求导函数：
输入一个原函数（即需要求导的函数），然后设置需要求导的变量和求导的阶数，点击“下一步”按钮，即可获得该函数相应阶数的导函数。
注意，输入的函数支持数学函数和其它常量。
输入一个原函数（即需要求导的函数），然后设置需要求导的变量和求导的阶数，点击“下一步”按钮，即可获得该函数相应阶数的导函数。
注意，输入的函数支持数学函数和其它常量。
本次共计算 1 个题目：每一题对 t 求 4 阶导数。
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数\frac{dn}{(4e^{\frac{1}{5}t}dt + 2t)} 关于 t 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{dn}{(4dte^{\frac{1}{5}t} + 2t)}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( \frac{dn}{(4dte^{\frac{1}{5}t} + 2t)}\right)}{dt}\\=&(\frac{-(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{2}})dn + 0\\=&\frac{-4d^{2}ne^{\frac{1}{5}t}}{(4dte^{\frac{1}{5}t} + 2t)^{2}} - \frac{4d^{2}nte^{\frac{1}{5}t}}{5(4dte^{\frac{1}{5}t} + 2t)^{2}} - \frac{2dn}{(4dte^{\frac{1}{5}t} + 2t)^{2}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{-4d^{2}ne^{\frac{1}{5}t}}{(4dte^{\frac{1}{5}t} + 2t)^{2}} - \frac{4d^{2}nte^{\frac{1}{5}t}}{5(4dte^{\frac{1}{5}t} + 2t)^{2}} - \frac{2dn}{(4dte^{\frac{1}{5}t} + 2t)^{2}}\right)}{dt}\\=&-4(\frac{-2(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{3}})d^{2}ne^{\frac{1}{5}t} - \frac{4d^{2}ne^{\frac{1}{5}t}*\frac{1}{5}}{(4dte^{\frac{1}{5}t} + 2t)^{2}} - \frac{4(\frac{-2(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{3}})d^{2}nte^{\frac{1}{5}t}}{5} - \frac{4d^{2}ne^{\frac{1}{5}t}}{5(4dte^{\frac{1}{5}t} + 2t)^{2}} - \frac{4d^{2}nte^{\frac{1}{5}t}*\frac{1}{5}}{5(4dte^{\frac{1}{5}t} + 2t)^{2}} - 2(\frac{-2(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{3}})dn + 0\\=&\frac{32d^{3}ne^{{\frac{1}{5}t}*{2}}}{(4dte^{\frac{1}{5}t} + 2t)^{3}} + \frac{64d^{3}nte^{{\frac{1}{5}t}*{2}}}{5(4dte^{\frac{1}{5}t} + 2t)^{3}} + \frac{32d^{2}ne^{\frac{1}{5}t}}{(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{8d^{2}ne^{\frac{1}{5}t}}{5(4dte^{\frac{1}{5}t} + 2t)^{2}} + \frac{32d^{3}nt^{2}e^{{\frac{1}{5}t}*{2}}}{25(4dte^{\frac{1}{5}t} + 2t)^{3}} + \frac{32d^{2}nte^{\frac{1}{5}t}}{5(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{4d^{2}nte^{\frac{1}{5}t}}{25(4dte^{\frac{1}{5}t} + 2t)^{2}} + \frac{8dn}{(4dte^{\frac{1}{5}t} + 2t)^{3}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{32d^{3}ne^{{\frac{1}{5}t}*{2}}}{(4dte^{\frac{1}{5}t} + 2t)^{3}} + \frac{64d^{3}nte^{{\frac{1}{5}t}*{2}}}{5(4dte^{\frac{1}{5}t} + 2t)^{3}} + \frac{32d^{2}ne^{\frac{1}{5}t}}{(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{8d^{2}ne^{\frac{1}{5}t}}{5(4dte^{\frac{1}{5}t} + 2t)^{2}} + \frac{32d^{3}nt^{2}e^{{\frac{1}{5}t}*{2}}}{25(4dte^{\frac{1}{5}t} + 2t)^{3}} + \frac{32d^{2}nte^{\frac{1}{5}t}}{5(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{4d^{2}nte^{\frac{1}{5}t}}{25(4dte^{\frac{1}{5}t} + 2t)^{2}} + \frac{8dn}{(4dte^{\frac{1}{5}t} + 2t)^{3}}\right)}{dt}\\=&32(\frac{-3(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{4}})d^{3}ne^{{\frac{1}{5}t}*{2}} + \frac{32d^{3}n*2e^{\frac{1}{5}t}e^{\frac{1}{5}t}*\frac{1}{5}}{(4dte^{\frac{1}{5}t} + 2t)^{3}} + \frac{64(\frac{-3(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{4}})d^{3}nte^{{\frac{1}{5}t}*{2}}}{5} + \frac{64d^{3}ne^{{\frac{1}{5}t}*{2}}}{5(4dte^{\frac{1}{5}t} + 2t)^{3}} + \frac{64d^{3}nt*2e^{\frac{1}{5}t}e^{\frac{1}{5}t}*\frac{1}{5}}{5(4dte^{\frac{1}{5}t} + 2t)^{3}} + 32(\frac{-3(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{4}})d^{2}ne^{\frac{1}{5}t} + \frac{32d^{2}ne^{\frac{1}{5}t}*\frac{1}{5}}{(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{8(\frac{-2(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{3}})d^{2}ne^{\frac{1}{5}t}}{5} - \frac{8d^{2}ne^{\frac{1}{5}t}*\frac{1}{5}}{5(4dte^{\frac{1}{5}t} + 