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.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ third\ derivative\ of\ function\ ({(z - \frac{a{e}^{i}arcsin(0)}{4})}^{4}){({z}^{4} + {a}^{4})}^{-1}\ with\ respect\ to\ z：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{z^{4}}{(z^{4} + a^{4})} - \frac{az^{3}{e}^{i}arcsin(0)}{(z^{4} + a^{4})} + \frac{\frac{3}{8}a^{2}z^{2}{e}^{(2(i))}arcsin^{2}(0)}{(z^{4} + a^{4})} - \frac{\frac{1}{16}a^{3}z{e}^{(3(i))}arcsin^{3}(0)}{(z^{4} + a^{4})} + \frac{\frac{1}{256}a^{4}{e}^{(4(i))}arcsin^{4}(0)}{(z^{4} + a^{4})}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{z^{4}}{(z^{4} + a^{4})} - \frac{az^{3}{e}^{i}arcsin(0)}{(z^{4} + a^{4})} + \frac{\frac{3}{8}a^{2}z^{2}{e}^{(2(i))}arcsin^{2}(0)}{(z^{4} + a^{4})} - \frac{\frac{1}{16}a^{3}z{e}^{(3(i))}arcsin^{3}(0)}{(z^{4} + a^{4})} + \frac{\frac{1}{256}a^{4}{e}^{(4(i))}arcsin^{4}(0)}{(z^{4} + a^{4})}\right)}{dz}\\=&(\frac{-(4z^{3} + 0)}{(z^{4} + a^{4})^{2}})z^{4} + \frac{4z^{3}}{(z^{4} + a^{4})} - (\frac{-(4z^{3} + 0)}{(z^{4} + a^{4})^{2}})az^{3}{e}^{i}arcsin(0) - \frac{a*3z^{2}{e}^{i}arcsin(0)}{(z^{4} + a^{4})} - \frac{az^{3}({e}^{i}((0)ln(e) + \frac{(i)(0)}{(e)}))arcsin(0)}{(z^{4} + a^{4})} - \frac{az^{3}{e}^{i}(\frac{(0)}{((1 - (0)^{2})^{\frac{1}{2}})})}{(z^{4} + a^{4})} + \frac{3}{8}(\frac{-(4z^{3} + 0)}{(z^{4} + a^{4})^{2}})a^{2}z^{2}{e}^{(2(i))}arcsin^{2}(0) + \frac{\frac{3}{8}a^{2}*2z{e}^{(2(i))}arcsin^{2}(0)}{(z^{4} + a^{4})} + \frac{\frac{3}{8}a^{2}z^{2}({e}^{(2(i))}((2(0))ln(e) + \frac{(2(i))(0)}{(e)}))arcsin^{2}(0)}{(z^{4} + a^{4})} + \frac{\frac{3}{8}a^{2}z^{2}{e}^{(2(i))}(\frac{2arcsin(0)(0)}{((1 - (0)^{2})^{\frac{1}{2}})})}{(z^{4} + a^{4})} - \frac{1}{16}(\frac{-(4z^{3} + 0)}{(z^{4} + a^{4})^{2}})a^{3}z{e}^{(3(i))}arcsin^{3}(0) - \frac{\frac{1}{16}a^{3}{e}^{(3(i))}arcsin^{3}(0)}{(z^{4} + a^{4})} - \frac{\frac{1}{16}a^{3}z({e}^{(3(i))}((3(0))ln(e) + \frac{(3(i))(0)}{(e)}))arcsin^{3}(0)}{(z^{4} + a^{4})} - \frac{\frac{1}{16}a^{3}z{e}^{(3(i))}(\frac{3arcsin^{2}(0)(0)}{((1 - (0)^{2})^{\frac{1}{2}})})}{(z^{4} + a^{4})} + \frac{1}{256}(\frac{-(4z^{3} + 0)}{(z^{4} + a^{4})^{2}})a^{4}{e}^{(4(i))}arcsin^{4}(0) + \frac{\frac{1}{256}a^{4}({e}^{(4(i))}((4(0))ln(e) + \frac{(4(i))(0)}{(e)}))arcsin^{4}(0)}{(z^{4} + a^{4})} + \frac{\frac{1}{256}a^{4}{e}^{(4(i))}(\frac{4arcsin^{3}(0)(0)}{((1 - (0)^{2})^{\frac{1}{2}})})}{(z^{4} + a^{4})}\\=&\frac{-4z^{7}}{(z^{4} + a^{4})^{2}} + \frac{4z^{3}}{(z^{4} + a^{4})} + \frac{4az^{6}{e}^{i}arcsin(0)}{(z^{4} + a^{4})^{2}} - \frac{3az^{2}{e}^{i}arcsin(0)}{(z^{4} + a^{4})} - \frac{3a^{2}z^{5}{e}^{(2i)}arcsin^{2}(0)}{2(z^{4} + a^{4})^{2}} + \frac{3a^{2}z{e}^{(2i)}arcsin^{2}(0)}{4(z^{4} + a^{4})} + \frac{a^{3}z^{4}{e}^{(3i)}arcsin^{3}(0)}{4(z^{4} + a^{4})^{2}} - \frac{a^{3}{e}^{(3i)}arcsin^{3}(0)}{