本次共计算 5 个题目：每一题对 x 求 4 阶导数。
    注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/5】求函数ln(ax + b) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( ln(ax + b)\right)}{dx}\\=&\frac{(a + 0)}{(ax + b)}\\=&\frac{a}{(ax + b)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{a}{(ax + b)}\right)}{dx}\\=&(\frac{-(a + 0)}{(ax + b)^{2}})a + 0\\=&\frac{-a^{2}}{(ax + b)^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-a^{2}}{(ax + b)^{2}}\right)}{dx}\\=&-(\frac{-2(a + 0)}{(ax + b)^{3}})a^{2} + 0\\=&\frac{2a^{3}}{(ax + b)^{3}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{2a^{3}}{(ax + b)^{3}}\right)}{dx}\\=&2(\frac{-3(a + 0)}{(ax + b)^{4}})a^{3} + 0\\=&\frac{-6a^{4}}{(ax + b)^{4}}\\ \end{split}\end{equation} \]

\[ \begin{equation}\begin{split}【2/5】求函数ln(a{x}^{2} + bx + c) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = ln(ax^{2} + bx + c)\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( ln(ax^{2} + bx + c)\right)}{dx}\\=&\frac{(a*2x + b + 0)}{(ax^{2} + bx + c)}\\=&\frac{2ax}{(ax^{2} + bx + c)} + \frac{b}{(ax^{2} + bx + c)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{2ax}{(ax^{2} + bx + c)} + \frac{b}{(ax^{2} + bx + c)}\right)}{dx}\\=&2(\frac{-(a*2x + b + 0)}{(ax^{2} + bx + c)^{2}})ax + \frac{2a}{(ax^{2} + bx + c)} + (\frac{-(a*2x + b + 0)}{(ax^{2} + bx + c)^{2}})b + 0\\=&\frac{-4a^{2}x^{2}}{(ax^{2} + bx + c)^{2}} - \frac{4abx}{(ax^{2} + bx + c)^{2}} + \frac{2a}{(ax^{2} + bx + c)} - \frac{b^{2}}{(ax^{2} + bx + c)^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-4a^{2}x^{2}}{(ax^{2} + bx + c)^{2}} - \frac{4abx}{(ax^{2} + bx + c)^{2}} + \frac{2a}{(ax^{2} + bx + c)} - \frac{b^{2}}{(ax^{2} + bx + c)^{2}}\right)}{dx}\\=&-4(\frac{-2(a*2x + b + 0)}{(ax^{2} + bx + c)^{3}})a^{2}x^{2} - \frac{4a^{2}*2x}{(ax^{2} + bx + c)^{2}} - 4(\frac{-2(a*2x + b + 0)}{(ax^{2} + bx + c)^{3}})abx - \frac{4ab}{(ax^{2} + bx + c)^{2}} + 2(\frac{-(a*2x + b + 0)}{(ax^{2} + bx + c)^{2}})a + 0 - (\frac{-2(a*2x + b + 0)}{(ax^{2} + bx + c)^{3}})b^{2} + 0\\=&\frac{16a^{3}x^{3}}{(ax^{2} + bx + c)^{3}} + \frac{24a^{2}bx^{2}}{(ax^{2} + bx + c)^{3}} - \frac{12a^{2}x}{(ax^{2} + bx + c)^{2}} + \frac{12ab^{2}x}{(ax^{2} + bx + c)^{3}} - \frac{6ab}{(ax^{2} + bx + c)^{2}} + \frac{2b^{3}}{(ax^{2} + bx + c)^{3}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{16a^{3}x^{3}}{(ax^{2} + bx + c)^{3}} + \frac{24a^{2}bx^{2}}{(ax^{2} + bx + c)^{3}} - \frac{12a^{2}x}{(ax^{2} + bx + c)^{2}} + \frac{12ab^{2}x}{(ax^{2} + bx + c)^{3}} - \frac{6ab}{(ax^{2} + bx + c)^{2}} + \frac{2b^{3}}{(ax^{2} + bx + c)^{3}}\right)}{dx}\\=&16(\frac{-3(a*2x + b + 0)}{(ax^{2} + bx + c)^{4}})a^{3}x^{3} + \frac{16a^{3}*3x^{2}}{(ax^{2} + bx + c)^{3}} + 24(\frac{-3(a*2x + b + 0)}{(ax^{2} + bx + c)^{4}})a^{2}bx^{2} + \frac{24a^{2}b*2x}{(ax^{2} + bx + c)^{3}} - 12(\frac{-2(a*2x + b + 0)}{(ax^{2} + bx + c)^{3}})a^{2}x - \frac{12a^{2}}{(ax^{2} + bx + c)^{2}} + 12(\frac{-3(a*2x + b + 0)}{(ax^{2} + bx + c)^{4}})ab^{2}x + \frac{12ab^{2}}{(ax^{2} + bx + c)^{3}} - 6(\frac{-2(a*2x + b + 0)}{(ax^{2} + bx + c)^{3}})ab + 0 + 2(\frac{-3(a*2x + b + 0)}{(ax^{2} + bx + c)^{4}})b^{3} + 0\\=&\frac{-96a^{4}x^{4}}{(ax^{2} + bx + c)^{4}} - \frac{192a^{3}bx^{3}}{(ax^{2} + bx + c)^{4}} + \frac{96a^{3}x^{2}}{(ax^{2} + bx + c)^{3}} - \frac{144a^{2}b^{2}x^{2}}{(ax^{2} + bx + c)^{4}} + \frac{96a^{2}bx}{(ax^{2} + bx + c)^{3}} - \frac{48ab^{3}x}{(ax^{2} + bx + c)^{4}} + \frac{24ab^{2}}{(ax^{2} + bx + c)^{3}} - \frac{12a^{2}}{(ax^{2} + bx + c)^{2}} - \frac{6b^{4}}{(ax^{2} + bx + c)^{4}}\\ \end{split}\end{equation} \]

\[ \begin{equation}\begin{split}【3/5】求函数ln(a{x}^{3} + b{x}^{2} + cx + d) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = ln(ax^{3} + bx^{2} + cx + d)\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( ln(ax^{3} + bx^{2} + cx + d)\right)}{dx}\\=&\frac{(a*3x^{2} + b*2x + c + 0)}{(ax^{3} + bx^{2} + cx + d)}\\=&\frac{3ax^{2}}{(ax^{3} + bx^{2} + cx + d)} + \frac{2bx}{(ax^{3} + bx^{2} + cx + d)} + \frac{c}{(ax^{3} + bx^{2} + cx + d)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{3ax^{2}}{(ax^{3} + bx^{2} + cx + d)} + \frac{2bx}{(ax^{3} + bx^{2} + cx + d)} + \frac{c}{(ax^{3} + bx^{2} + cx + d)}\right)}{dx}\\=&3(\frac{-(a*3x^{2} + b*2x + c + 0)}{(ax^{3} + bx^{2} + cx + d)^{2}})ax^{2} + \frac{3a*2x}{(ax^{3} + bx^{2} + cx + d)} + 2(\frac{-(a*3x^{2} + b*2x + c + 0)}{(ax^{3} + bx^{2} + cx + d)^{2}})bx + \frac{2b}{(ax^{3} + bx^{2} + cx + d)} + (\frac{-(a*3x^{2} + b*2x + c + 0)}{(ax^{3} + bx^{2} + cx + d)^{2}})c + 0\\=&\frac{-9a^{2}x^{4}}{(ax^{3} + bx^{2} + cx + d)^{2}} - \frac{12abx^{3}}{(ax^{3} + bx^{2} + cx + d)^{2}} - \frac{6acx^{2}}{(ax^{3} + bx^{2} + cx + d)^{2}} + \frac{6ax}{(ax^{3} + bx^{2} + cx + d)} - \frac{4b^{2}x^{2}}{(ax^{3} + bx^{2} + cx + d)^{2}} - \frac{4bcx}{(ax^{3} + bx^{2} + cx + d)^{2}} + \frac{2b}{(ax^{3} + bx^{2} + cx + d)} - \frac{c^{2}}{(ax^{3} + bx^{2} + cx + d)^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-9a^{2}x^{4}}{(ax^{3} + bx^{2} + cx + d)^{2}} - \frac{12abx^{3}}{(ax^{3} + bx^{2} + cx + d)^{2}} - \frac{6acx^{2}}{(ax^{3} + bx^{2} + cx + d)^{2}} + \frac{6ax}{(ax^{3} + bx^{2} + cx + d)} - \frac{4b^{2}x^{2}}{(ax^{3} + bx^{2} + cx + d)^{2}} - \frac{4bcx}{(ax^{3} + bx^{2} + cx + d)^{2}} + \frac{2b}{(ax^{3} + bx^{2} + cx + d)} - \frac{c^{2}}{(ax^{3} + bx^{2} + cx + d)^{2}}\right)}{dx}\\=&-9(\frac{-2(a*3x^{2} + b*2x + c + 0)}{(ax^{3} + bx^{2} + cx + d)^{3}})a^{2}x^{4} - \frac{9a^{2}*4x^{3}}{(ax^{3} + bx^{2} + cx + d)^{2}} - 12(\frac{-2(a*3x^{2} + b*2x + c + 0)}{(ax^{3} + bx^{2} + cx + d)^{3}})abx^{3} - \frac{12ab*3x^{2}}{(ax^{3} + bx^{2} + cx + d)^{2}} - 6(\frac{-2(a*3x^{2} + b*2x + c + 0)}{(ax^{3} + bx^{2} + cx + d)^{3}})acx^{2} - \frac{6ac*2x}{(ax^{3} + bx^{2} + cx + d)^{2}} + 6(\frac{-(a*3x^{2} + b*2x + c + 0)}{(ax^{3} + bx^{2} + cx + d)^{2}})ax + \frac{6a}{(ax^{3} + bx^{2} + cx + d)} - 4(\frac{-2(a*3x^{2} + b*2x + c + 0)}{(ax^{3} + bx^{2} + cx + d)^{3}})b^{2}x^{2} - \frac{4b^{2}*2x}{(ax^{3} + bx^{2} + cx + d)^{2}} - 4(\frac{-2(a*3x^{2} + b*2x + c + 0)}{(ax^{3} + bx^{2} + cx + d)^{3}})bcx - \frac{4bc}{(ax^{3} + bx^{2} + cx + d)^{2}} + 2(\frac{-(a*3x^{2} + b*2x + c + 0)}{(ax^{3} + bx^{2} + cx + d)^{2}})b + 0 - (\frac{-2(a*3x^{2} + b*2x + c + 0)}{(ax^{3} + bx^{2} + cx + d)^{3}})c^{2} + 0\\=&\frac{54a^{3}x^{6}}{(ax^{3} + bx^{2} + cx + d)^{3}} + \frac{108a^{2}bx^{5}}{(ax^{3} + bx^{2} + cx + d)^{3}} + \frac{54a^{2}cx^{4}}{(ax^{3} + bx^{2} + cx + d)^{3}} - \frac{54a^{2}x^{3}}{(ax^{3} + bx^{2} + cx + d)^{2}} + \frac{72ab^{2}x^{4}}{(ax^{3} + bx^{2} + cx + d)^{3}} + \frac{72abcx^{3}}{(ax^{3} + bx^{2} + cx + d)^{3}} - \frac{54abx^{2}}{(ax^{3} + bx^{2} + cx + d)^{2}} + \frac{18ac^{2}x^{2}}{(ax^{3} + bx^{2} + cx + d)^{3}} - \frac{18acx}{(ax^{3} + bx^{2} + cx + d)^{2}} + \frac{6a}{(ax^{3} + bx^{2} + cx + d)} + \frac{16b^{3}x^{3}}{(ax^{3} + bx^{2} + cx + d)^{3}} + \frac{24b^{2}cx^{2}}{(ax^{3} + bx^{2} + cx + d)^{3}} - \frac{12b^{2}x}{(ax^{3} + bx^{2} + cx + d)^{2}} + \frac{12bc^{2}x}{(ax^{3} + bx^{2} + cx + d)^{3}} - \frac{6bc}{(ax^{3} + bx^{2} + cx + d)^{2}} + \frac{2c^{3}}{(ax^{3} + bx^{2} + cx + d)^{3}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{54a^{3}x^{6}}{(ax^{3} + bx^{2} + cx + d)^{3}} + \frac{108a^{2}bx^{5}}{(ax^{3} + bx^{2} + cx + d)^{3}} + \frac{54a^{2}cx^{4}}{(ax^{3} + bx^{2} + cx + d)^{3}} - \frac{54a^{2}x^{3}}{(ax^{3} + bx^{2} + cx + d)^{2}} + \frac{72ab^{2}x^{4}}{(ax^{3} + bx^{2} + cx + d)^{3}} + \frac{72abcx^{3}}{(ax^{3} + bx^{2} + cx + d)^{3}} - \frac{54abx^{2}}{(ax^{3} + bx^{2} + cx + d)^{2}} + \frac{18ac^{2}x^{2}}{(ax^{3} + bx^{2} + cx + d)^{3}} - \frac{18acx}{(ax^{3} + bx^{2} + cx + d)^{2}} + \frac{6a}{(ax^{3} + bx^{2} + cx + d)} + \frac{16b^{3}x^{3}}{(ax^{3} + bx^{2} + cx + d)^{3}} + \frac{24b^{2}cx^{2}}{(ax^{3} + bx^{2} + cx + d)^{3}} - \frac{12b^{2}x}{(ax^{3} + bx^{2} + cx + d)^{2}} + \frac{12bc^{2}x}{(ax^{3} + bx^{2} + cx + d)^{3}} - \frac{6bc}{(ax^{3} + bx^{2} + cx + d)^{2}} + \frac{2c^{3}}{(ax^{3} + bx^{2} + cx + d)^{3}}\right)}{dx}\\=&54(\frac{-3(a*3x^{2} + b*2x + c + 0)}{(ax^{3} + bx^{2} + cx + d)^{4}})a^{3}x^{6} + \frac{54a^{3}*6x^{5}}{(ax^{3} + bx^{2} + cx + d)^{3}} + 108(\frac{-3(a*3x^{2} + b*2x + c + 0)}{(ax^{3} + bx^{2} + cx + d)^{4}})a^{2}bx^{5} + \frac{108a^{2}b*5x^{4}}{(ax^{3} + bx^{2} + cx + d)^{3}} + 54(\frac{-3(a*3x^{2} + b*2x + c + 0)}{(ax^{3} + bx^{2} + cx + d)^{4}})a^{2}cx^{4} + \frac{54a^{2}c*4x^{3}}{(ax^{3} + bx^{2} + cx + d)^{3}} - 54(\frac{-2(a*3x^{2} + b*2x + c + 0)}{(ax^{3} + bx^{2} + cx + d)^{3}})a^{2}x^{3} - \frac{54a^{2}*3x^{2}}{(ax^{3} + bx^{2} + cx + d)^{2}} + 72(\frac{-3(a*3x^{2} + b*2x + c + 0)}{(ax^{3} + bx^{2} + cx + d)^{4}})ab^{2}x^{4} + \frac{72ab^{2}*4x^{3}}{(ax^{3} + bx^{2} + cx + d)^{3}} + 72(\frac{-3(a*3x^{2} + b*2x + c + 0)}{(ax^{3} + bx^{2} + cx + d)^{4}})abcx^{3} + \frac{72abc*3x^{2}}{(ax^{3} + bx^{2} + cx + d)^{3}} - 54(\frac{-2(a*3x^{2} + b*2x + c + 0)}{(ax^{3} + bx^{2} + cx + d)^{3}})abx^{2} - \frac{54ab*2x}{(ax^{3} + bx^{2} + cx + d)^{2}} + 18(\frac{-3(a*3x^{2} + b*2x + c + 0)}{(ax^{3} + bx^{2} + cx + d)^{4}})ac^{2}x^{2} + \frac{18ac^{2}*2x}{(ax^{3} + bx^{2} + cx + d)^{3}} - 18(\frac{-2(a*3x^{2} + b*2x + c + 0)}{(ax^{3} + bx^{2} + cx + d)^{3}})acx - \frac{18ac}{(ax^{3} + bx^{2} + cx + d)^{2}} + 6(\frac{-(a*3x^{2} + b*2x + c + 0)}{(ax^{3} + bx^{2} + cx + d)^{2}})a + 0 + 16(\frac{-3(a*3x^{2} + b*2x + c + 0)}{(ax^{3} + bx^{2} + cx + d)^{4}})b^{3}x^{3} + \frac{16b^{3}*3x^{2}}{(ax^{3} + bx^{2} + cx + d)^{3}} + 24(\frac{-3(a*3x^{2} + b*2x + c + 0)}{(ax^{3} + bx^{2} + cx + d)^{4}})b^{2}cx^{2} + \frac{24b^{2}c*2x}{(ax^{3} + bx^{2} + cx + d)^{3}} - 12(\frac{-2(a*3x^{2} + b*2x + c + 0)}{(ax^{3} + bx^{2} + cx + d)^{3}})b^{2}x - \frac{12b^{2}}{(ax^{3} + bx^{2} + cx + d)^{2}} + 12(\frac{-3(a*3x^{2} + b*2x + c + 0)}{(ax^{3} + bx^{2} + cx + d)^{4}})bc^{2}x + \frac{12bc^{2}}{(ax^{3} + bx^{2} + cx + d)^{3}} - 6(\frac{-2(a*3x^{2} + b*2x + c + 0)}{(ax^{3} + bx^{2} + cx + d)^{3}})bc + 0 + 2(\frac{-3(a*3x^{2} + b*2x + c + 0)}{(ax^{3} + bx^{2} + cx + d)^{4}})c^{3} + 0\\=&\frac{-486a^{4}x^{8}}{(ax^{3} + bx^{2} + cx + d)^{4}} - \frac{1296a^{3}bx^{7}}{(ax^{3} + bx^{2} + cx + d)^{4}} - \frac{648a^{3}cx^{6}}{(ax^{3} + bx^{2} + cx + d)^{4}} + \frac{648a^{3}x^{5}}{(ax^{3} + bx^{2} + cx + d)^{3}} - \frac{1296a^{2}b^{2}x^{6}}{(ax^{3} + bx^{2} + cx + d)^{4}} - \frac{1296a^{2}bcx^{5}}{(ax^{3} + bx^{2} + cx + d)^{4}} + \frac{1080a^{2}bx^{4}}{(ax^{3} + bx^{2} + cx + d)^{3}} - \frac{432abc^{2}x^{3}}{(ax^{3} + bx^{2} + cx + d)^{4}} - \frac{324a^{2}c^{2}x^{4}}{(ax^{3} + bx^{2} + cx + d)^{4}} + \frac{432a^{2}cx^{3}}{(ax^{3} + bx^{2} + cx + d)^{3}} - \frac{180a^{2}x^{2}}{(ax^{3} + bx^{2} + cx + d)^{2}} - \frac{576ab^{3}x^{5}}{(ax^{3} + bx^{2} + cx + d)^{4}} - \frac{864ab^{2}cx^{4}}{(ax^{3} + bx^{2} + cx + d)^{4}} + \frac{576ab^{2}x^{3}}{(ax^{3} + bx^{2} + cx + d)^{3}} + \frac{432abcx^{2}}{(ax^{3} + bx^{2} + cx + d)^{3}} - \frac{120abx}{(ax^{3} + bx^{2} + cx + d)^{2}} - \frac{72ac^{3}x^{2}}{(ax^{3} + bx^{2} + cx + d)^{4}} + \frac{72ac^{2}x}{(ax^{3} + bx^{2} + cx + d)^{3}} - \frac{24ac}{(ax^{3} + bx^{2} + cx + d)^{2}} - \frac{96b^{4}x^{4}}{(ax^{3} + bx^{2} + cx + d)^{4}} - \frac{192b^{3}cx^{3}}{(ax^{3} + bx^{2} + cx + d)^{4}} + \frac{96b^{3}x^{2}}{(ax^{3} + bx^{2} + cx + d)^{3}} - \frac{144b^{2}c^{2}x^{2}}{(ax^{3} + bx^{2} + cx + d)^{4}} + \frac{96b^{2}cx}{(ax^{3} + bx^{2} + cx + d)^{3}} - \frac{48bc^{3}x}{(ax^{3} + bx^{2} + cx + d)^{4}} + \frac{24bc^{2}}{(ax^{3} + bx^{2} + cx + d)^{3}} - \frac{12b^{2}}{(ax^{3} + bx^{2} + cx + d)^{2}} - \frac{6c^{4}}{(ax^{3} + bx^{2} + cx + d)^{4}}\\ \end{split}\end{equation} \]

\[ \begin{equation}\begin{split}【4/5】求函数ln(a{x}^{4} + b{x}^{3} + c{x}^{2} + dx + f) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = ln(ax^{4} + bx^{3} + cx^{2} + dx + f)\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( ln(ax^{4} + bx^{3} + cx^{2} + dx + f)\right)}{dx}\\=&\frac{(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)}\\=&\frac{4ax^{3}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)} + \frac{3bx^{2}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)} + \frac{2cx}{(ax^{4} + bx^{3} + cx^{2} + dx + f)} + \frac{d}{(ax^{4} + bx^{3} + cx^{2} + dx + f)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{4ax^{3}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)} + \frac{3bx^{2}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)} + \frac{2cx}{(ax^{4} + bx^{3} + cx^{2} + dx + f)} + \frac{d}{(ax^{4} + bx^{3} + cx^{2} + dx + f)}\right)}{dx}\\=&4(\frac{-(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}})ax^{3} + \frac{4a*3x^{2}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)} + 3(\frac{-(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}})bx^{2} + \frac{3b*2x}{(ax^{4} + bx^{3} + cx^{2} + dx + f)} + 2(\frac{-(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}})cx + \frac{2c}{(ax^{4} + bx^{3} + cx^{2} + dx + f)} + (\frac{-(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}})d + 0\\=&\frac{-16a^{2}x^{6}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} - \frac{24abx^{5}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} - \frac{16acx^{4}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} - \frac{8adx^{3}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} + \frac{12ax^{2}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)} - \frac{9b^{2}x^{4}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} - \frac{12bcx^{3}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} - \frac{6bdx^{2}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} + \frac{6bx}{(ax^{4} + bx^{3} + cx^{2} + dx + f)} - \frac{4c^{2}x^{2}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} - \frac{4cdx}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} + \frac{2c}{(ax^{4} + bx^{3} + cx^{2} + dx + f)} - \frac{d^{2}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-16a^{2}x^{6}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} - \frac{24abx^{5}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} - \frac{16acx^{4}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} - \frac{8adx^{3}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} + \frac{12ax^{2}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)} - \frac{9b^{2}x^{4}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} - \frac{12bcx^{3}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} - \frac{6bdx^{2}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} + \frac{6bx}{(ax^{4} + bx^{3} + cx^{2} + dx + f)} - \frac{4c^{2}x^{2}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} - \frac{4cdx}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} + \frac{2c}{(ax^{4} + bx^{3} + cx^{2} + dx + f)} - \frac{d^{2}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}}\right)}{dx}\\=&-16(\frac{-2(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}})a^{2}x^{6} - \frac{16a^{2}*6x^{5}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} - 24(\frac{-2(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}})abx^{5} - \frac{24ab*5x^{4}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} - 16(\frac{-2(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}})acx^{4} - \frac{16ac*4x^{3}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} - 8(\frac{-2(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}})adx^{3} - \frac{8ad*3x^{2}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} + 12(\frac{-(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}})ax^{2} + \frac{12a*2x}{(ax^{4} + bx^{3} + cx^{2} + dx + f)} - 