2t)^{2}} + \frac{32(\frac{-3(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{4}})d^{3}nt^{2}e^{{\frac{1}{5}t}*{2}}}{25} + \frac{32d^{3}n*2te^{{\frac{1}{5}t}*{2}}}{25(4dte^{\frac{1}{5}t} + 2t)^{3}} + \frac{32d^{3}nt^{2}*2e^{\frac{1}{5}t}e^{\frac{1}{5}t}*\frac{1}{5}}{25(4dte^{\frac{1}{5}t} + 2t)^{3}} + \frac{32(\frac{-3(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{4}})d^{2}nte^{\frac{1}{5}t}}{5} + \frac{32d^{2}ne^{\frac{1}{5}t}}{5(4dte^{\frac{1}{5}t} + 2t)^{3}} + \frac{32d^{2}nte^{\frac{1}{5}t}*\frac{1}{5}}{5(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{4(\frac{-2(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{3}})d^{2}nte^{\frac{1}{5}t}}{25} - \frac{4d^{2}ne^{\frac{1}{5}t}}{25(4dte^{\frac{1}{5}t} + 2t)^{2}} - \frac{4d^{2}nte^{\frac{1}{5}t}*\frac{1}{5}}{25(4dte^{\frac{1}{5}t} + 2t)^{2}} + 8(\frac{-3(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{4}})dn + 0\\=&\frac{-384d^{4}ne^{{\frac{1}{5}t}*{3}}}{(4dte^{\frac{1}{5}t} + 2t)^{4}} - \frac{1152d^{4}nte^{{\frac{1}{5}t}*{3}}}{5(4dte^{\frac{1}{5}t} + 2t)^{4}} - \frac{576d^{3}ne^{{\frac{1}{5}t}*{2}}}{(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{192d^{3}ne^{{\frac{1}{5}t}*{2}}}{5(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{1152d^{4}nt^{2}e^{{\frac{1}{5}t}*{3}}}{25(4dte^{\frac{1}{5}t} + 2t)^{4}} - \frac{1152d^{3}nte^{{\frac{1}{5}t}*{2}}}{5(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{288d^{3}nte^{{\frac{1}{5}t}*{2}}}{25(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{288d^{2}ne^{\frac{1}{5}t}}{(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{96d^{2}ne^{\frac{1}{5}t}}{5(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{12d^{2}ne^{\frac{1}{5}t}}{25(4dte^{\frac{1}{5}t} + 2t)^{2}} - \frac{384d^{4}nt^{3}e^{{\frac{1}{5}t}*{3}}}{125(4dte^{\frac{1}{5}t} + 2t)^{4}} - \frac{576d^{3}nt^{2}e^{{\frac{1}{5}t}*{2}}}{25(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{96d^{3}nt^{2}e^{{\frac{1}{5}t}*{2}}}{125(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{288d^{2}nte^{\frac{1}{5}t}}{5(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{48d^{2}nte^{\frac{1}{5}t}}{25(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{4d^{2}nte^{\frac{1}{5}t}}{125(4dte^{\frac{1}{5}t} + 2t)^{2}} - \frac{48dn}{(4dte^{\frac{1}{5}t} + 2t)^{4}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{-384d^{4}ne^{{\frac{1}{5}t}*{3}}}{(4dte^{\frac{1}{5}t} + 2t)^{4}} - \frac{1152d^{4}nte^{{\frac{1}{5}t}*{3}}}{5(4dte^{\frac{1}{5}t} + 2t)^{4}} - \frac{576d^{3}ne^{{\frac{1}{5}t}*{2}}}{(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{192d^{3}ne^{{\frac{1}{5}t}*{2}}}{5(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{1152d^{4}nt^{2}e^{{\frac{1}{5}t}*{3}}}{25(4dte^{\frac{1}{5}t} + 2t)^{4}} - \frac{1152d^{3}nte^{{\frac{1}{5}t}*{2}}}{5(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{288d^{3}nte^{{\frac{1}{5}t}*{2}}}{25(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{288d^{2}ne^{\frac{1}{5}t}}{(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{96d^{2}ne^{\frac{1}{5}t}}{5(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{12d^{2}ne^{\frac{1}{5}t}}{25(4dte^{\frac{1}{5}t} + 2t)^{2}} - \frac{384d^{4}nt^{3}e^{{\frac{1}{5}t}*{3}}}{125(4dte^{\frac{1}{5}t} + 2t)^{4}} - \frac{576d^{3}nt^{2}e^{{\frac{1}{5}t}*{2}}}{25(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{96d^{3}nt^{2}e^{{\frac{1}{5}t}*{2}}}{125(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{288d^{2}nte^{\frac{1}{5}t}}{5(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{48d^{2}nte^{\frac{1}{5}t}}{25(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{4d^{2}nte^{\frac{1}{5}t}}{125(4dte^{\frac{1}{5}t} + 