a^{4})}\right)}{dz}\\=&(\frac{-(4z^{3} + 0)}{(z^{4} + a^{4})^{2}})z^{4} + \frac{4z^{3}}{(z^{4} + a^{4})} - 4(\frac{-(4z^{3} + 0)}{(z^{4} + a^{4})^{2}})az^{3}{e}^{(\frac{1}{4}ip)} - \frac{4a*3z^{2}{e}^{(\frac{1}{4}ip)}}{(z^{4} + a^{4})} - \frac{4az^{3}({e}^{(\frac{1}{4}ip)}((0)ln(e) + \frac{(\frac{1}{4}ip)(0)}{(e)}))}{(z^{4} + a^{4})} + 6(\frac{-(4z^{3} + 0)}{(z^{4} + a^{4})^{2}})a^{2}z^{2}{e}^{(2(\frac{1}{4}ip))} + \frac{6a^{2}*2z{e}^{(2(\frac{1}{4}ip))}}{(z^{4} + a^{4})} + \frac{6a^{2}z^{2}({e}^{(2(\frac{1}{4}ip))}((2(0))ln(e) + \frac{(2(\frac{1}{4}ip))(0)}{(e)}))}{(z^{4} + a^{4})} - 4(\frac{-(4z^{3} + 0)}{(z^{4} + a^{4})^{2}})a^{3}z{e}^{(3(\frac{1}{4}ip))} - \frac{4a^{3}{e}^{(3(\frac{1}{4}ip))}}{(z^{4} + a^{4})} - \frac{4a^{3}z({e}^{(3(\frac{1}{4}ip))}((3(0))ln(e) + \frac{(3(\frac{1}{4}ip))(0)}{(e)}))}{(z^{4} + a^{4})} + (\frac{-(4z^{3} + 0)}{(z^{4} + a^{4})^{2}})a^{4}{e}^{(4(\frac{1}{4}ip))} + \frac{a^{4}({e}^{(4(\frac{1}{4}ip))}((4(0))ln(e) + \frac{(4(\frac{1}{4}ip))(0)}{(e)}))}{(z^{4} + a^{4})}\\=&\frac{-4z^{7}}{(z^{4} + a^{4})^{2}} + \frac{4z^{3}}{(z^{4} + a^{4})} + \frac{16az^{6}{e}^{(\frac{1}{4}ip)}}{(z^{4} + a^{4})^{2}} - \frac{12az^{2}{e}^{(\frac{1}{4}ip)}}{(z^{4} + a^{4})} - \frac{24a^{2}z^{5}{e}^{(\frac{1}{2}ip)}}{(z^{4} + a^{4})^{2}} + \frac{12a^{2}z{e}^{(\frac{1}{2}ip)}}{(z^{4} + a^{4})} + \frac{16a^{3}z^{4}{e}^{(\frac{3}{4}ip)}}{(z^{4} + a^{4})^{2}} - \frac{4a^{3}{e}^{(\frac{3}{4}ip)}}{(z^{4} + a^{4})} - \frac{4a^{4}z^{3}{e}^{(ip)}}{(z^{4} + a^{4})^{2}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-4z^{7}}{(z^{4} + a^{4})^{2}} + \frac{4z^{3}}{(z^{4} + a^{4})} + \frac{16az^{6}{e}^{(\frac{1}{4}ip)}}{(z^{4} + a^{4})^{2}} - \frac{12az^{2}{e}^{(\frac{1}{4}ip)}}{(z^{4} + a^{4})} - \frac{24a^{2}z^{5}{e}^{(\frac{1}{2}ip)}}{(z^{4} + a^{4})^{2}} + \frac{12a^{2}z{e}^{(\frac{1}{2}ip)}}{(z^{4} + a^{4})} + \frac{16a^{3}z^{4}{e}^{(\frac{3}{4}ip)}}{(z^{4} + a^{4})^{2}} - \frac{4a^{3}{e}^{(\frac{3}{4}ip)}}{(z^{4} + a^{4})} - \frac{4a^{4}z^{3}{e}^{(ip)}}{(z^{4} + a^{4})^{2}}\right)}{dz}\\=&-4(\frac{-2(4z^{3} + 0)}{(z^{4} + a^{4})^{3}})z^{7} - \frac{4*7z^{6}}{(z^{4} + a^{4})^{2}} + 4(\frac{-(4z^{3} + 0)}{(z^{4} + a^{4})^{2}})z^{3} + \frac{4*3z^{2}}{(z^{4} + a^{4})} + 