9(\frac{-2(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}})b^{2}x^{4} - \frac{9b^{2}*4x^{3}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} - 12(\frac{-2(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}})bcx^{3} - \frac{12bc*3x^{2}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} - 6(\frac{-2(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}})bdx^{2} - \frac{6bd*2x}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} + 6(\frac{-(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}})bx + \frac{6b}{(ax^{4} + bx^{3} + cx^{2} + dx + f)} - 4(\frac{-2(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}})c^{2}x^{2} - \frac{4c^{2}*2x}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} - 4(\frac{-2(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}})cdx - \frac{4cd}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} + 2(\frac{-(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}})c + 0 - (\frac{-2(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}})d^{2} + 0\\=&\frac{128a^{3}x^{9}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} + \frac{288a^{2}bx^{8}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} + \frac{192a^{2}cx^{7}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} + \frac{96a^{2}dx^{6}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - \frac{144a^{2}x^{5}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} + \frac{216ab^{2}x^{7}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} + \frac{288abcx^{6}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} + \frac{144abdx^{5}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - \frac{180abx^{4}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} + \frac{96ac^{2}x^{5}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} + \frac{96acdx^{4}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - \frac{96acx^{3}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} + \frac{24ad^{2}x^{3}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - \frac{36adx^{2}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} + \frac{24ax}{(ax^{4} + bx^{3} + cx^{2} + dx + f)} + \frac{54b^{3}x^{6}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} + \frac{108b^{2}cx^{5}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} + \frac{54b^{2}dx^{4}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - \frac{54b^{2}x^{3}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} + \frac{72bc^{2}x^{4}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} + \frac{72bcdx^{3}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - \frac{54bcx^{2}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} + \frac{18bd^{2}x^{2}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - \frac{18bdx}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} + \frac{6b}{(ax^{4} + bx^{3} + cx^{2} + dx + f)} + \frac{16c^{3}x^{3}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} + \frac{24c^{2}dx^{2}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - \frac{12c^{2}x}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} + \frac{12cd^{2}x}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - \frac{6cd}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} + \frac{2d^{3}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{128a^{3}x^{9}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} + \frac{288a^{2}bx^{8}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} + \frac{192a^{2}cx^{7}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} + \frac{96a^{2}dx^{6}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - \frac{144a^{2}x^{5}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} + \frac{216ab^{2}x^{7}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} + \frac{288abcx^{6}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} + \frac{144abdx^{5}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - \frac{180abx^{4}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} + \frac{96ac^{2}x^{5}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} + \frac{96acdx^{4}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - \frac{96acx^{3}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} + \frac{24ad^{2}x^{3}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - \frac{36adx^{2}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} + \frac{24ax}{(ax^{4} + bx^{3} + cx^{2} + dx + f)} + \frac{54b^{3}x^{6}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} + \frac{108b^{2}cx^{5}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} + \frac{54b^{2}dx^{4}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - \frac{54b^{2}x^{3}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} + \frac{72bc^{2}x^{4}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} + \frac{72bcdx^{3}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - \frac{54bcx^{2}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} + \frac{18bd^{2}x^{2}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - \frac{18bdx}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} + \frac{6b}{(ax^{4} + bx^{3} + cx^{2} + dx + f)} + \frac{16c^{3}x^{3}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} + \frac{24c^{2}dx^{2}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - \frac{12c^{2}x}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} + \frac{12cd^{2}x}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - \frac{6cd}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} + \frac{2d^{3}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}}\right)}{dx}\\=&128(\frac{-3(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}})a^{3}x^{9} + \frac{128a^{3}*9x^{8}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} + 288(\frac{-3(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}})a^{2}bx^{8} + \frac{288a^{2}b*8x^{7}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} + 192(\frac{-3(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}})a^{2}cx^{7} + \frac{192a^{2}c*7x^{6}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} + 96(\frac{-3(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}})a^{2}dx^{6} + \frac{96a^{2}d*6x^{5}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - 144(\frac{-2(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}})a^{2}x^{5} - \frac{144a^{2}*5x^{4}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} + 216(\frac{-3(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}})ab^{2}x^{7} + \frac{216ab^{2}*7x^{6}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} + 288(\frac{-3(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}})abcx^{6} + \frac{288abc*6x^{5}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} + 144(\frac{-3(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}})abdx^{5} + \frac{144abd*5x^{4}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - 180(\frac{-2(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}})abx^{4} - \frac{180ab*4x^{3}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} + 96(\frac{-3(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}})ac^{2}x^{5} + \frac{96ac^{2}*5x^{4}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} + 96(\frac{-3(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}})acdx^{4} + \frac{96acd*4x^{3}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - 96(\frac{-2(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}})acx^{3} - \frac{96ac*3x^{2}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} + 24(\frac{-3(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}})ad^{2}x^{3} + \frac{24ad^{2}*3x^{2}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - 36(\frac{-2(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}})adx^{2} - \frac{36ad*2x}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} + 