2t)^{2}} - \frac{48dn}{(4dte^{\frac{1}{5}t} + 2t)^{4}}\right)}{dt}\\=&-384(\frac{-4(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{5}})d^{4}ne^{{\frac{1}{5}t}*{3}} - \frac{384d^{4}n*3e^{{\frac{1}{5}t}*{2}}e^{\frac{1}{5}t}*\frac{1}{5}}{(4dte^{\frac{1}{5}t} + 2t)^{4}} - \frac{1152(\frac{-4(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{5}})d^{4}nte^{{\frac{1}{5}t}*{3}}}{5} - \frac{1152d^{4}ne^{{\frac{1}{5}t}*{3}}}{5(4dte^{\frac{1}{5}t} + 2t)^{4}} - \frac{1152d^{4}nt*3e^{{\frac{1}{5}t}*{2}}e^{\frac{1}{5}t}*\frac{1}{5}}{5(4dte^{\frac{1}{5}t} + 2t)^{4}} - 576(\frac{-4(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{5}})d^{3}ne^{{\frac{1}{5}t}*{2}} - \frac{576d^{3}n*2e^{\frac{1}{5}t}e^{\frac{1}{5}t}*\frac{1}{5}}{(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{192(\frac{-3(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{4}})d^{3}ne^{{\frac{1}{5}t}*{2}}}{5} + \frac{192d^{3}n*2e^{\frac{1}{5}t}e^{\frac{1}{5}t}*\frac{1}{5}}{5(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{1152(\frac{-4(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{5}})d^{4}nt^{2}e^{{\frac{1}{5}t}*{3}}}{25} - \frac{1152d^{4}n*2te^{{\frac{1}{5}t}*{3}}}{25(4dte^{\frac{1}{5}t} + 2t)^{4}} - \frac{1152d^{4}nt^{2}*3e^{{\frac{1}{5}t}*{2}}e^{\frac{1}{5}t}*\frac{1}{5}}{25(4dte^{\frac{1}{5}t} + 2t)^{4}} - \frac{1152(\frac{-4(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{5}})d^{3}nte^{{\frac{1}{5}t}*{2}}}{5} - \frac{1152d^{3}ne^{{\frac{1}{5}t}*{2}}}{5(4dte^{\frac{1}{5}t} + 2t)^{4}} - \frac{1152d^{3}nt*2e^{\frac{1}{5}t}e^{\frac{1}{5}t}*\frac{1}{5}}{5(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{288(\frac{-3(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{4}})d^{3}nte^{{\frac{1}{5}t}*{2}}}{25} + \frac{288d^{3}ne^{{\frac{1}{5}t}*{2}}}{25(4dte^{\frac{1}{5}t} + 2t)^{3}} + \frac{288d^{3}nt*2e^{\frac{1}{5}t}e^{\frac{1}{5}t}*\frac{1}{5}}{25(4dte^{\frac{1}{5}t} + 2t)^{3}} - 288(\frac{-4(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{5}})d^{2}ne^{\frac{1}{5}t} - \frac{288d^{2}ne^{\frac{1}{5}t}*\frac{1}{5}}{(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{96(\frac{-3(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{4}})d^{2}ne^{\frac{1}{5}t}}{5} + \frac{96d^{2}ne^{\frac{1}{5}t}*\frac{1}{5}}{5(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{12(\frac{-2(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{3}})d^{2}ne^{\frac{1}{5}t}}{25} - \frac{12d^{2}ne^{\frac{1}{5}t}*\frac{1}{5}}{25(4dte^{\frac{1}{5}t} + 2t)^{2}} - \frac{384(\frac{-4(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{5}})d^{4}nt^{3}e^{{\frac{1}{5}t}*{3}}}{125} - \frac{384d^{4}n*3t^{2}e^{{\frac{1}{5}t}*{3}}}{125(4dte^{\frac{1}{5}t} + 2t)^{4}} - \frac{384d^{4}nt^{3}*3e^{{\frac{1}{5}t}*{2}}e^{\frac{1}{5}t}*\frac{1}{5}}{125(4dte^{\frac{1}{5}t} + 2t)^{4}} - \frac{576(\frac{-4(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{5}})d^{3}nt^{2}e^{{\frac{1}{5}t}*{2}}}{25} - \frac{576d^{3}n*2te^{{\frac{1}{5}t}*{2}}}{25(4dte^{\frac{1}{5}t} + 2t)^{4}} - \frac{576d^{3}nt^{2}*2e^{\frac{1}{5}t}e^{\frac{1}{5}t}*\frac{1}{5}}{25(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{96(\frac{-3(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{4}})d^{3}nt^{2}e^{{\frac{1}{5}t}*{2}}}{125} + \frac{96d^{3}n*2te^{{\frac{1}{5}t}*{2}}}{125(4dte^{\frac{1}{5}t} + 