本次共计算 1 个题目：每一题对 z 求 3 阶导数。
    注意，变量是区分大小写的。
$$\begin{equation}\begin{split}【1/1】求函数((z - \frac{a(sqrt(2)){(1 + i)}^{4}}{2}){({z}^{4} + {a}^{4})}^{-1}) 关于 z 的 3 阶导数：\\\end{split}\end{equation}$$ $$\begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{z}{(z^{4} + a^{4})} - \frac{\frac{1}{2}ai^{4}sqrt(2)}{(z^{4} + a^{4})} - \frac{2ai^{3}sqrt(2)}{(z^{4} + a^{4})} - \frac{3ai^{2}sqrt(2)}{(z^{4} + a^{4})} - \frac{2aisqrt(2)}{(z^{4} + a^{4})} - \frac{\frac{1}{2}asqrt(2)}{(z^{4} + a^{4})}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( \frac{z}{(z^{4} + a^{4})} - \frac{\frac{1}{2}ai^{4}sqrt(2)}{(z^{4} + a^{4})} - \frac{2ai^{3}sqrt(2)}{(z^{4} + a^{4})} - \frac{3ai^{2}sqrt(2)}{(z^{4} + a^{4})} - \frac{2aisqrt(2)}{(z^{4} + a^{4})} - \frac{\frac{1}{2}asqrt(2)}{(z^{4} + a^{4})}\right)}{dz}\\=&(\frac{-(4z^{3} + 0)}{(z^{4} + a^{4})^{2}})z + \frac{1}{(z^{4} + a^{4})} - \frac{1}{2}(\frac{-(4z^{3} + 0)}{(z^{4} + a^{4})^{2}})ai^{4}sqrt(2) - \frac{\frac{1}{2}ai^{4}*0*\frac{1}{2}*2^{\frac{1}{2}}}{(z^{4} + a^{4})} - 2(\frac{-(4z^{3} + 0)}{(z^{4} + a^{4})^{2}})ai^{3}sqrt(2) - \frac{2ai^{3}*0*\frac{1}{2}*2^{\frac{1}{2}}}{(z^{4} + a^{4})} - 3(\frac{-(4z^{3} + 0)}{(z^{4} + a^{4})^{2}})ai^{2}sqrt(2) - \frac{3ai^{2}*0*\frac{1}{2}*2^{\frac{1}{2}}}{(z^{4} + a^{4})} - 2(\frac{-(4z^{3} + 0)}{(z^{4} + a^{4})^{2}})aisqrt(2) - \frac{2ai*0*\frac{1}{2}*2^{\frac{1}{2}}}{(z^{4} + a^{4})} - \frac{1}{2}(\frac{-(4z^{3} + 0)}{(z^{4} + a^{4})^{2}})asqrt(2) - \frac{\frac{1}{2}a*0*\frac{1}{2}*2^{\frac{1}{2}}}{(z^{4} + a^{4})}\\=&\frac{-4z^{4}}{(z^{4} + a^{4})^{2}} + \frac{2ai^{4}z^{3}sqrt(2)}{(z^{4} + a^{4})^{2}} + \frac{8ai^{3}z^{3}sqrt(2)}{(z^{4} + a^{4})^{2}} + \frac{12ai^{2}z^{3}sqrt(2)}{(z^{4} + a^{4})^{2}} + \frac{8aiz^{3}sqrt(2)}{(z^{4} + a^{4})^{2}} + \frac{2az^{3}sqrt(2)}{(z^{4} + a^{4})^{2}} + \frac{1}{(z^{4} + a^{4})}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{-4z^{4}}{(z^{4} + a^{4})^{2}} + \frac{2ai^{4}z^{3}sqrt(2)}{(z^{4} + a^{4})^{2}} + \frac{8ai^{3}z^{3}sqrt(2)}{(z^{4} + a^{4})^{2}} + \frac{12ai^{2}z^{3}sqrt(2)}{(z^{4} + a^{4})^{2}} + \frac{8aiz^{3}sqrt(2)}{(z^{4} + a^{4})^{2}} + \frac{2az^{3}sqrt(2)}{(z^{4} + a^{4})^{2}} + \frac{1}{(z^{4} + a^{4})}\right)}{dz}\\=&-4(\frac{-2(4z^{3} + 0)}{(z^{4} + a^{4})^{3}})z^{4} - \frac{4*4z^{3}}{(z^{4} + a^{4})^{2}} + 2(\frac{-2(4z^{3} + 0)}{(z^{4} + a^{4})^{3}})ai^{4}z^{3}sqrt(2) + \frac{2ai^{4}*3z^{2}sqrt(2)}{(z^{4} + a^{4})^{2}} + \frac{2ai^{4}z^{3}*0*\frac{1}{2}*2^{\frac{1}{2}}}{(z^{4} + a^{4})^{2}} + 