24(\frac{-(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}})ax + \frac{24a}{(ax^{4} + bx^{3} + cx^{2} + dx + f)} + 54(\frac{-3(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}})b^{3}x^{6} + \frac{54b^{3}*6x^{5}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} + 108(\frac{-3(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}})b^{2}cx^{5} + \frac{108b^{2}c*5x^{4}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} + 54(\frac{-3(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}})b^{2}dx^{4} + \frac{54b^{2}d*4x^{3}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - 54(\frac{-2(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}})b^{2}x^{3} - \frac{54b^{2}*3x^{2}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} + 72(\frac{-3(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}})bc^{2}x^{4} + \frac{72bc^{2}*4x^{3}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} + 72(\frac{-3(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}})bcdx^{3} + \frac{72bcd*3x^{2}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - 54(\frac{-2(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}})bcx^{2} - \frac{54bc*2x}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} + 18(\frac{-3(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}})bd^{2}x^{2} + \frac{18bd^{2}*2x}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - 18(\frac{-2(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}})bdx - \frac{18bd}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} + 6(\frac{-(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}})b + 0 + 16(\frac{-3(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}})c^{3}x^{3} + \frac{16c^{3}*3x^{2}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} + 24(\frac{-3(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}})c^{2}dx^{2} + \frac{24c^{2}d*2x}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - 12(\frac{-2(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}})c^{2}x - \frac{12c^{2}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} + 12(\frac{-3(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}})cd^{2}x + \frac{12cd^{2}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - 6(\frac{-2(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}})cd + 0 + 2(\frac{-3(a*4x^{3} + b*3x^{2} + c*2x + d + 0)}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}})d^{3} + 0\\=&\frac{-1536a^{4}x^{12}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}} - \frac{4608a^{3}bx^{11}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}} - \frac{3072a^{3}cx^{10}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}} - \frac{1536a^{3}dx^{9}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}} + \frac{2304a^{3}x^{8}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - \frac{5184a^{2}b^{2}x^{10}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}} - \frac{6912a^{2}bcx^{9}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}} - \frac{3456a^{2}bdx^{8}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}} + \frac{4608a^{2}bx^{7}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - \frac{2304a^{2}c^{2}x^{8}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}} - \frac{2304a^{2}cdx^{7}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}} + \frac{2688a^{2}cx^{6}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - \frac{864abd^{2}x^{5}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}} - \frac{576acd^{2}x^{4}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}} - \frac{576a^{2}d^{2}x^{6}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}} + \frac{1152a^{2}dx^{5}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - \frac{816a^{2}x^{4}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} - \frac{2592ab^{3}x^{9}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}} - \frac{5184ab^{2}cx^{8}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}} - \frac{2592ab^{2}dx^{7}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}} + \frac{3024ab^{2}x^{6}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - \frac{3456abc^{2}x^{7}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}} - \frac{3456abcdx^{6}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}} + \frac{3456abcx^{5}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} + \frac{1440abdx^{4}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - \frac{816abx^{3}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} - \frac{768ac^{3}x^{6}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}} - \frac{1152ac^{2}dx^{5}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}} + \frac{960ac^{2}x^{4}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} + \frac{768acdx^{3}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - \frac{336acx^{2}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} - \frac{96ad^{3}x^{3}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}} + \frac{144ad^{2}x^{2}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - \frac{96adx}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} + \frac{24a}{(ax^{4} + bx^{3} + cx^{2} + dx + f)} - \frac{486b^{4}x^{8}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}} - \frac{1296b^{3}cx^{7}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}} - \frac{648b^{3}dx^{6}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}} + \frac{648b^{3}x^{5}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - \frac{1296b^{2}c^{2}x^{6}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}} - \frac{1296b^{2}cdx^{5}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}} + \frac{1080b^{2}cx^{4}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - \frac{432bcd^{2}x^{3}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}} - \frac{324b^{2}d^{2}x^{4}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}} + \frac{432b^{2}dx^{3}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - \frac{180b^{2}x^{2}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} - \frac{576bc^{3}x^{5}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}} - \frac{864bc^{2}dx^{4}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}} + \frac{576bc^{2}x^{3}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} + \frac{432bcdx^{2}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - \frac{120bcx}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} - \frac{72bd^{3}x^{2}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}} + \frac{72bd^{2}x}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - \frac{24bd}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} - \frac{96c^{4}x^{4}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}} - \frac{192c^{3}dx^{3}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}} + \frac{96c^{3}x^{2}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - \frac{144c^{2}d^{2}x^{2}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}} + \frac{96c^{2}dx}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - \frac{48cd^{3}x}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}} + \frac{24cd^{2}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{3}} - \frac{12c^{2}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{2}} - \frac{6d^{4}}{(ax^{4} + bx^{3} + cx^{2} + dx + f)^{4}}\\ \end{split}\end{equation} \]