2t)^{3}} + \frac{96d^{3}nt^{2}*2e^{\frac{1}{5}t}e^{\frac{1}{5}t}*\frac{1}{5}}{125(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{288(\frac{-4(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{5}})d^{2}nte^{\frac{1}{5}t}}{5} - \frac{288d^{2}ne^{\frac{1}{5}t}}{5(4dte^{\frac{1}{5}t} + 2t)^{4}} - \frac{288d^{2}nte^{\frac{1}{5}t}*\frac{1}{5}}{5(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{48(\frac{-3(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{4}})d^{2}nte^{\frac{1}{5}t}}{25} + \frac{48d^{2}ne^{\frac{1}{5}t}}{25(4dte^{\frac{1}{5}t} + 2t)^{3}} + \frac{48d^{2}nte^{\frac{1}{5}t}*\frac{1}{5}}{25(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{4(\frac{-2(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{3}})d^{2}nte^{\frac{1}{5}t}}{125} - \frac{4d^{2}ne^{\frac{1}{5}t}}{125(4dte^{\frac{1}{5}t} + 2t)^{2}} - \frac{4d^{2}nte^{\frac{1}{5}t}*\frac{1}{5}}{125(4dte^{\frac{1}{5}t} + 2t)^{2}} - 48(\frac{-4(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{5}})dn + 0\\=&\frac{6144d^{5}ne^{{\frac{1}{5}t}*{4}}}{(4dte^{\frac{1}{5}t} + 2t)^{5}} + \frac{24576d^{5}nte^{{\frac{1}{5}t}*{4}}}{5(4dte^{\frac{1}{5}t} + 2t)^{5}} + \frac{12288d^{4}ne^{{\frac{1}{5}t}*{3}}}{(4dte^{\frac{1}{5}t} + 2t)^{5}} - \frac{4608d^{4}ne^{{\frac{1}{5}t}*{3}}}{5(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{36864d^{5}nt^{2}e^{{\frac{1}{5}t}*{4}}}{25(4dte^{\frac{1}{5}t} + 2t)^{5}} + \frac{36864d^{4}nte^{{\frac{1}{5}t}*{3}}}{5(4dte^{\frac{1}{5}t} + 2t)^{5}} - \frac{2304d^{4}nte^{{\frac{1}{5}t}*{3}}}{5(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{9216d^{3}ne^{{\frac{1}{5}t}*{2}}}{(4dte^{\frac{1}{5}t} + 2t)^{5}} - \frac{4608d^{3}ne^{{\frac{1}{5}t}*{2}}}{5(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{768d^{3}ne^{{\frac{1}{5}t}*{2}}}{25(4dte^{\frac{1}{5}t} + 2t)^{3}} + \frac{24576d^{5}nt^{3}e^{{\frac{1}{5}t}*{4}}}{125(4dte^{\frac{1}{5}t} + 2t)^{5}} + \frac{36864d^{4}nt^{2}e^{{\frac{1}{5}t}*{3}}}{25(4dte^{\frac{1}{5}t} + 2t)^{5}} - \frac{9216d^{4}nt^{2}e^{{\frac{1}{5}t}*{3}}}{125(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{18432d^{3}nte^{{\frac{1}{5}t}*{2}}}{5(4dte^{\frac{1}{5}t} + 2t)^{5}} - \frac{6912d^{3}nte^{{\frac{1}{5}t}*{2}}}{25(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{896d^{3}nte^{{\frac{1}{5}t}*{2}}}{125(4dte^{\frac{1}{5}t} + 2t)^{3}} + \frac{3072d^{2}ne^{\frac{1}{5}t}}{(4dte^{\frac{1}{5}t} + 2t)^{5}} - \frac{1152d^{2}ne^{\frac{1}{5}t}}{5(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{192d^{2}ne^{\frac{1}{5}t}}{25(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{16d^{2}ne^{\frac{1}{5}t}}{125(4dte^{\frac{1}{5}t} + 2t)^{2}} + \frac{6144d^{5}nt^{4}e^{{\frac{1}{5}t}*{4}}}{625(4dte^{\frac{1}{5}t} + 2t)^{5}} + \frac{12288d^{4}nt^{3}e^{{\frac{1}{5}t}*{3}}}{125(4dte^{\frac{1}{5}t} + 2t)^{5}} - \frac{2304d^{4}nt^{3}e^{{\frac{1}{5}t}*{3}}}{625(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{9216d^{3}nt^{2}e^{{\frac{1}{5}t}*{2}}}{25(4dte^{\frac{1}{5}t} + 2t)^{5}} - \frac{2304d^{3}nt^{2}e^{{\frac{1}{5}t}*{2}}}{125(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{224d^{3}nt^{2}e^{{\frac{1}{5}t}*{2}}}{625(4dte^{\frac{1}{5}t} + 2t)^{3}} + \frac{3072d^{2}nte^{\frac{1}{5}t}}{5(4dte^{\frac{1}{5}t} + 2t)^{5}} - \frac{576d^{2}nte^{\frac{1}{5}t}}{25(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{64d^{2}nte^{\frac{1}{5}t}}{125(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{4d^{2}nte^{\frac{1}{5}t}}{625(4dte^{\frac{1}{5}t} + 2t)^{2}} + \frac{384dn}{(4dte^{\frac{1}{5}t} + 2t)^{5}}\\ \end{split}\end{equation} \]