8(\frac{-2(4z^{3} + 0)}{(z^{4} + a^{4})^{3}})ai^{3}z^{3}sqrt(2) + \frac{8ai^{3}*3z^{2}sqrt(2)}{(z^{4} + a^{4})^{2}} + \frac{8ai^{3}z^{3}*0*\frac{1}{2}*2^{\frac{1}{2}}}{(z^{4} + a^{4})^{2}} + 12(\frac{-2(4z^{3} + 0)}{(z^{4} + a^{4})^{3}})ai^{2}z^{3}sqrt(2) + \frac{12ai^{2}*3z^{2}sqrt(2)}{(z^{4} + a^{4})^{2}} + \frac{12ai^{2}z^{3}*0*\frac{1}{2}*2^{\frac{1}{2}}}{(z^{4} + a^{4})^{2}} + 8(\frac{-2(4z^{3} + 0)}{(z^{4} + a^{4})^{3}})aiz^{3}sqrt(2) + \frac{8ai*3z^{2}sqrt(2)}{(z^{4} + a^{4})^{2}} + \frac{8aiz^{3}*0*\frac{1}{2}*2^{\frac{1}{2}}}{(z^{4} + a^{4})^{2}} + 2(\frac{-2(4z^{3} + 0)}{(z^{4} + a^{4})^{3}})az^{3}sqrt(2) + \frac{2a*3z^{2}sqrt(2)}{(z^{4} + a^{4})^{2}} + \frac{2az^{3}*0*\frac{1}{2}*2^{\frac{1}{2}}}{(z^{4} + a^{4})^{2}} + (\frac{-(4z^{3} + 0)}{(z^{4} + a^{4})^{2}})\\=&\frac{32z^{7}}{(z^{4} + a^{4})^{3}} - \frac{20z^{3}}{(z^{4} + a^{4})^{2}} - \frac{16ai^{4}z^{6}sqrt(2)}{(z^{4} + a^{4})^{3}} + \frac{6ai^{4}z^{2}sqrt(2)}{(z^{4} + a^{4})^{2}} - \frac{64ai^{3}z^{6}sqrt(2)}{(z^{4} + a^{4})^{3}} + \frac{24ai^{3}z^{2}sqrt(2)}{(z^{4} + a^{4})^{2}} - \frac{96ai^{2}z^{6}sqrt(2)}{(z^{4} + a^{4})^{3}} + \frac{36ai^{2}z^{2}sqrt(2)}{(z^{4} + a^{4})^{2}} - \frac{64aiz^{6}sqrt(2)}{(z^{4} + a^{4})^{3}} + \frac{24aiz^{2}sqrt(2)}{(z^{4} + a^{4})^{2}} - \frac{16az^{6}sqrt(2)}{(z^{4} + a^{4})^{3}} + \frac{6az^{2}sqrt(2)}{(z^{4} + a^{4})^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{32z^{7}}{(z^{4} + a^{4})^{3}} - \frac{20z^{3}}{(z^{4} + a^{4})^{2}} - \frac{16ai^{4}z^{6}sqrt(2)}{(z^{4} + a^{4})^{3}} + \frac{6ai^{4}z^{2}sqrt(2)}{(z^{4} + a^{4})^{2}} - \frac{64ai^{3}z^{6}sqrt(2)}{(z^{4} + a^{4})^{3}} + \frac{24ai^{3}z^{2}sqrt(2)}{(z^{4} + a^{4})^{2}} - \frac{96ai^{2}z^{6}sqrt(2)}{(z^{4} + a^{4})^{3}} + \frac{36ai^{2}z^{2}sqrt(2)}{(z^{4} + a^{4})^{2}} - \frac{64aiz^{6}sqrt(2)}{(z^{4} + a^{4})^{3}} + \frac{24aiz^{2}sqrt(2)}{(z^{4} + a^{4})^{2}} - \frac{16az^{6}sqrt(2)}{(z^{4} + a^{4})^{3}} + \frac{6az^{2}sqrt(2)}{(z^{4} + a^{4})^{2}}\right)}{dz}\\=&32(\frac{-3(4z^{3} + 0)}{(z^{4} + a^{4})^{4}})z^{7} + \frac{32*7z^{6}}{(z^{4} + a^{4})^{3}} - 20(\frac{-2(4z^{3} + 0)}{(z^{4} + a^{4})^{3}})z^{3} - \frac{20*3z^{2}}{(z^{4} + a^{4})^{2}} - 16(\frac{-3(4z^{3} + 0)}{(z^{4} + a^{4})^{4}})ai^{4}z^{6}sqrt(2) - \frac{16ai^{4}*6z^{5}sqrt(2)}{(z^{4} + a^{4})^{3}} - \frac{16ai^{4}z^{6}*0*\frac{1}{2}*2^{\frac{1}{2}}}{(z^{4} + a^{4})^{3}} + 6(\frac{-2(4z^{3} + 0)}{(z^{4} + a^{4})^{3}})ai^{4}z^{2}sqrt(2) + \frac{6ai^{4}*2zsqrt(2)}{(z^{4} + a^{4})^{2}} + \frac{6ai^{4}z^{2}*0*\frac{1}{2}*2^{\frac{1}{2}}}{(z^{4} + a^{4})^{2}} - 