\[ \begin{equation}\begin{split}【5/5】求函数ln(a{x}^{5} + b{x}^{4} + c{x}^{3} + d{x}^{2} + fx + g) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = ln(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( ln(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)\right)}{dx}\\=&\frac{(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)}\\=&\frac{5ax^{4}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)} + \frac{4bx^{3}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)} + \frac{3cx^{2}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)} + \frac{2dx}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)} + \frac{f}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{5ax^{4}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)} + \frac{4bx^{3}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)} + \frac{3cx^{2}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)} + \frac{2dx}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)} + \frac{f}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)}\right)}{dx}\\=&5(\frac{-(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}})ax^{4} + \frac{5a*4x^{3}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)} + 4(\frac{-(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}})bx^{3} + \frac{4b*3x^{2}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)} + 3(\frac{-(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}})cx^{2} + \frac{3c*2x}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)} + 2(\frac{-(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}})dx + \frac{2d}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)} + (\frac{-(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}})f + 0\\=&\frac{-25a^{2}x^{8}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} - \frac{40abx^{7}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} - \frac{30acx^{6}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} - \frac{20adx^{5}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} - \frac{10afx^{4}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + \frac{20ax^{3}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)} - \frac{16b^{2}x^{6}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} - \frac{24bcx^{5}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} - \frac{16bdx^{4}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} - \frac{8bfx^{3}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + \frac{12bx^{2}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)} - \frac{9c^{2}x^{4}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} - \frac{12cdx^{3}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} - \frac{6cfx^{2}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + \frac{6cx}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)} - \frac{4d^{2}x^{2}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} - \frac{4dfx}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + \frac{2d}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)} - \frac{f^{2}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-25a^{2}x^{8}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} - \frac{40abx^{7}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} - \frac{30acx^{6}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} - \frac{20adx^{5}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} - \frac{10afx^{4}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + \frac{20ax^{3}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)} - \frac{16b^{2}x^{6}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} - \frac{24bcx^{5}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} - \frac{16bdx^{4}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} - \frac{8bfx^{3}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + \frac{12bx^{2}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)} - \frac{9c^{2}x^{4}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} - \frac{12cdx^{3}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} - \frac{6cfx^{2}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + \frac{6cx}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)} - \frac{4d^{2}x^{2}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} - \frac{4dfx}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + \frac{2d}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)} - \frac{f^{2}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}}\right)}{dx}\\=&-25(\frac{-2(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}})a^{2}x^{8} - \frac{25a^{2}*8x^{7}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} - 40(\frac{-2(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}})abx^{7} - \frac{40ab*7x^{6}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} - 30(\frac{-2(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}})acx^{6} - \frac{30ac*6x^{5}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} - 20(\frac{-2(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}})adx^{5} - \frac{20ad*5x^{4}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} - 10(\frac{-2(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}})afx^{4} - \frac{10af*4x^{3}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + 20(\frac{-(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}})ax^{3} + \frac{20a*3x^{2}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)} - 16(\frac{-2(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}})b^{2}x^{6} - \frac{16b^{2}*6x^{5}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} - 24(\frac{-2(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}})bcx^{5} - \frac{24bc*5x^{4}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} - 16(\frac{-2(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}})bdx^{4} - \frac{16bd*4x^{3}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} - 8(\frac{-2(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}})bfx^{3} - \frac{8bf*3x^{2}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + 12(\frac{-(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}})bx^{2} + \frac{12b*2x}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)} - 9(\frac{-2(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}})c^{2}x^{4} - \frac{9c^{2}*4x^{3}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} - 12(\frac{-2(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}})cdx^{3} - \frac{12cd*3x^{2}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} - 6(\frac{-2(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}})cfx^{2} - \frac{6cf*2x}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + 6(\frac{-(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}})cx + \frac{6c}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)} - 4(\frac{-2(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}})d^{2}x^{2} - \frac{4d^{2}*2x}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} - 4(\frac{-2(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}})dfx - \frac{4df}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + 2(\frac{-(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}})d + 0 - (\frac{-2(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}})f^{2} + 0\\=&\frac{250a^{3}x^{12}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{600a^{2}bx^{11}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{450a^{2}cx^{10}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{300a^{2}dx^{9}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{150a^{2}fx^{8}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{300a^{2}x^{7}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + \frac{480ab^{2}x^{10}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{720abcx^{9}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{480abdx^{8}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{240abfx^{7}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{420abx^{6}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + \frac{270ac^{2}x^{8}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{360acdx^{7}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{180acfx^{6}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{270acx^{5}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + \frac{120ad^{2}x^{6}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{120adfx^{5}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{150adx^{4}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + \frac{30af^{2}x^{4}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{60afx^{3}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + \frac{60ax^{2}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)} + \frac{128b^{3}x^{9}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{288b^{2}cx^{8}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{192b^{2}dx^{7}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{96b^{2}fx^{6}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{144b^{2}x^{5}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + \frac{216bc^{2}x^{7}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{288bcdx^{6}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{144bcfx^{5}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{180bcx^{4}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + \frac{96bd^{2}x^{5}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{96bdfx^{4}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{96bdx^{3}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + \frac{24bf^{2}x^{3}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{36bfx^{2}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + \frac{24bx}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)} + \frac{54c^{3}x^{6}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{108c^{2}dx^{5}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{54c^{2}fx^{4}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{54c^{2}x^{3}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + \frac{72cd^{2}x^{4}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{72cdfx^{3}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{54cdx^{2}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + \frac{18cf^{2}x^{2}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{18cfx}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + \frac{6c}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)} + \frac{16d^{3}x^{3}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{24d^{2}fx^{2}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{12d^{2}x}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + \frac{12df^{2}x}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{6df}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + \frac{2f^{3}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{250a^{3}x^{12}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{600a^{2}bx^{11}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{450a^{2}cx^{10}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{300a^{2}dx^{9}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{150a^{2}fx^{8}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{300a^{2}x^{7}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + \frac{480ab^{2}x^{10}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{720abcx^{9}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{480abdx^{8}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{240abfx^{7}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{420abx^{6}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + \frac{270ac^{2}x^{8}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{360acdx^{7}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{180acfx^{6}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{270acx^{5}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + \frac{120ad^{2}x^{6}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{120adfx^{5}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{150adx^{4}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + \frac{30af^{2}x^{4}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{60afx^{3}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + \frac{60ax^{2}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)} + \frac{128b^{3}x^{9}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{288b^{2}cx^{8}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{192b^{2}dx^{7}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{96b^{2}fx^{6}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{144b^{2}x^{5}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + \frac{216bc^{2}x^{7}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{288bcdx^{6}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{144bcfx^{5}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{180bcx^{4}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + \frac{96bd^{2}x^{5}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{96bdfx^{4}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{96bdx^{3}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + \frac{24bf^{2}x^{3}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{36bfx^{2}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + \frac{24bx}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)} + \frac{54c^{3}x^{6}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{108c^{2}dx^{5}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{54c^{2}fx^{4}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{54c^{2}x^{3}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + \frac{72cd^{2}x^{4}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{72cdfx^{3}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{54cdx^{2}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + \frac{18cf^{2}x^{2}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{18cfx}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + \frac{6c}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)} + \frac{16d^{3}x^{3}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{24d^{2}fx^{2}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{12d^{2}x}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + \frac{12df^{2}x}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{6df}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + \frac{2f^{3}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}}\right)}{dx}\\=&250(\frac{-3(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}})a^{3}x^{12} + \frac{250a^{3}*12x^{11}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + 600(\frac{-3(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}})a^{2}bx^{11} + \frac{600a^{2}b*11x^{10}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + 450(\frac{-3(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}})a^{2}cx^{10} + \frac{450a^{2}c*10x^{9}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + 300(\frac{-3(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}})a^{2}dx^{9} + \frac{300a^{2}d*9x^{8}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + 150(\frac{-3(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}})a^{2}fx^{8} + \frac{150a^{2}f*8x^{7}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - 300(\frac{-2(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}})a^{2}x^{7} - \frac{300a^{2}*7x^{6}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + 480(\frac{-3(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}})ab^{2}x^{10} + \frac{480ab^{2}*10x^{9}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + 720(\frac{-3(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}})abcx^{9} + \frac{720abc*9x^{8}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + 480(\frac{-3(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}})abdx^{8} + \frac{480abd*8x^{7}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + 240(\frac{-3(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}})abfx^{7} + \frac{240abf*7x^{6}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - 420(\frac{-2(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}})abx^{6} - \frac{420ab*6x^{5}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + 270(\frac{-3(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}})ac^{2}x^{8} + \frac{270ac^{2}*8x^{7}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + 360(\frac{-3(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}})acdx^{7} + \frac{360acd*7x^{6}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + 180(\frac{-3(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}})acfx^{6} + \frac{180acf*6x^{5}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - 270(\frac{-2(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}})acx^{5} - \frac{270ac*5x^{4}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + 120(\frac{-3(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}})ad^{2}x^{6} + \frac{120ad^{2}*6x^{5}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + 120(\frac{-3(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}})adfx^{5} + \frac{120adf*5x^{4}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - 150(\frac{-2(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}})adx^{4} - \frac{150ad*4x^{3}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + 30(\frac{-3(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}})af^{2}x^{4} + \frac{30af^{2}*4x^{3}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - 60(\frac{-2(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}})afx^{3} - \frac{60af*3x^{2}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + 60(\frac{-(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}})ax^{2} + \frac{60a*2x}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)} + 128(\frac{-3(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}})b^{3}x^{9} + \frac{128b^{3}*9x^{8}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + 288(\frac{-3(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}})b^{2}cx^{8} + \frac{288b^{2}c*8x^{7}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + 192(\frac{-3(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}})b^{2}dx^{7} + \frac{192b^{2}d*7x^{6}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + 96(\frac{-3(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}})b^{2}fx^{6} + \frac{96b^{2}f*6x^{5}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - 144(\frac{-2(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}})b^{2}x^{5} - \frac{144b^{2}*5x^{4}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + 216(\frac{-3(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}})bc^{2}x^{7} + \frac{216bc^{2}*7x^{6}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + 288(\frac{-3(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}})bcdx^{6} + \frac{288bcd*6x^{5}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + 144(\frac{-3(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}})bcfx^{5} + \frac{144bcf*5x^{4}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - 180(\frac{-2(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}})bcx^{4} - \frac{180bc*4x^{3}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + 96(\frac{-3(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}})bd^{2}x^{5} + \frac{96bd^{2}*5x^{4}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + 96(\frac{-3(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}})bdfx^{4} + \frac{96bdf*4x^{3}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - 96(\frac{-2(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}})bdx^{3} - \frac{96bd*3x^{2}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + 24(\frac{-3(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}})bf^{2}x^{3} + \frac{24bf^{2}*3x^{2}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - 36(\frac{-2(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}})bfx^{2} - \frac{36bf*2x}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + 24(\frac{-(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}})bx + \frac{24b}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)} + 54(\frac{-3(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}})c^{3}x^{6} + \frac{54c^{3}*6x^{5}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + 108(\frac{-3(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}})c^{2}dx^{5} + \frac{108c^{2}d*5x^{4}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + 54(\frac{-3(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}})c^{2}fx^{4} + \frac{54c^{2}f*4x^{3}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - 54(\frac{-2(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}})c^{2}x^{3} - \frac{54c^{2}*3x^{2}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + 72(\frac{-3(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}})cd^{2}x^{4} + \frac{72cd^{2}*4x^{3}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + 72(\frac{-3(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}})cdfx^{3} + \frac{72cdf*3x^{2}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - 54(\frac{-2(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}})cdx^{2} - \frac{54cd*2x}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + 18(\frac{-3(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}})cf^{2}x^{2} + \frac{18cf^{2}*2x}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - 18(\frac{-2(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}})cfx - \frac{18cf}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + 6(\frac{-(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}})c + 0 + 16(\frac{-3(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}})d^{3}x^{3} + \frac{16d^{3}*3x^{2}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + 24(\frac{-3(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}})d^{2}fx^{2} + \frac{24d^{2}f*2x}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - 12(\frac{-2(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}})d^{2}x - \frac{12d^{2}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + 12(\frac{-3(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}})df^{2}x + \frac{12df^{2}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - 6(\frac{-2(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}})df + 0 + 2(\frac{-3(a*5x^{4} + b*4x^{3} + c*3x^{2} + d*2x + f + 0)}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}})f^{3} + 0\\=&\frac{-3750a^{4}x^{16}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} - \frac{12000a^{3}bx^{15}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} - \frac{9000a^{3}cx^{14}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} - \frac{6000a^{3}dx^{13}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} - \frac{3000a^{3}fx^{12}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} + \frac{6000a^{3}x^{11}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{14400a^{2}b^{2}x^{14}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} - \frac{21600a^{2}bcx^{13}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} - \frac{14400a^{2}bdx^{12}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} - \frac{7200a^{2}bfx^{11}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} + \frac{13200a^{2}bx^{10}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{8100a^{2}c^{2}x^{12}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} - \frac{10800a^{2}cdx^{11}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} - \frac{5400a^{2}cfx^{10}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} + \frac{9000a^{2}cx^{9}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{3600a^{2}d^{2}x^{10}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} - \frac{3600a^{2}dfx^{9}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} + \frac{5400a^{2}dx^{8}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{1440abf^{2}x^{7}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} - \frac{1080acf^{2}x^{6}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} - \frac{720adf^{2}x^{5}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} - \frac{900a^{2}f^{2}x^{8}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} + \frac{2400a^{2}fx^{7}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{2400a^{2}x^{6}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} - \frac{7680ab^{3}x^{13}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} - \frac{17280ab^{2}cx^{12}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} - \frac{11520ab^{2}dx^{11}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} - \frac{5760ab^{2}fx^{10}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} + \frac{9600ab^{2}x^{9}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{12960abc^{2}x^{11}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} - \frac{17280abcdx^{10}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} - \frac{8640abcfx^{9}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} + \frac{12960abcx^{8}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{5760abd^{2}x^{9}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} - \frac{5760abdfx^{8}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} + \frac{7680abdx^{7}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{3360abfx^{6}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{2880abx^{5}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} - \frac{3240ac^{3}x^{10}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} - \frac{6480ac^{2}dx^{9}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} - \frac{3240ac^{2}fx^{8}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} + \frac{4320ac^{2}x^{7}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{4320acd^{2}x^{8}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} - \frac{4320acdfx^{7}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} + \frac{5040acdx^{6}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{2160acfx^{5}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{1560acx^{4}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} - \frac{960ad^{3}x^{7}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} - \frac{1440ad^{2}fx^{6}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} + \frac{1440ad^{2}x^{5}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{1200adfx^{4}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{720adx^{3}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} - \frac{120af^{3}x^{4}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} + \frac{240af^{2}x^{3}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{240afx^{2}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + \frac{120ax}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)} - \frac{1536b^{4}x^{12}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} - \frac{4608b^{3}cx^{11}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} - \frac{3072b^{3}dx^{10}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} - \frac{1536b^{3}fx^{9}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} + \frac{2304b^{3}x^{8}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{5184b^{2}c^{2}x^{10}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} - \frac{6912b^{2}cdx^{9}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} - \frac{3456b^{2}cfx^{8}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} + \frac{4608b^{2}cx^{7}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{2304b^{2}d^{2}x^{8}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} - \frac{2304b^{2}dfx^{7}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} + \frac{2688b^{2}dx^{6}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{864bcf^{2}x^{5}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} - \frac{576bdf^{2}x^{4}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} - \frac{576b^{2}f^{2}x^{6}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} + \frac{1152b^{2}fx^{5}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{816b^{2}x^{4}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} - \frac{2592bc^{3}x^{9}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} - \frac{5184bc^{2}dx^{8}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} - \frac{2592bc^{2}fx^{7}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} + \frac{3024bc^{2}x^{6}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{3456bcd^{2}x^{7}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} - \frac{3456bcdfx^{6}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} + \frac{3456bcdx^{5}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{1440bcfx^{4}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{816bcx^{3}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} - \frac{768bd^{3}x^{6}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} - \frac{1152bd^{2}fx^{5}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} + \frac{960bd^{2}x^{4}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{768bdfx^{3}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{336bdx^{2}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} - \frac{96bf^{3}x^{3}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} + \frac{144bf^{2}x^{2}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{96bfx}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} + \frac{24b}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)} - \frac{486c^{4}x^{8}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} - \frac{1296c^{3}dx^{7}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} - \frac{648c^{3}fx^{6}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} + \frac{648c^{3}x^{5}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{1296c^{2}d^{2}x^{6}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} - \frac{1296c^{2}dfx^{5}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} + \frac{1080c^{2}dx^{4}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{432cdf^{2}x^{3}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} - \frac{324c^{2}f^{2}x^{4}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} + \frac{432c^{2}fx^{3}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{180c^{2}x^{2}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} - \frac{576cd^{3}x^{5}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} - \frac{864cd^{2}fx^{4}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} + \frac{576cd^{2}x^{3}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} + \frac{432cdfx^{2}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{120cdx}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} - \frac{72cf^{3}x^{2}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} + \frac{72cf^{2}x}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{24cf}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} - \frac{96d^{4}x^{4}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} - \frac{192d^{3}fx^{3}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} + \frac{96d^{3}x^{2}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{144d^{2}f^{2}x^{2}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} + \frac{96d^{2}fx}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{48df^{3}x}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}} + \frac{24df^{2}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{3}} - \frac{12d^{2}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{2}} - \frac{6f^{4}}{(ax^{5} + bx^{4} + cx^{3} + dx^{2} + fx + g)^{4}}\\ \end{split}\end{equation} \]

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