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数\frac{dn}{(4e^{\frac{1}{5}t}dt + 2t)} 关于 t 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{dn}{(4dte^{\frac{1}{5}t} + 2t)}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( \frac{dn}{(4dte^{\frac{1}{5}t} + 2t)}\right)}{dt}\\=&(\frac{-(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{2}})dn + 0\\=&\frac{-4d^{2}ne^{\frac{1}{5}t}}{(4dte^{\frac{1}{5}t} + 2t)^{2}} - \frac{4d^{2}nte^{\frac{1}{5}t}}{5(4dte^{\frac{1}{5}t} + 2t)^{2}} - \frac{2dn}{(4dte^{\frac{1}{5}t} + 2t)^{2}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{-4d^{2}ne^{\frac{1}{5}t}}{(4dte^{\frac{1}{5}t} + 2t)^{2}} - \frac{4d^{2}nte^{\frac{1}{5}t}}{5(4dte^{\frac{1}{5}t} + 2t)^{2}} - \frac{2dn}{(4dte^{\frac{1}{5}t} + 2t)^{2}}\right)}{dt}\\=&-4(\frac{-2(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{3}})d^{2}ne^{\frac{1}{5}t} - \frac{4d^{2}ne^{\frac{1}{5}t}*\frac{1}{5}}{(4dte^{\frac{1}{5}t} + 2t)^{2}} - \frac{4(\frac{-2(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{3}})d^{2}nte^{\frac{1}{5}t}}{5} - \frac{4d^{2}ne^{\frac{1}{5}t}}{5(4dte^{\frac{1}{5}t} + 2t)^{2}} - \frac{4d^{2}nte^{\frac{1}{5}t}*\frac{1}{5}}{5(4dte^{\frac{1}{5}t} + 2t)^{2}} - 2(\frac{-2(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{3}})dn + 0\\=&\frac{32d^{3}ne^{{\frac{1}{5}t}*{2}}}{(4dte^{\frac{1}{5}t} + 2t)^{3}} + \frac{64d^{3}nte^{{\frac{1}{5}t}*{2}}}{5(4dte^{\frac{1}{5}t} + 2t)^{3}} + \frac{32d^{2}ne^{\frac{1}{5}t}}{(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{8d^{2}ne^{\frac{1}{5}t}}{5(4dte^{\frac{1}{5}t} + 2t)^{2}} + \frac{32d^{3}nt^{2}e^{{\frac{1}{5}t}*{2}}}{25(4dte^{\frac{1}{5}t} + 2t)^{3}} + \frac{32d^{2}nte^{\frac{1}{5}t}}{5(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{4d^{2}nte^{\frac{1}{5}t}}{25(4dte^{\frac{1}{5}t} + 2t)^{2}} + \frac{8dn}{(4dte^{\frac{1}{5}t} + 2t)^{3}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{32d^{3}ne^{{\frac{1}{5}t}*{2}}}{(4dte^{\frac{1}{5}t} + 2t)^{3}} + \frac{64d^{3}nte^{{\frac{1}{5}t}*{2}}}{5(4dte^{\frac{1}{5}t} + 2t)^{3}} + \frac{32d^{2}ne^{\frac{1}{5}t}}{(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{8d^{2}ne^{\frac{1}{5}t}}{5(4dte^{\frac{1}{5}t} + 2t)^{2}} + \frac{32d^{3}nt^{2}e^{{\frac{1}{5}t}*{2}}}{25(4dte^{\frac{1}{5}t} + 2t)^{3}} + \frac{32d^{2}nte^{\frac{1}{5}t}}{5(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{4d^{2}nte^{\frac{1}{5}t}}{25(4dte^{\frac{1}{5}t} + 2t)^{2}} + \frac{8dn}{(4dte^{\frac{1}{5}t} + 2t)^{3}}\right)}{dt}\\=&32(\frac{-3(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{4}})d^{3}ne^{{\frac{1}{5}t}*{2}} + \frac{32d^{3}n*2e^{\frac{1}{5}t}e^{\frac{1}{5}t}*\frac{1}{5}}{(4dte^{\frac{1}{5}t} + 2t)^{3}} + \frac{64(\frac{-3(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{4}})d^{3}nte^{{\frac{1}{5}t}*{2}}}{5} + \frac{64d^{3}ne^{{\frac{1}{5}t}*{2}}}{5(4dte^{\frac{1}{5}t} + 2t)^{3}} + \frac{64d^{3}nt*2e^{\frac{1}{5}t}e^{\frac{1}{5}t}*\frac{1}{5}}{5(4dte^{\frac{1}{5}t} + 2t)^{3}} + 32(\frac{-3(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{4}})d^{2}ne^{\frac{1}{5}t} + \frac{32d^{2}ne^{\frac{1}{5}t}*\frac{1}{5}}{(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{8(\frac{-2(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{3}})d^{2}ne^{\frac{1}{5}t}}{5} - \frac{8d^{2}ne^{\frac{1}{5}t}*\frac{1}{5}}{5(4dte^{\frac{1}{5}t} + 