16(z^{4} + a^{4})} - \frac{a^{4}z^{3}{e}^{(4i)}arcsin^{4}(0)}{64(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{4az^{6}{e}^{i}arcsin(0)}{(z^{4} + a^{4})^{2}} - \frac{3az^{2}{e}^{i}arcsin(0)}{(z^{4} + a^{4})} - \frac{3a^{2}z^{5}{e}^{(2i)}arcsin^{2}(0)}{2(z^{4} + a^{4})^{2}} + \frac{3a^{2}z{e}^{(2i)}arcsin^{2}(0)}{4(z^{4} + a^{4})} + \frac{a^{3}z^{4}{e}^{(3i)}arcsin^{3}(0)}{4(z^{4} + a^{4})^{2}} - \frac{a^{3}{e}^{(3i)}arcsin^{3}(0)}{16(z^{4} + a^{4})} - \frac{a^{4}z^{3}{e}^{(4i)}arcsin^{4}(0)}{64(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})} + 4(\frac{-2(4z^{3} + 0)}{(z^{4} + a^{4})^{3}})az^{6}{e}^{i}arcsin(0) + \frac{4a*6z^{5}{e}^{i}arcsin(0)}{(z^{4} + a^{4})^{2}} + \frac{4az^{6}({e}^{i}((0)ln(e) + \frac{(i)(0)}{(e)}))arcsin(0)}{(z^{4} + a^{4})^{2}} + \frac{4az^{6}{e}^{i}(\frac{(0)}{((1 - (0)^{2})^{\frac{1}{2}})})}{(z^{4} + a^{4})^{2}} - 3(\frac{-(4z^{3} + 0)}{(z^{4} + a^{4})^{2}})az^{2}{e}^{i}arcsin(0) - \frac{3a*2z{e}^{i}arcsin(0)}{(z^{4} + a^{4})} - \frac{3az^{2}({e}^{i}((0)ln(e) + \frac{(i)(0)}{(e)}))arcsin(0)}{(z^{4} + a^{4})} - \frac{3az^{2}{e}^{i}(\frac{(0)}{((1 - (0)^{2})^{\frac{1}{2}})})}{(z^{4} + a^{4})} - \frac{3(\frac{-2(4z^{3} + 0)}{(z^{4} + a^{4})^{3}})a^{2}z^{5}{e}^{(2i)}arcsin^{2}(0)}{2} - \frac{3a^{2}*5z^{4}{e}^{(2i)}arcsin^{2}(0)}{2(z^{4} + a^{4})^{2}} - \frac{3a^{2}z^{5}({e}^{(2i)}((0)ln(e) + \frac{(2i)(0)}{(e)}))arcsin^{2}(0)}{2(z^{4} + a^{4})^{2}} - \frac{3a^{2}z^{5}{e}^{(2i)}(\frac{2arcsin(0)(0)}{((1 - (0)^{2})^{\frac{1}{2}})})}{2(z^{4} + a^{4})^{2}} + \frac{3(\frac{-(4z^{3} + 0)}{(z^{4} + a^{4})^{2}})a^{2}z{e}^{(2i)}arcsin^{2}(0)}{4} + \frac{3a^{2}{e}^{(2i)}arcsin^{2}(0)}{4(z^{4} + a^{4})} + \frac{3a^{2}z({e}^{(2i)}((0)ln(e) + \frac{(2i)(0)}{(e)}))arcsin^{2}(0)}{4(z^{4} + a^{4})} + \frac{3a^{2}z{e}^{(2i)}(\frac{2arcsin(0)(0)}{((1 - (0)^{2})^{\frac{1}{2}})})}{4(z^{4} + a^{4})} + \frac{(\frac{-2(4z^{3} + 0)}{(z^{4} + a^{4})^{3}})a^{3}z^{4}{e}^{(3i)}arcsin^{3}(0)}{4} + \frac{a^{3}*4z^{3}{e}^{(3i)}arcsin^{3}(0)}{4(z^{4} + a^{4})^{2}} + \frac{a^{3}z^{4}({e}^{(3i)}((0)ln(e) + \frac{(3i)(0)}{(e)}))arcsin^{3}(0)}{4(z^{4} + a^{4})^{2}} + \frac{a^{3}z^{4}{e}^{(3i)}(\frac{3arcsin^{2}(0)(0)}{((1 - (0)^{2})^{\frac{1}{2}})})}{4(z^{4} + a^{4})^{2}} - \frac{(\frac{-(4z^{3} + 0)}{(z^{4} + a^{4})^{2}})a^{3}{e}^{(3i)}arcsin^{3}(0)}{16} - \frac{a^{3}({e}^{(3i)}((0)ln(e) + \frac{(3i)(0)}{(e)}))arcsin^{3}(0)}{16(z^{4} + a^{4})} - \frac{a^{3}{e}^{(3i)}(\frac{3arcsin^{2}(0)(0)}{((1 - (0)^{2})^{\frac{1}{2}})})}{16(z^{4} + a^{4})} - \frac{(\frac{-2(4z^{3} + 0)}{(z^{4} + a^{4})^{3}})a^{4}z^{3}{e}^{(4i)}arcsin^{4}(0)}{64} - \frac{a^{4}*3z^{2}{e}^{(4i)}arcsin^{4}(0)}{64(z^{4} + a^{4})^{2}} - \frac{a^{4}z^{3}({e}^{(4i)}((0)ln(e) + \frac{(4i)(0)}{(e)}))arcsin^{4}(0)}{64(z^{4} + a^{4})^{2}} - \frac{a^{4}z^{3}{e}^{(4i)}(\frac{4arcsin^{3}(0)(0)}{((1 - (0)^{2})^{\frac{1}{2}})})}{64(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{32az^{9}{e}^{i}arcsin(0)}{(z^{4} + a^{4})^{3}} + \frac{36az^{5}{e}^{i}arcsin(0)}{(z^{4} + a^{4})^{2}} - \frac{6az{e}^{i}arcsin(0)}{(z^{4} + a^{4})} + \frac{12a^{2}z^{8}{e}^{(2i)}arcsin^{2}(0)}{(z^{4} + a^{4})^{3}} - \frac{21a^{2}z^{4}{e}^{(2i)}arcsin^{2}(0)}{2(z^{4} + a^{4})^{2}} + \frac{3a^{2}{e}^{(2i)}arcsin^{2}(0)}{4(z^{4} + a^{4})} - \frac{2a^{3}z^{7}{e}^{(3i)}arcsin^{3}(0)}{(z^{4} + a^{4})^{3}} + \frac{5a^{3}z^{3}{e}^{(3i)}arcsin^{3}(0)}{4(z^{4} + a^{4})^{2}} + \frac{a^{4}z^{6}{e}^{(4i)}arcsin^{4}(0)}{8(z^{4} + a^{4})^{3}} - \frac{3a^{4}z^{2}{e}^{(4i)}arcsin^{4}(0)}{64(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{32az^{9}{e}^{i}arcsin(0)}{(z^{4} + a^{4})^{3}} + \frac{36az^{5}{e}^{i}arcsin(0)}{(z^{4} + a^{4})^{2}} - \frac{6az{e}^{i}arcsin(0)}{(z^{4} + a^{4})} + \frac{12a^{2}z^{8}{e}^{(2i)}arcsin^{2}(0)}{(z^{4} + a^{4})^{3}} - \frac{21a^{2}z^{4}{e}^{(2i)}arcsin^{2}(0)}{2(z^{4} + a^{4})^{2}} + \frac{3a^{2}{e}^{(2i)}arcsin^{2}(0)}{4(z^{4} + a^{4})} - \frac{2a^{3}z^{7}{e}^{(3i)}arcsin^{3}(0)}{(z^{4} + a^{4})^{3}} + \frac{5a^{3}z^{3}{e}^{(3i)}arcsin^{3}(0)}{4(z^{4} + a^{4})^{2}} + \frac{a^{4}z^{6}{e}^{(4i)}arcsin^{4}(0)}{8(z^{4} + a^{4})^{3}} - \frac{3a^{4}z^{2}{e}^{(4i)}arcsin^{4}(0)}{64(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})} - 32(\frac{-3(4z^{3} + 0)}{(z^{4} + a^{4})^{4}})az^{9}{e}^{i}arcsin(0) - \frac{32a*9z^{8}{e}^{i}arcsin(0)}{(z^{4} + a^{4})^{3}} - \frac{32az^{9}({e}^{i}((0)ln(e) + \frac{(i)(0)}{(e)}))arcsin(0)}{(z^{4} + a^{4})^{3}} - \frac{32az^{9}{e}^{i}(\frac{(0)}{((1 - (0)^{2})^{\frac{1}{2}})})}{(z^{4} + a^{4})^{3}} + 36(\frac{-2(4z^{3} + 0)}{(z^{4} + a^{4})^{3}})az^{5}{e}^{i}arcsin(0) + \frac{36a*5z^{4}{e}^{i}arcsin(0)}{(z^{4} + a^{4})^{2}} + \frac{36az^{5}({e}^{i}((0)ln(e) + \frac{(i)(0)}{(e)}))arcsin(0)}{(z^{4} + a^{4})^{2}} + \frac{36az^{5}{e}^{i}(\frac{(0)}{((1 - (0)^{2})^{\frac{1}{2}})})}{(z^{4} + a^{4})^{2}} - 6(\frac{-(4z^{3} + 0)}{(z^{4} + a^{4})^{2}})az{e}^{i}arcsin(0) - \frac{6a{e}^{i}arcsin(0)}{(z^{4} + a^{4})} - \frac{6az({e}^{i}((0)ln(e) + \frac{(i)(0)}{(e)}))arcsin(0)}{(z^{4} + a^{4})} - \frac{6az{e}^{i}(\frac{(0)}{((1 - (0)^{2})^{\frac{1}{2}})})}{(z^{4} + a^{4})} + 12(\frac{-3(4z^{3} + 0)}{(z^{4} + a^{4})^{4}})a^{2}z^{8}{e}^{(2i)}arcsin^{2}(0) + \frac{12a^{2}*8z^{7}{e}^{(2i)}arcsin^{2}(0)}{(z^{4} + a^{4})^{3}} + \frac{12a^{2}z^{8}({e}^{(2i)}((0)ln(e) + \frac{(2i)(0)}{(e)}))arcsin^{2}(0)}{(z^{4} + a^{4})^{3}} + \frac{12a^{2}z^{8}{e}^{(2i)}(\frac{2arcsin(0)(0)}{((1 - (0)^{2})^{\frac{1}{2}})})}{(z^{4} + a^{4})^{3}} - \frac{21(\frac{-2(4z^{3} + 