16(\frac{-2(4z^{3} + 0)}{(z^{4} + a^{4})^{3}})az^{6}{e}^{(\frac{1}{4}ip)} + \frac{16a*6z^{5}{e}^{(\frac{1}{4}ip)}}{(z^{4} + a^{4})^{2}} + \frac{16az^{6}({e}^{(\frac{1}{4}ip)}((0)ln(e) + \frac{(\frac{1}{4}ip)(0)}{(e)}))}{(z^{4} + a^{4})^{2}} - 12(\frac{-(4z^{3} + 0)}{(z^{4} + a^{4})^{2}})az^{2}{e}^{(\frac{1}{4}ip)} - \frac{12a*2z{e}^{(\frac{1}{4}ip)}}{(z^{4} + a^{4})} - \frac{12az^{2}({e}^{(\frac{1}{4}ip)}((0)ln(e) + \frac{(\frac{1}{4}ip)(0)}{(e)}))}{(z^{4} + a^{4})} - 24(\frac{-2(4z^{3} + 0)}{(z^{4} + a^{4})^{3}})a^{2}z^{5}{e}^{(\frac{1}{2}ip)} - \frac{24a^{2}*5z^{4}{e}^{(\frac{1}{2}ip)}}{(z^{4} + a^{4})^{2}} - \frac{24a^{2}z^{5}({e}^{(\frac{1}{2}ip)}((0)ln(e) + \frac{(\frac{1}{2}ip)(0)}{(e)}))}{(z^{4} + a^{4})^{2}} + 12(\frac{-(4z^{3} + 0)}{(z^{4} + a^{4})^{2}})a^{2}z{e}^{(\frac{1}{2}ip)} + \frac{12a^{2}{e}^{(\frac{1}{2}ip)}}{(z^{4} + a^{4})} + \frac{12a^{2}z({e}^{(\frac{1}{2}ip)}((0)ln(e) + \frac{(\frac{1}{2}ip)(0)}{(e)}))}{(z^{4} + a^{4})} + 16(\frac{-2(4z^{3} + 0)}{(z^{4} + a^{4})^{3}})a^{3}z^{4}{e}^{(\frac{3}{4}ip)} + \frac{16a^{3}*4z^{3}{e}^{(\frac{3}{4}ip)}}{(z^{4} + a^{4})^{2}} + \frac{16a^{3}z^{4}({e}^{(\frac{3}{4}ip)}((0)ln(e) + \frac{(\frac{3}{4}ip)(0)}{(e)}))}{(z^{4} + a^{4})^{2}} - 4(\frac{-(4z^{3} + 0)}{(z^{4} + a^{4})^{2}})a^{3}{e}^{(\frac{3}{4}ip)} - \frac{4a^{3}({e}^{(\frac{3}{4}ip)}((0)ln(e) + \frac{(\frac{3}{4}ip)(0)}{(e)}))}{(z^{4} + a^{4})} - 4(\frac{-2(4z^{3} + 0)}{(z^{4} + a^{4})^{3}})a^{4}z^{3}{e}^{(ip)} - \frac{4a^{4}*3z^{2}{e}^{(ip)}}{(z^{4} + a^{4})^{2}} - \frac{4a^{4}z^{3}({e}^{(ip)}((0)ln(e) + \frac{(ip)(0)}{(e)}))}{(z^{4} + a^{4})^{2}}\\=&\frac{32z^{10}}{(z^{4} + a^{4})^{3}} - \frac{44z^{6}}{(z^{4} + a^{4})^{2}} + \frac{12z^{2}}{(z^{4} + a^{4})} - \frac{128az^{9}{e}^{(\frac{1}{4}ip)}}{(z^{4} + a^{4})^{3}} + \frac{144az^{5}{e}^{(\frac{1}{4}ip)}}{(z^{4} + a^{4})^{2}} - \frac{24az{e}^{(\frac{1}{4}ip)}}{(z^{4} + a^{4})} + \frac{192a^{2}z^{8}{e}^{(\frac{1}{2}ip)}}{(z^{4} + a^{4})^{3}} - \frac{168a^{2}z^{4}{e}^{(\frac{1}{2}ip)}}{(z^{4} + a^{4})^{2}} + \frac{12a^{2}{e}^{(\frac{1}{2}ip)}}{(z^{4} + a^{4})} - \frac{128a^{3}z^{7}{e}^{(\frac{3}{4}ip)}}{(z^{4} + a^{4})^{3}} + \frac{80a^{3}z^{3}{e}^{(\frac{3}{4}ip)}}{(z^{4} + a^{4})^{2}} + \frac{32a^{4}z^{6}