2t)^{2}} + \frac{32(\frac{-3(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{4}})d^{3}nt^{2}e^{{\frac{1}{5}t}*{2}}}{25} + \frac{32d^{3}n*2te^{{\frac{1}{5}t}*{2}}}{25(4dte^{\frac{1}{5}t} + 2t)^{3}} + \frac{32d^{3}nt^{2}*2e^{\frac{1}{5}t}e^{\frac{1}{5}t}*\frac{1}{5}}{25(4dte^{\frac{1}{5}t} + 2t)^{3}} + \frac{32(\frac{-3(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{4}})d^{2}nte^{\frac{1}{5}t}}{5} + \frac{32d^{2}ne^{\frac{1}{5}t}}{5(4dte^{\frac{1}{5}t} + 2t)^{3}} + \frac{32d^{2}nte^{\frac{1}{5}t}*\frac{1}{5}}{5(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{4(\frac{-2(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{3}})d^{2}nte^{\frac{1}{5}t}}{25} - \frac{4d^{2}ne^{\frac{1}{5}t}}{25(4dte^{\frac{1}{5}t} + 2t)^{2}} - \frac{4d^{2}nte^{\frac{1}{5}t}*\frac{1}{5}}{25(4dte^{\frac{1}{5}t} + 2t)^{2}} + 8(\frac{-3(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{4}})dn + 0\\=&\frac{-384d^{4}ne^{{\frac{1}{5}t}*{3}}}{(4dte^{\frac{1}{5}t} + 2t)^{4}} - \frac{1152d^{4}nte^{{\frac{1}{5}t}*{3}}}{5(4dte^{\frac{1}{5}t} + 2t)^{4}} - \frac{576d^{3}ne^{{\frac{1}{5}t}*{2}}}{(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{192d^{3}ne^{{\frac{1}{5}t}*{2}}}{5(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{1152d^{4}nt^{2}e^{{\frac{1}{5}t}*{3}}}{25(4dte^{\frac{1}{5}t} + 2t)^{4}} - \frac{1152d^{3}nte^{{\frac{1}{5}t}*{2}}}{5(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{288d^{3}nte^{{\frac{1}{5}t}*{2}}}{25(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{288d^{2}ne^{\frac{1}{5}t}}{(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{96d^{2}ne^{\frac{1}{5}t}}{5(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{12d^{2}ne^{\frac{1}{5}t}}{25(4dte^{\frac{1}{5}t} + 2t)^{2}} - \frac{384d^{4}nt^{3}e^{{\frac{1}{5}t}*{3}}}{125(4dte^{\frac{1}{5}t} + 2t)^{4}} - \frac{576d^{3}nt^{2}e^{{\frac{1}{5}t}*{2}}}{25(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{96d^{3}nt^{2}e^{{\frac{1}{5}t}*{2}}}{125(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{288d^{2}nte^{\frac{1}{5}t}}{5(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{48d^{2}nte^{\frac{1}{5}t}}{25(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{4d^{2}nte^{\frac{1}{5}t}}{125(4dte^{\frac{1}{5}t} + 2t)^{2}} - \frac{48dn}{(4dte^{\frac{1}{5}t} + 2t)^{4}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{-384d^{4}ne^{{\frac{1}{5}t}*{3}}}{(4dte^{\frac{1}{5}t} + 2t)^{4}} - \frac{1152d^{4}nte^{{\frac{1}{5}t}*{3}}}{5(4dte^{\frac{1}{5}t} + 2t)^{4}} - \frac{576d^{3}ne^{{\frac{1}{5}t}*{2}}}{(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{192d^{3}ne^{{\frac{1}{5}t}*{2}}}{5(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{1152d^{4}nt^{2}e^{{\frac{1}{5}t}*{3}}}{25(4dte^{\frac{1}{5}t} + 2t)^{4}} - \frac{1152d^{3}nte^{{\frac{1}{5}t}*{2}}}{5(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{288d^{3}nte^{{\frac{1}{5}t}*{2}}}{25(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{288d^{2}ne^{\frac{1}{5}t}}{(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{96d^{2}ne^{\frac{1}{5}t}}{5(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{12d^{2}ne^{\frac{1}{5}t}}{25(4dte^{\frac{1}{5}t} + 2t)^{2}} - \frac{384d^{4}nt^{3}e^{{\frac{1}{5}t}*{3}}}{125(4dte^{\frac{1}{5}t} + 2t)^{4}} - \frac{576d^{3}nt^{2}e^{{\frac{1}{5}t}*{2}}}{25(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{96d^{3}nt^{2}e^{{\frac{1}{5}t}*{2}}}{125(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{288d^{2}nte^{\frac{1}{5}t}}{5(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{48d^{2}nte^{\frac{1}{5}t}}{25(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{4d^{2}nte^{\frac{1}{5}t}}{125(4dte^{\frac{1}{5}t} + 