0)}{(z^{4} + a^{4})^{3}})a^{2}z^{4}{e}^{(2i)}arcsin^{2}(0)}{2} - \frac{21a^{2}*4z^{3}{e}^{(2i)}arcsin^{2}(0)}{2(z^{4} + a^{4})^{2}} - \frac{21a^{2}z^{4}({e}^{(2i)}((0)ln(e) + \frac{(2i)(0)}{(e)}))arcsin^{2}(0)}{2(z^{4} + a^{4})^{2}} - \frac{21a^{2}z^{4}{e}^{(2i)}(\frac{2arcsin(0)(0)}{((1 - (0)^{2})^{\frac{1}{2}})})}{2(z^{4} + a^{4})^{2}} + \frac{3(\frac{-(4z^{3} + 0)}{(z^{4} + a^{4})^{2}})a^{2}{e}^{(2i)}arcsin^{2}(0)}{4} + \frac{3a^{2}({e}^{(2i)}((0)ln(e) + \frac{(2i)(0)}{(e)}))arcsin^{2}(0)}{4(z^{4} + a^{4})} + \frac{3a^{2}{e}^{(2i)}(\frac{2arcsin(0)(0)}{((1 - (0)^{2})^{\frac{1}{2}})})}{4(z^{4} + a^{4})} - 2(\frac{-3(4z^{3} + 0)}{(z^{4} + a^{4})^{4}})a^{3}z^{7}{e}^{(3i)}arcsin^{3}(0) - \frac{2a^{3}*7z^{6}{e}^{(3i)}arcsin^{3}(0)}{(z^{4} + a^{4})^{3}} - \frac{2a^{3}z^{7}({e}^{(3i)}((0)ln(e) + \frac{(3i)(0)}{(e)}))arcsin^{3}(0)}{(z^{4} + a^{4})^{3}} - \frac{2a^{3}z^{7}{e}^{(3i)}(\frac{3arcsin^{2}(0)(0)}{((1 - (0)^{2})^{\frac{1}{2}})})}{(z^{4} + a^{4})^{3}} + \frac{5(\frac{-2(4z^{3} + 0)}{(z^{4} + a^{4})^{3}})a^{3}z^{3}{e}^{(3i)}arcsin^{3}(0)}{4} + \frac{5a^{3}*3z^{2}{e}^{(3i)}arcsin^{3}(0)}{4(z^{4} + a^{4})^{2}} + \frac{5a^{3}z^{3}({e}^{(3i)}((0)ln(e) + \frac{(3i)(0)}{(e)}))arcsin^{3}(0)}{4(z^{4} + a^{4})^{2}} + \frac{5a^{3}z^{3}{e}^{(3i)}(\frac{3arcsin^{2}(0)(0)}{((1 - (0)^{2})^{\frac{1}{2}})})}{4(z^{4} + a^{4})^{2}} + \frac{(\frac{-3(4z^{3} + 0)}{(z^{4} + a^{4})^{4}})a^{4}z^{6}{e}^{(4i)}arcsin^{4}(0)}{8} + \frac{a^{4}*6z^{5}{e}^{(4i)}arcsin^{4}(0)}{8(z^{4} + a^{4})^{3}} + \frac{a^{4}z^{6}({e}^{(4i)}((0)ln(e) + \frac{(4i)(0)}{(e)}))arcsin^{4}(0)}{8(z^{4} + a^{4})^{3}} + \frac{a^{4}z^{6}{e}^{(4i)}(\frac{4arcsin^{3}(0)(0)}{((1 - (0)^{2})^{\frac{1}{2}})})}{8(z^{4} + a^{4})^{3}} - \frac{3(\frac{-2(4z^{3} + 0)}{(z^{4} + a^{4})^{3}})a^{4}z^{2}{e}^{(4i)}arcsin^{4}(0)}{64} - \frac{3a^{4}*2z{e}^{(4i)}arcsin^{4}(0)}{64(z^{4} + a^{4})^{2}} - \frac{3a^{4}z^{2}({e}^{(4i)}((0)ln(e) + \frac{(4i)(0)}{(e)}))arcsin^{4}(0)}{64(z^{4} + a^{4})^{2}} - \frac{3a^{4}z^{2}{e}^{(4i)}(\frac{4arcsin^{3}(0)(0)}{((1 - (0)^{2})^{\frac{1}{2}})})}{64(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{384az^{12}{e}^{i}arcsin(0)}{(z^{4} + a^{4})^{4}} - \frac{576az^{8}{e}^{i}arcsin(0)}{(z^{4} + a^{4})^{3}} + \frac{204az^{4}{e}^{i}arcsin(0)}{(z^{4} + a^{4})^{2}} - \frac{6a{e}^{i}arcsin(0)}{(z^{4} + a^{4})} - \frac{144a^{2}z^{11}{e}^{(2i)}arcsin^{2}(0)}{(z^{4} + a^{4})^{4}} + \frac{180a^{2}z^{7}{e}^{(2i)}arcsin^{2}(0)}{(z^{4} + a^{4})^{3}} - \frac{45a^{2}z^{3}{e}^{(2i)}arcsin^{2}(0)}{(z^{4} + a^{4})^{2}} + \frac{24a^{3}z^{10}{e}^{(3i)}arcsin^{3}(0)}{(z^{4} + a^{4})^{4}} - \frac{24a^{3}z^{6}{e}^{(3i)}arcsin^{3}(0)}{(z^{4} + a^{4})^{3}} + \frac{15a^{3}z^{2}{e}^{(3i)}arcsin^{3}(0)}{4(z^{4} + a^{4})^{2}} - \frac{3a^{4}z^{9}{e}^{(4i)}arcsin^{4}(0)}{2(z^{4} + a^{4})^{4}} + \frac{9a^{4}z^{5}{e}^{(4i)}arcsin^{4}(0)}{8(z^{4} + a^{4})^{3}} - \frac{3a^{4}z{e}^{(4i)}arcsin^{4}(0)}{32(z^{4} + a^{4})^{2}}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ third\ derivative\ of\ function\ ({(z - \frac{a{e}^{i}arcsin(0)}{4})}^{4}){({z}^{4} + {a}^{4})}^{-1}\ with\ respect\ to\ z：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{z^{4}}{(z^{4} + a^{4})} - \frac{az^{3}{e}^{i}arcsin(0)}{(z^{4} + a^{4})} + \frac{\frac{3}{8}a^{2}z^{2}{e}^{(2(i))}arcsin^{2}(0)}{(z^{4} + a^{4})} - \frac{\frac{1}{16}a^{3}z{e}^{(3(i))}arcsin^{3}(0)}{(z^{4} + a^{4})} + \frac{\frac{1}{256}a^{4}{e}^{(4(i))}arcsin^{4}(0)}{(z^{4} + a^{4})}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{z^{4}}{(z^{4} + a^{4})} - \frac{az^{3}{e}^{i}arcsin(0)}{(z^{4} + a^{4})} + \frac{\frac{3}{8}a^{2}z^{2}{e}^{(2(i))}arcsin^{2}(0)}{(z^{4} + a^{4})} - \frac{\frac{1}{16}a^{3}z{e}^{(3(i))}arcsin^{3}(0)}{(z^{4} + a^{4})} + \frac{\frac{1}{256}a^{4}{e}^{(4(i))}arcsin^{4}(0)}{(z^{4} + a^{4})}\right)}{dz}\\=&(\frac{-(4z^{3} + 0)}{(z^{4} + a^{4})^{2}})z^{4} + \frac{4z^{3}}{(z^{4} + a^{4})} - (\frac{-(4z^{3} + 0)}{(z^{4} + a^{4})^{2}})az^{3}{e}^{i}arcsin(0) - \frac{a*3z^{2}{e}^{i}arcsin(0)}{(z^{4} + a^{4})} - \frac{az^{3}({e}^{i}((0)ln(e) + \frac{(i)(0)}{(e)}))arcsin(0)}{(z^{4} + a^{4})} - \frac{az^{3}{e}^{i}(\frac{(0)}{((1 - (0)^{2})^{\frac{1}{2}})})}{(z^{4} + a^{4})} + \frac{3}{8}(\frac{-(4z^{3} + 0)}{(z^{4} + a^{4})^{2}})a^{2}z^{2}{e}^{(2(i))}arcsin^{2}(0) + \frac{\frac{3}{8}a^{2}*2z{e}^{(2(i))}arcsin^{2}(0)}{(z^{4} + a^{4})} + \frac{\frac{3}{8}a^{2}z^{2}({e}^{(2(i))}((2(0))ln(e) + \frac{(2(i))(0)}{(e)}))arcsin^{2}(0)}{(z^{4} + a^{4})} + \frac{\frac{3}{8}a^{2}z^{2}{e}^{(2(i))}(\frac{2arcsin(0)(0)}{((1 - (0)^{2})^{\frac{1}{2}})})}{(z^{4} + a^{4})} - \frac{1}{16}(\frac{-(4z^{3} + 0)}{(z^{4} + a^{4})^{2}})a^{3}z{e}^{(3(i))}arcsin^{3}(0) - \frac{\frac{1}{16}a^{3}{e}^{(3(i))}arcsin^{3}(0)}{(z^{4} + a^{4})} - \frac{\frac{1}{16}a^{3}z({e}^{(3(i))}((3(0))ln(e) + \frac{(3(i))(0)}{(e)}))arcsin^{3}(0)}{(z^{4} + a^{4})} - \frac{\frac{1}{16}a^{3}z{e}^{(3(i))}(\frac{3arcsin^{2}(0)(0)}{((1 - (0)^{2})^{\frac{1}{2}})})}{(z^{4} + a^{4})} + \frac{1}{256}(\frac{-(4z^{3} + 0)}{(z^{4} + a^{4})^{2}})a^{4}{e}^{(4(i))}arcsin^{4}(0) + \frac{\frac{1}{256}a^{4}({e}^{(4(i))}((4(0))ln(e) + \frac{(4(i))(0)}{(e)}))arcsin^{4}(0)}{(z^{4} + a^{4})} + \frac{\frac{1}{256}a^{4}{e}^{(4(i))}(\frac{4arcsin^{3}(0)(0)}{((1 - (0)^{2})^{\frac{1}{2}})})}{(z^{4} + a^{4})}\\=&\frac{-4z^{7}}{(z^{4} + a^{4})^{2}} + \frac{4z^{3}}{(z^{4} + a^{4})} + \frac{4az^{6}{e}^{i}arcsin(0)}{(z^{4} + a^{4})^{2}} - \frac{3az^{2}{e}^{i}arcsin(0)}{(z^{4} + a^{4})} - \frac{3a^{2}z^{5}{e}^{(2i)}arcsin^{2}(0)}{2(z^{4} + a^{4})^{2}} + \frac{3a^{2}z{e}^{(2i)}arcsin^{2}(0)}{4(z^{4} + a^{4})} + \frac{a^{3}z^{4}{e}^{(3i)}arcsin^{3}(0)}{4(z^{4} + a^{4})^{2}} - \frac{a^{3}{e}^{(3i)}arcsin^{3}(0)}{16(z^{4} + a^{4})} - \frac{a^{4}z^{3}{e}^{