{e}^{(ip)}}{(z^{4} + a^{4})^{3}} - \frac{12a^{4}z^{2}{e}^{(ip)}}{(z^{4} + a^{4})^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{32z^{10}}{(z^{4} + a^{4})^{3}} - \frac{44z^{6}}{(z^{4} + a^{4})^{2}} + \frac{12z^{2}}{(z^{4} + a^{4})} - \frac{128az^{9}{e}^{(\frac{1}{4}ip)}}{(z^{4} + a^{4})^{3}} + \frac{144az^{5}{e}^{(\frac{1}{4}ip)}}{(z^{4} + a^{4})^{2}} - \frac{24az{e}^{(\frac{1}{4}ip)}}{(z^{4} + a^{4})} + \frac{192a^{2}z^{8}{e}^{(\frac{1}{2}ip)}}{(z^{4} + a^{4})^{3}} - \frac{168a^{2}z^{4}{e}^{(\frac{1}{2}ip)}}{(z^{4} + a^{4})^{2}} + \frac{12a^{2}{e}^{(\frac{1}{2}ip)}}{(z^{4} + a^{4})} - \frac{128a^{3}z^{7}{e}^{(\frac{3}{4}ip)}}{(z^{4} + a^{4})^{3}} + \frac{80a^{3}z^{3}{e}^{(\frac{3}{4}ip)}}{(z^{4} + a^{4})^{2}} + \frac{32a^{4}z^{6}{e}^{(ip)}}{(z^{4} + a^{4})^{3}} - \frac{12a^{4}z^{2}{e}^{(ip)}}{(z^{4} + a^{4})^{2}}\right)}{dz}\\=&32(\frac{-3(4z^{3} + 0)}{(z^{4} + a^{4})^{4}})z^{10} + \frac{32*10z^{9}}{(z^{4} + a^{4})^{3}} - 44(\frac{-2(4z^{3} + 0)}{(z^{4} + a^{4})^{3}})z^{6} - \frac{44*6z^{5}}{(z^{4} + a^{4})^{2}} + 12(\frac{-(4z^{3} + 0)}{(z^{4} + a^{4})^{2}})z^{2} + \frac{12*2z}{(z^{4} + a^{4})} - 128(\frac{-3(4z^{3} + 0)}{(z^{4} + a^{4})^{4}})az^{9}{e}^{(\frac{1}{4}ip)} - \frac{128a*9z^{8}{e}^{(\frac{1}{4}ip)}}{(z^{4} + a^{4})^{3}} - \frac{128az^{9}({e}^{(\frac{1}{4}ip)}((0)ln(e) + \frac{(\frac{1}{4}ip)(0)}{(e)}))}{(z^{4} + a^{4})^{3}} + 144(\frac{-2(4z^{3} + 0)}{(z^{4} + a^{4})^{3}})az^{5}{e}^{(\frac{1}{4}ip)} + \frac{144a*5z^{4}{e}^{(\frac{1}{4}ip)}}{(z^{4} + a^{4})^{2}} + \frac{144az^{5}({e}^{(\frac{1}{4}ip)}((0)ln(e) + \frac{(\frac{1}{4}ip)(0)}{(e)}))}{(z^{4} + a^{4})^{2}} - 24(\frac{-(4z^{3} + 0)}{(z^{4} + a^{4})^{2}})az{e}^{(\frac{1}{4}ip)} - \frac{24a{e}^{(\frac{1}{4}ip)}}{(z^{4} + a^{4})} - \frac{24az({e}^{(\frac{1}{4}ip)}((0)ln(e) + \frac{(\frac{1}{4}ip)(0)}{(e)}))}{(z^{4} + a^{4})} + 192(\frac{-3(4z^{3} + 0)}{(z^{4} + a^{4})^{4}})a^{2}z^{8}{e}^{(\frac{1}{2}ip)} + \frac{192a^{2}*8z^{7}{e}^{(\frac{1}{2}ip)}}{(z^{4} + a^{4})^{3}} + \frac{192a^{2}z^{8}({e}^{(\frac{1}{2}ip)}((0)ln(e) + \frac{(\frac{1}{2}ip)(0)}{(e)}))}{(z^{4} + a^{4})^{3}} - 168(\frac{-2(4z^{3} + 