2t)^{2}} - \frac{48dn}{(4dte^{\frac{1}{5}t} + 2t)^{4}}\right)}{dt}\\=&-384(\frac{-4(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{5}})d^{4}ne^{{\frac{1}{5}t}*{3}} - \frac{384d^{4}n*3e^{{\frac{1}{5}t}*{2}}e^{\frac{1}{5}t}*\frac{1}{5}}{(4dte^{\frac{1}{5}t} + 2t)^{4}} - \frac{1152(\frac{-4(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{5}})d^{4}nte^{{\frac{1}{5}t}*{3}}}{5} - \frac{1152d^{4}ne^{{\frac{1}{5}t}*{3}}}{5(4dte^{\frac{1}{5}t} + 2t)^{4}} - \frac{1152d^{4}nt*3e^{{\frac{1}{5}t}*{2}}e^{\frac{1}{5}t}*\frac{1}{5}}{5(4dte^{\frac{1}{5}t} + 2t)^{4}} - 576(\frac{-4(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{5}})d^{3}ne^{{\frac{1}{5}t}*{2}} - \frac{576d^{3}n*2e^{\frac{1}{5}t}e^{\frac{1}{5}t}*\frac{1}{5}}{(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{192(\frac{-3(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{4}})d^{3}ne^{{\frac{1}{5}t}*{2}}}{5} + \frac{192d^{3}n*2e^{\frac{1}{5}t}e^{\frac{1}{5}t}*\frac{1}{5}}{5(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{1152(\frac{-4(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{5}})d^{4}nt^{2}e^{{\frac{1}{5}t}*{3}}}{25} - \frac{1152d^{4}n*2te^{{\frac{1}{5}t}*{3}}}{25(4dte^{\frac{1}{5}t} + 2t)^{4}} - \frac{1152d^{4}nt^{2}*3e^{{\frac{1}{5}t}*{2}}e^{\frac{1}{5}t}*\frac{1}{5}}{25(4dte^{\frac{1}{5}t} + 2t)^{4}} - \frac{1152(\frac{-4(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{5}})d^{3}nte^{{\frac{1}{5}t}*{2}}}{5} - \frac{1152d^{3}ne^{{\frac{1}{5}t}*{2}}}{5(4dte^{\frac{1}{5}t} + 2t)^{4}} - \frac{1152d^{3}nt*2e^{\frac{1}{5}t}e^{\frac{1}{5}t}*\frac{1}{5}}{5(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{288(\frac{-3(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{4}})d^{3}nte^{{\frac{1}{5}t}*{2}}}{25} + \frac{288d^{3}ne^{{\frac{1}{5}t}*{2}}}{25(4dte^{\frac{1}{5}t} + 2t)^{3}} + \frac{288d^{3}nt*2e^{\frac{1}{5}t}e^{\frac{1}{5}t}*\frac{1}{5}}{25(4dte^{\frac{1}{5}t} + 2t)^{3}} - 288(\frac{-4(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{5}})d^{2}ne^{\frac{1}{5}t} - \frac{288d^{2}ne^{\frac{1}{5}t}*\frac{1}{5}}{(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{96(\frac{-3(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{4}})d^{2}ne^{\frac{1}{5}t}}{5} + \frac{96d^{2}ne^{\frac{1}{5}t}*\frac{1}{5}}{5(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{12(\frac{-2(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{3}})d^{2}ne^{\frac{1}{5}t}}{25} - \frac{12d^{2}ne^{\frac{1}{5}t}*\frac{1}{5}}{25(4dte^{\frac{1}{5}t} + 2t)^{2}} - \frac{384(\frac{-4(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{5}})d^{4}nt^{3}e^{{\frac{1}{5}t}*{3}}}{125} - \frac{384d^{4}n*3t^{2}e^{{\frac{1}{5}t}*{3}}}{125(4dte^{\frac{1}{5}t} + 2t)^{4}} - \frac{384d^{4}nt^{3}*3e^{{\frac{1}{5}t}*{2}}e^{\frac{1}{5}t}*\frac{1}{5}}{125(4dte^{\frac{1}{5}t} + 2t)^{4}} - \frac{576(\frac{-4(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{5}})d^{3}nt^{2}e^{{\frac{1}{5}t}*{2}}}{25} - \frac{576d^{3}n*2te^{{\frac{1}{5}t}*{2}}}{25(4dte^{\frac{1}{5}t} + 2t)^{4}} - \frac{576d^{3}nt^{2}*2e^{\frac{1}{5}t}e^{\frac{1}{5}t}*\frac{1}{5}}{25(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{96(\frac{-3(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{4}})d^{3}nt^{2}e^{{\frac{1}{5}t}*{2}}}{125} + \frac{96d^{3}n*2te^{{\frac{1}{5}t}*{2}}}{125(4dte^{\frac{1}{5}t} + 