64(\frac{-3(4z^{3} + 0)}{(z^{4} + a^{4})^{4}})ai^{3}z^{6}sqrt(2) - \frac{64ai^{3}*6z^{5}sqrt(2)}{(z^{4} + a^{4})^{3}} - \frac{64ai^{3}z^{6}*0*\frac{1}{2}*2^{\frac{1}{2}}}{(z^{4} + a^{4})^{3}} + 24(\frac{-2(4z^{3} + 0)}{(z^{4} + a^{4})^{3}})ai^{3}z^{2}sqrt(2) + \frac{24ai^{3}*2zsqrt(2)}{(z^{4} + a^{4})^{2}} + \frac{24ai^{3}z^{2}*0*\frac{1}{2}*2^{\frac{1}{2}}}{(z^{4} + a^{4})^{2}} - 96(\frac{-3(4z^{3} + 0)}{(z^{4} + a^{4})^{4}})ai^{2}z^{6}sqrt(2) - \frac{96ai^{2}*6z^{5}sqrt(2)}{(z^{4} + a^{4})^{3}} - \frac{96ai^{2}z^{6}*0*\frac{1}{2}*2^{\frac{1}{2}}}{(z^{4} + a^{4})^{3}} + 36(\frac{-2(4z^{3} + 0)}{(z^{4} + a^{4})^{3}})ai^{2}z^{2}sqrt(2) + \frac{36ai^{2}*2zsqrt(2)}{(z^{4} + a^{4})^{2}} + \frac{36ai^{2}z^{2}*0*\frac{1}{2}*2^{\frac{1}{2}}}{(z^{4} + a^{4})^{2}} - 64(\frac{-3(4z^{3} + 0)}{(z^{4} + a^{4})^{4}})aiz^{6}sqrt(2) - \frac{64ai*6z^{5}sqrt(2)}{(z^{4} + a^{4})^{3}} - \frac{64aiz^{6}*0*\frac{1}{2}*2^{\frac{1}{2}}}{(z^{4} + a^{4})^{3}} + 24(\frac{-2(4z^{3} + 0)}{(z^{4} + a^{4})^{3}})aiz^{2}sqrt(2) + \frac{24ai*2zsqrt(2)}{(z^{4} + a^{4})^{2}} + \frac{24aiz^{2}*0*\frac{1}{2}*2^{\frac{1}{2}}}{(z^{4} + a^{4})^{2}} - 16(\frac{-3(4z^{3} + 0)}{(z^{4} + a^{4})^{4}})az^{6}sqrt(2) - \frac{16a*6z^{5}sqrt(2)}{(z^{4} + a^{4})^{3}} - \frac{16az^{6}*0*\frac{1}{2}*2^{\frac{1}{2}}}{(z^{4} + a^{4})^{3}} + 6(\frac{-2(4z^{3} + 0)}{(z^{4} + a^{4})^{3}})az^{2}sqrt(2) + \frac{6a*2zsqrt(2)}{(z^{4} + a^{4})^{2}} + \frac{6az^{2}*0*\frac{1}{2}*2^{\frac{1}{2}}}{(z^{4} + a^{4})^{2}}\\=&\frac{-384z^{10}}{(z^{4} + a^{4})^{4}} + \frac{384z^{6}}{(z^{4} + a^{4})^{3}} - \frac{60z^{2}}{(z^{4} + a^{4})^{2}} + \frac{192ai^{4}z^{9}sqrt(2)}{(z^{4} + a^{4})^{4}} - \frac{144ai^{4}z^{5}sqrt(2)}{(z^{4} + a^{4})^{3}} + \frac{12ai^{4}zsqrt(2)}{(z^{4} + a^{4})^{2}} + \frac{768ai^{3}z^{9}sqrt(2)}{(z^{4} + a^{4})^{4}} - \frac{576ai^{3}z^{5}sqrt(2)}{(z^{4} + a^{4})^{3}} + \frac{48ai^{3}zsqrt(2)}{(z^{4} + a^{4})^{2}} + \frac{1152ai^{2}z^{9}sqrt(2)}{(z^{4} + a^{4})^{4}} - \frac{864ai^{2}z^{5}sqrt(2)}{(z^{4} + a^{4})^{3}} + \frac{72ai^{2}zsqrt(2)}{(z^{4} + a^{4})^{2}} + \frac{768aiz^{9}sqrt(2)}{(z^{4} + a^{4})^{4}} - \frac{576aiz^{5}sqrt(2)}{(z^{4} + a^{4})^{3}} + \frac{48aizsqrt(2)}{(z^{4} + a^{4})^{2}} + \frac{192az^{9}sqrt(2)}{(z^{4} + a^{4})^{4}} - \frac{144az^{5}sqrt(2)}{(z^{4} + a^{4})^{3}} + \frac{12azsqrt(2)}{(z^{4} + a^{4})^{2}}\\ \end{split}\end{equation}$$

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