0)}{(z^{4} + a^{4})^{3}})a^{2}z^{4}{e}^{(\frac{1}{2}ip)} - \frac{168a^{2}*4z^{3}{e}^{(\frac{1}{2}ip)}}{(z^{4} + a^{4})^{2}} - \frac{168a^{2}z^{4}({e}^{(\frac{1}{2}ip)}((0)ln(e) + \frac{(\frac{1}{2}ip)(0)}{(e)}))}{(z^{4} + a^{4})^{2}} + 12(\frac{-(4z^{3} + 0)}{(z^{4} + a^{4})^{2}})a^{2}{e}^{(\frac{1}{2}ip)} + \frac{12a^{2}({e}^{(\frac{1}{2}ip)}((0)ln(e) + \frac{(\frac{1}{2}ip)(0)}{(e)}))}{(z^{4} + a^{4})} - 128(\frac{-3(4z^{3} + 0)}{(z^{4} + a^{4})^{4}})a^{3}z^{7}{e}^{(\frac{3}{4}ip)} - \frac{128a^{3}*7z^{6}{e}^{(\frac{3}{4}ip)}}{(z^{4} + a^{4})^{3}} - \frac{128a^{3}z^{7}({e}^{(\frac{3}{4}ip)}((0)ln(e) + \frac{(\frac{3}{4}ip)(0)}{(e)}))}{(z^{4} + a^{4})^{3}} + 80(\frac{-2(4z^{3} + 0)}{(z^{4} + a^{4})^{3}})a^{3}z^{3}{e}^{(\frac{3}{4}ip)} + \frac{80a^{3}*3z^{2}{e}^{(\frac{3}{4}ip)}}{(z^{4} + a^{4})^{2}} + \frac{80a^{3}z^{3}({e}^{(\frac{3}{4}ip)}((0)ln(e) + \frac{(\frac{3}{4}ip)(0)}{(e)}))}{(z^{4} + a^{4})^{2}} + 32(\frac{-3(4z^{3} + 0)}{(z^{4} + a^{4})^{4}})a^{4}z^{6}{e}^{(ip)} + \frac{32a^{4}*6z^{5}{e}^{(ip)}}{(z^{4} + a^{4})^{3}} + \frac{32a^{4}z^{6}({e}^{(ip)}((0)ln(e) + \frac{(ip)(0)}{(e)}))}{(z^{4} + a^{4})^{3}} - 12(\frac{-2(4z^{3} + 0)}{(z^{4} + a^{4})^{3}})a^{4}z^{2}{e}^{(ip)} - \frac{12a^{4}*2z{e}^{(ip)}}{(z^{4} + a^{4})^{2}} - \frac{12a^{4}z^{2}({e}^{(ip)}((0)ln(e) + \frac{(ip)(0)}{(e)}))}{(z^{4} + a^{4})^{2}}\\=&\frac{-384z^{13}}{(z^{4} + a^{4})^{4}} + \frac{672z^{9}}{(z^{4} + a^{4})^{3}} - \frac{312z^{5}}{(z^{4} + a^{4})^{2}} + \frac{24z}{(z^{4} + a^{4})} + \frac{1536az^{12}{e}^{(\frac{1}{4}ip)}}{(z^{4} + a^{4})^{4}} - \frac{2304az^{8}{e}^{(\frac{1}{4}ip)}}{(z^{4} + a^{4})^{3}} + \frac{816az^{4}{e}^{(\frac{1}{4}ip)}}{(z^{4} + a^{4})^{2}} - \frac{24a{e}^{(\frac{1}{4}ip)}}{(z^{4} + a^{4})} - \frac{2304a^{2}z^{11}{e}^{(\frac{1}{2}ip)}}{(z^{4} + a^{4})^{4}} + \frac{2880a^{2}z^{7}{e}^{(\frac{1}{2}ip)}}{(z^{4} + a^{4})^{3}} - \frac{720a^{2}z^{3}{e}^{(\frac{1}{2}ip)}}{(z^{4} + a^{4})^{2}} + \frac{1536a^{3}z^{10}{e}^{(\frac{3}{4}ip)}}{(z^{4} + a^{4})^{4}} - \frac{1536a^{3}z^{6}{e}^{(\frac{3}{4}ip)}}{(z^{4} + a^{4})^{3}} + \frac{240a^{3}z^{2}{e}^{(\frac{3}{4}ip)}}{(z^{4} + a^{4})^{2}} - \frac{384a^{4}z^{9}{e}^{(ip)}}{(z^{4} + a^{4})^{4}} + \frac{288a^{4}z^{5}{e}^{(ip)}}{(z^{4} + a^{4})^{3}} - \frac{24a^{4}z{e}^{(ip)}}{(z^{4} + a^{4})^{2}}\\ \end{split}\end{equation}