(4i)}arcsin^{4}(0)}{64(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{4az^{6}{e}^{i}arcsin(0)}{(z^{4} + a^{4})^{2}} - \frac{3az^{2}{e}^{i}arcsin(0)}{(z^{4} + a^{4})} - \frac{3a^{2}z^{5}{e}^{(2i)}arcsin^{2}(0)}{2(z^{4} + a^{4})^{2}} + \frac{3a^{2}z{e}^{(2i)}arcsin^{2}(0)}{4(z^{4} + a^{4})} + \frac{a^{3}z^{4}{e}^{(3i)}arcsin^{3}(0)}{4(z^{4} + a^{4})^{2}} - \frac{a^{3}{e}^{(3i)}arcsin^{3}(0)}{16(z^{4} + a^{4})} - \frac{a^{4}z^{3}{e}^{(4i)}arcsin^{4}(0)}{64(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})} + 4(\frac{-2(4z^{3} + 0)}{(z^{4} + a^{4})^{3}})az^{6}{e}^{i}arcsin(0) + \frac{4a*6z^{5}{e}^{i}arcsin(0)}{(z^{4} + a^{4})^{2}} + \frac{4az^{6}({e}^{i}((0)ln(e) + \frac{(i)(0)}{(e)}))arcsin(0)}{(z^{4} + a^{4})^{2}} + \frac{4az^{6}{e}^{i}(\frac{(0)}{((1 - (0)^{2})^{\frac{1}{2}})})}{(z^{4} + a^{4})^{2}} - 3(\frac{-(4z^{3} + 0)}{(z^{4} + a^{4})^{2}})az^{2}{e}^{i}arcsin(0) - \frac{3a*2z{e}^{i}arcsin(0)}{(z^{4} + a^{4})} - \frac{3az^{2}({e}^{i}((0)ln(e) + \frac{(i)(0)}{(e)}))arcsin(0)}{(z^{4} + a^{4})} - \frac{3az^{2}{e}^{i}(\frac{(0)}{((1 - (0)^{2})^{\frac{1}{2}})})}{(z^{4} + a^{4})} - \frac{3(\frac{-2(4z^{3} + 0)}{(z^{4} + a^{4})^{3}})a^{2}z^{5}{e}^{(2i)}arcsin^{2}(0)}{2} - \frac{3a^{2}*5z^{4}{e}^{(2i)}arcsin^{2}(0)}{2(z^{4} + a^{4})^{2}} - \frac{3a^{2}z^{5}({e}^{(2i)}((0)ln(e) + \frac{(2i)(0)}{(e)}))arcsin^{2}(0)}{2(z^{4} + a^{4})^{2}} - \frac{3a^{2}z^{5}{e}^{(2i)}(\frac{2arcsin(0)(0)}{((1 - (0)^{2})^{\frac{1}{2}})})}{2(z^{4} + a^{4})^{2}} + \frac{3(\frac{-(4z^{3} + 0)}{(z^{4} + a^{4})^{2}})a^{2}z{e}^{(2i)}arcsin^{2}(0)}{4} + \frac{3a^{2}{e}^{(2i)}arcsin^{2}(0)}{4(z^{4} + a^{4})} + \frac{3a^{2}z({e}^{(2i)}((0)ln(e) + \frac{(2i)(0)}{(e)}))arcsin^{2}(0)}{4(z^{4} + a^{4})} + \frac{3a^{2}z{e}^{(2i)}(\frac{2arcsin(0)(0)}{((1 - (0)^{2})^{\frac{1}{2}})})}{4(z^{4} + a^{4})} + \frac{(\frac{-2(4z^{3} + 0)}{(z^{4} + a^{4})^{3}})a^{3}z^{4}{e}^{(3i)}arcsin^{3}(0)}{4} + \frac{a^{3}*4z^{3}{e}^{(3i)}arcsin^{3}(0)}{4(z^{4} + a^{4})^{2}} + \frac{a^{3}z^{4}({e}^{(3i)}((0)ln(e) + \frac{(3i)(0)}{(e)}))arcsin^{3}(0)}{4(z^{4} + a^{4})^{2}} + \frac{a^{3}z^{4}{e}^{(3i)}(\frac{3arcsin^{2}(0)(0)}{((1 - (0)^{2})^{\frac{1}{2}})})}{4(z^{4} + a^{4})^{2}} - \frac{(\frac{-(4z^{3} + 0)}{(z^{4} + a^{4})^{2}})a^{3}{e}^{(3i)}arcsin^{3}(0)}{16} - \frac{a^{3}({e}^{(3i)}((0)ln(e) + \frac{(3i)(0)}{(e)}))arcsin^{3}(0)}{16(z^{4} + a^{4})} - \frac{a^{3}{e}^{(3i)}(\frac{3arcsin^{2}(0)(0)}{((1 - (0)^{2})^{\frac{1}{2}})})}{16(z^{4} + a^{4})} - \frac{(\frac{-2(4z^{3} + 0)}{(z^{4} + a^{4})^{3}})a^{4}z^{3}{e}^{(4i)}arcsin^{4}(0)}{64} - \frac{a^{4}*3z^{2}{e}^{(4i)}arcsin^{4}(0)}{64(z^{4} + a^{4})^{2}} - \frac{a^{4}z^{3}({e}^{(4i)}((0)ln(e) + \frac{(4i)(0)}{(e)}))arcsin^{4}(0)}{64(z^{4} + a^{4})^{2}} - \frac{a^{4}z^{3}{e}^{(4i)}(\frac{4arcsin^{3}(0)(0)}{((1 - (0)^{2})^{\frac{1}{2}})})}{64(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{32az^{9}{e}^{i}arcsin(0)}{(z^{4} + a^{4})^{3}} + \frac{36az^{5}{e}^{i}arcsin(0)}{(z^{4} + a^{4})^{2}} - \frac{6az{e}^{i}arcsin(0)}{(z^{4} + a^{4})} + \frac{12a^{2}z^{8}{e}^{(2i)}arcsin^{2}(0)}{(z^{4} + a^{4})^{3}} - \frac{21a^{2}z^{4}{e}^{(2i)}arcsin^{2}(0)}{2(z^{4} + a^{4})^{2}} + \frac{3a^{2}{e}^{(2i)}arcsin^{2}(0)}{4(z^{4} + a^{4})} - \frac{2a^{3}z^{7}{e}^{(3i)}arcsin^{3}(0)}{(z^{4} + a^{4})^{3}} + \frac{5a^{3}z^{3}{e}^{(3i)}arcsin^{3}(0)}{4(z^{4} + a^{4})^{2}} + \frac{a^{4}z^{6}{e}^{(4i)}arcsin^{4}(0)}{8(z^{4} + a^{4})^{3}} - \frac{3a^{4}z^{2}{e}^{(4i)}arcsin^{4}(0)}{64(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{32az^{9}{e}^{i}arcsin(0)}{(z^{4} + a^{4})^{3}} + \frac{36az^{5}{e}^{i}arcsin(0)}{(z^{4} + a^{4})^{2}} - \frac{6az{e}^{i}arcsin(0)}{(z^{4} + a^{4})} + \frac{12a^{2}z^{8}{e}^{(2i)}arcsin^{2}(0)}{(z^{4} + a^{4})^{3}} - \frac{21a^{2}z^{4}{e}^{(2i)}arcsin^{2}(0)}{2(z^{4} + a^{4})^{2}} + \frac{3a^{2}{e}^{(2i)}arcsin^{2}(0)}{4(z^{4} + a^{4})} - \frac{2a^{3}z^{7}{e}^{(3i)}arcsin^{3}(0)}{(z^{4} + a^{4})^{3}} + \frac{5a^{3}z^{3}{e}^{(3i)}arcsin^{3}(0)}{4(z^{4} + a^{4})^{2}} + \frac{a^{4}z^{6}{e}^{(4i)}arcsin^{4}(0)}{8(z^{4} + a^{4})^{3}} - \frac{3a^{4}z^{2}{e}^{(4i)}arcsin^{4}(0)}{64(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})} - 32(\frac{-3(4z^{3} + 0)}{(z^{4} + a^{4})^{4}})az^{9}{e}^{i}arcsin(0) - \frac{32a*9z^{8}{e}^{i}arcsin(0)}{(z^{4} + a^{4})^{3}} - \frac{32az^{9}({e}^{i}((0)ln(e) + \frac{(i)(0)}{(e)}))arcsin(0)}{(z^{4} + a^{4})^{3}} - \frac{32az^{9}{e}^{i}(\frac{(0)}{((1 - (0)^{2})^{\frac{1}{2}})})}{(z^{4} + a^{4})^{3}} + 36(\frac{-2(4z^{3} + 0)}{(z^{4} + a^{4})^{3}})az^{5}{e}^{i}arcsin(0) + \frac{36a*5z^{4}{e}^{i}arcsin(0)}{(z^{4} + a^{4})^{2}} + \frac{36az^{5}({e}^{i}((0)ln(e) + \frac{(i)(0)}{(e)}))arcsin(0)}{(z^{4} + a^{4})^{2}} + \frac{36az^{5}{e}^{i}(\frac{(0)}{((1 - (0)^{2})^{\frac{1}{2}})})}{(z^{4} + a^{4})^{2}} - 6(\frac{-(4z^{3} + 0)}{(z^{4} + a^{4})^{2}})az{e}^{i}arcsin(0) - \frac{6a{e}^{i}arcsin(0)}{(z^{4} + a^{4})} - \frac{6az({e}^{i}((0)ln(e) + \frac{(i)(0)}{(e)}))arcsin(0)}{(z^{4} + a^{4})} - \frac{6az{e}^{i}(\frac{(0)}{((1 - (0)^{2})^{\frac{1}{2}})})}{(z^{4} + a^{4})} + 12(\frac{-3(4z^{3} + 0)}{(z^{4} + a^{4})^{4}})a^{2}z^{8}{e}^{(2i)}arcsin^{2}(0) + \frac{12a^{2}*8z^{7}{e}^{(2i)}arcsin^{2}(0)}{(z^{4} + a^{4})^{3}} + \frac{12a^{2}z^{8}({e}^{(2i)}((0)ln(e) + \frac{(2i)(0)}{(e)}))arcsin^{2}(0)}{(z^{4} + a^{4})^{3}} + \frac{12a^{2}z^{8}{e}^{(2i)}(\frac{2arcsin(0)(0)}{((1 - (0)^{2})^{\frac{1}{2}})})}{(z^{4} + a^{4})^{3}} - \frac{21(\frac{-2(4z^{3} + 0)}{(z^{4} + a^{4})^{3}})a^{2}z^{4}{e}^{(2i)}arcsin^{2}(0)}{2} - \frac{21a^{2}*4z^{3}{e}^{(2i)}arcsin^{2}(0)}{2(z^{4} + a^{4})^{2}} - \frac{21a^{2}z^{4}({e}^{(2i)}((0)ln(e) + \frac{(2i)(0)}{(e)}))arcsin^{2}(0)}{2(z^{4} + a^{4})^{2}} - \frac{21a^{2}z^{4}{e}^{(2i)}(\frac{2arcsin(0)(0)}{((1 - (0)^{2})^{\frac{1}{2}})})}{2(z^{4} + a^{4})^{2}} + \frac{3(\frac{-(4z^{3} + 0)}{(z^{4} + a^{4})^{2}})a^{2}{e}^{(2i)}arcsin^{2}(0)}{4} + \frac{3a^{2}({e}^{(2i)}((0)ln(e) + \frac{(2i)(0)}{(e)}))arcsin^{2}(0)}{4(z^{4} + a^{4})} + \frac{3a^{2}{e}^{(2i)}(\frac{2arcsin(0)(0)}{((1 - (0)^{2})^{\frac{1}{2}})})}{4(z^{4} + a^{4})} - 2(\frac{-3(4z^{3} + 0)}{(z^{4} + a^{4})^{4}})a^{3}z^{7}{e}^{(3i)}arcsin^{3}(0) - \frac{2a^{3}*7z^{6}{e}^{(3i)}arcsin^{3}(0)}{(z^{4} + a^{4})^{3}} - \frac{2a^{3}z^{7}({e}^{(3i)}((0)ln(e) + \frac{(3i)(0)}{(e)}))arcsin^{3}(0)}{(z^{4} + a^{4})^{3}} - \frac{2a^{3}z^{7}{e}^{(3i)}(\frac{3arcsin^{2}(0)(0)}{((1 - (0)^{2})^{\frac{1}{2}})})}{(z^{4} + a^{4})^{3}} + \frac{5(\frac{-2(4z^{3} + 0)}{(z^{4} + a^{4})^{3}})a^{3}z^{3}{e}^{(3i)}arcsin^{3}(0)}{4} + \frac{5a^{3}*3z^{2}{e}^{(3i)}arcsin^{3}(0)}{4(z^{4} + a^{4})^{2}} + \frac{5a^{3}z^{3}({e}^{(3i)}((0)ln(e) + \frac{(3i)(0)}{(e)}))arcsin^{3}(0)}{4(z^{4} + a^{4})^{2}} + \frac{5a^{3}z^{3}{e}^{(3i)}(\frac{3arcsin^{2}(0)(0)}{((1 - (0)^{2})^{\frac{1}{2}})})}{4(z^{4} + a^{4})^{2}} + \frac{(\frac{-3(4z^{3} + 0)}{(z^{4} + a^{4})^{4}})a^{4}z^{6}{e}^{(4i)}arcsin^{4}(0)}{8} + \frac{a^{4}*6z^{5}{e}^{(4i)}arcsin^{4}(0)}{8(z^{4} + a^{4})^{3}} + \frac{a^{4}z^{6}({e}^{(4i)}((0)ln(e) + \frac{(4i)(0)}{(e)}))arcsin^{4}(0)}{8(z^{4} + a^{4})^{3}} + \frac{a^{4}z^{6}{e}^{(4i)}(\frac{4arcsin^{3}(0)(0)}{((1 - (0)^{2})^{\frac{1}{2}})})}{8(z^{4} + a^{4})^{3}} - \frac{3(\frac{-2(4z^{3} + 0)}{(z^{4} + a^{4})^{3}})a^{4}z^{2}{e}^{(4i)}arcsin^{4}(0)}{64} - \frac{3a^{4}*2z{e}^{(4i)}arcsin^{4}(0)}{64(z^{4} + a^{4})^{2}} - \frac{3a^{4}z^{2}({e}^{(4i)}((0)ln(e) + \frac{(4i)(0)}{(e)}))arcsin^{4}(0)}{64(z^{4} + a^{4})^{2}} - \frac{3a^{4}z^{2}{e}^{(4i)}(\frac{4arcsin^{3}(0)(0)}{((1 - (0)^{2})^{\frac{1}{2}})})}{64(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{384az^{12}{e}^{i}arcsin(0)}{(z^{4} + a^{4})^{4}} - \frac{576az^{8}{e}^{i}arcsin(0)}{(z^{4} + a^{4})^{3}} + \frac{204az^{4}{e}^{i}arcsin(0)}{(z^{4} + a^{4})^{2}} - \frac{6a{e}^{i}arcsin(0)}{(z^{4} + a^{4})} - \frac{144a^{2}z^{11}{e}^{(2i)}arcsin^{2}(0)}{(z^{4} + a^{4})^{4}} + \frac{180a^{2}z^{7}{e}^{(2i)}arcsin^{2}(0)}{(z^{4} + a^{4})^{3}} - \frac{45a^{2}z^{3}{e}^{(2i)}arcsin^{2}(0)}{(z^{4} + a^{4})^{2}} + \frac{24a^{3}z^{10}{e}^{(3i)}arcsin^{3}(0)}{(z^{4} + a^{4})^{4}} - \frac{24a^{3}z^{6}{e}^{(3i)}arcsin^{3}(0)}{(z^{4} + a^{4})^{3}} + \frac{15a^{3}z^{2}{e}^{(3i)}arcsin^{3}(0)}{4(z^{4} + a^{4})^{2}} - \frac{3a^{4}z^{9}{e}^{(4i)}arcsin^{4}(0)}{2(z^{4} + a^{4})^{4}} + \frac{9a^{4}z^{5}{e}^{(4i)}arcsin^{4}(0)}{8(z^{4} + a^{4})^{3}} - \frac{3a^{4}z{e}^{(4i)}arcsin^{4}(0)}{32(z^{4} + a^{4})^{2}}\\ \end{split}\end{equation} \]