2t)^{3}} + \frac{96d^{3}nt^{2}*2e^{\frac{1}{5}t}e^{\frac{1}{5}t}*\frac{1}{5}}{125(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{288(\frac{-4(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{5}})d^{2}nte^{\frac{1}{5}t}}{5} - \frac{288d^{2}ne^{\frac{1}{5}t}}{5(4dte^{\frac{1}{5}t} + 2t)^{4}} - \frac{288d^{2}nte^{\frac{1}{5}t}*\frac{1}{5}}{5(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{48(\frac{-3(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{4}})d^{2}nte^{\frac{1}{5}t}}{25} + \frac{48d^{2}ne^{\frac{1}{5}t}}{25(4dte^{\frac{1}{5}t} + 2t)^{3}} + \frac{48d^{2}nte^{\frac{1}{5}t}*\frac{1}{5}}{25(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{4(\frac{-2(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{3}})d^{2}nte^{\frac{1}{5}t}}{125} - \frac{4d^{2}ne^{\frac{1}{5}t}}{125(4dte^{\frac{1}{5}t} + 2t)^{2}} - \frac{4d^{2}nte^{\frac{1}{5}t}*\frac{1}{5}}{125(4dte^{\frac{1}{5}t} + 2t)^{2}} - 48(\frac{-4(4de^{\frac{1}{5}t} + 4dte^{\frac{1}{5}t}*\frac{1}{5} + 2)}{(4dte^{\frac{1}{5}t} + 2t)^{5}})dn + 0\\=&\frac{6144d^{5}ne^{{\frac{1}{5}t}*{4}}}{(4dte^{\frac{1}{5}t} + 2t)^{5}} + \frac{24576d^{5}nte^{{\frac{1}{5}t}*{4}}}{5(4dte^{\frac{1}{5}t} + 2t)^{5}} + \frac{12288d^{4}ne^{{\frac{1}{5}t}*{3}}}{(4dte^{\frac{1}{5}t} + 2t)^{5}} - \frac{4608d^{4}ne^{{\frac{1}{5}t}*{3}}}{5(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{36864d^{5}nt^{2}e^{{\frac{1}{5}t}*{4}}}{25(4dte^{\frac{1}{5}t} + 2t)^{5}} + \frac{36864d^{4}nte^{{\frac{1}{5}t}*{3}}}{5(4dte^{\frac{1}{5}t} + 2t)^{5}} - \frac{2304d^{4}nte^{{\frac{1}{5}t}*{3}}}{5(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{9216d^{3}ne^{{\frac{1}{5}t}*{2}}}{(4dte^{\frac{1}{5}t} + 2t)^{5}} - \frac{4608d^{3}ne^{{\frac{1}{5}t}*{2}}}{5(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{768d^{3}ne^{{\frac{1}{5}t}*{2}}}{25(4dte^{\frac{1}{5}t} + 2t)^{3}} + \frac{24576d^{5}nt^{3}e^{{\frac{1}{5}t}*{4}}}{125(4dte^{\frac{1}{5}t} + 2t)^{5}} + \frac{36864d^{4}nt^{2}e^{{\frac{1}{5}t}*{3}}}{25(4dte^{\frac{1}{5}t} + 2t)^{5}} - \frac{9216d^{4}nt^{2}e^{{\frac{1}{5}t}*{3}}}{125(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{18432d^{3}nte^{{\frac{1}{5}t}*{2}}}{5(4dte^{\frac{1}{5}t} + 2t)^{5}} - \frac{6912d^{3}nte^{{\frac{1}{5}t}*{2}}}{25(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{896d^{3}nte^{{\frac{1}{5}t}*{2}}}{125(4dte^{\frac{1}{5}t} + 2t)^{3}} + \frac{3072d^{2}ne^{\frac{1}{5}t}}{(4dte^{\frac{1}{5}t} + 2t)^{5}} - \frac{1152d^{2}ne^{\frac{1}{5}t}}{5(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{192d^{2}ne^{\frac{1}{5}t}}{25(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{16d^{2}ne^{\frac{1}{5}t}}{125(4dte^{\frac{1}{5}t} + 2t)^{2}} + \frac{6144d^{5}nt^{4}e^{{\frac{1}{5}t}*{4}}}{625(4dte^{\frac{1}{5}t} + 2t)^{5}} + \frac{12288d^{4}nt^{3}e^{{\frac{1}{5}t}*{3}}}{125(4dte^{\frac{1}{5}t} + 2t)^{5}} - \frac{2304d^{4}nt^{3}e^{{\frac{1}{5}t}*{3}}}{625(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{9216d^{3}nt^{2}e^{{\frac{1}{5}t}*{2}}}{25(4dte^{\frac{1}{5}t} + 2t)^{5}} - \frac{2304d^{3}nt^{2}e^{{\frac{1}{5}t}*{2}}}{125(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{224d^{3}nt^{2}e^{{\frac{1}{5}t}*{2}}}{625(4dte^{\frac{1}{5}t} + 2t)^{3}} + \frac{3072d^{2}nte^{\frac{1}{5}t}}{5(4dte^{\frac{1}{5}t} + 2t)^{5}} - \frac{576d^{2}nte^{\frac{1}{5}t}}{25(4dte^{\frac{1}{5}t} + 2t)^{4}} + \frac{64d^{2}nte^{\frac{1}{5}t}}{125(4dte^{\frac{1}{5}t} + 2t)^{3}} - \frac{4d^{2}nte^{\frac{1}{5}t}}{625(4dte^{\frac{1}{5}t} + 2t)^{2}} + \frac{384dn}{(4dte^{\frac{1}{5}t} + 2t)^{5}}\\ \end{split}\end{equation} \]
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