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 N 求 4 阶导数。
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数{(1 + \frac{x}{N})}^{N} - e^{N} 关于 N 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = (\frac{x}{N} + 1)^{N} - e^{N}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( (\frac{x}{N} + 1)^{N} - e^{N}\right)}{dN}\\=&((\frac{x}{N} + 1)^{N}((1)ln(\frac{x}{N} + 1) + \frac{(N)(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)})) - e^{N}\\=&(\frac{x}{N} + 1)^{N}ln(\frac{x}{N} + 1) - \frac{x(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)N} - e^{N}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( (\frac{x}{N} + 1)^{N}ln(\frac{x}{N} + 1) - \frac{x(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)N} - e^{N}\right)}{dN}\\=&((\frac{x}{N} + 1)^{N}((1)ln(\frac{x}{N} + 1) + \frac{(N)(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)}))ln(\frac{x}{N} + 1) + \frac{(\frac{x}{N} + 1)^{N}(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)} - \frac{(\frac{-(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)^{2}})x(\frac{x}{N} + 1)^{N}}{N} - \frac{x*-(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)N^{2}} - \frac{x((\frac{x}{N} + 1)^{N}((1)ln(\frac{x}{N} + 1) + \frac{(N)(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)}))}{(\frac{x}{N} + 1)N} - e^{N}\\=&(\frac{x}{N} + 1)^{N}ln^{2}(\frac{x}{N} + 1) - \frac{2x(\frac{x}{N} + 1)^{N}ln(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)N} - \frac{x^{2}(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{2}N^{3}} + \frac{x^{2}(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{2}N^{2}} - e^{N}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( (\frac{x}{N} + 1)^{N}ln^{2}(\frac{x}{N} + 1) - \frac{2x(\frac{x}{N} + 1)^{N}ln(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)N} - \frac{x^{2}(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{2}N^{3}} + \frac{x^{2}(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{2}N^{2}} - e^{N}\right)}{dN}\\=&((\frac{x}{N} + 1)^{N}((1)ln(\frac{x}{N} + 1) + \frac{(N)(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)}))ln^{2}(\frac{x}{N} + 1) + \frac{(\frac{x}{N} + 1)^{N}*2ln(\frac{x}{N} + 1)(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)} - \frac{2(\frac{-(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)^{2}})x(\frac{x}{N} + 1)^{N}ln(\frac{x}{N} + 1)}{N} - \frac{2x*-(\frac{x}{N} + 1)^{N}ln(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)N^{2}} - \frac{2x((\frac{x}{N} + 1)^{N}((1)ln(\frac{x}{N} + 1) + \frac{(N)(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)}))ln(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)N} - \frac{2x(\frac{x}{N} + 1)^{N}(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)N(\frac{x}{N} + 1)} - \frac{(\frac{-2(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)^{3}})x^{2}(\frac{x}{N} + 1)^{N}}{N^{3}} - \frac{x^{2}*-3(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{2}N^{4}} - \frac{x^{2}((\frac{x}{N} + 1)^{N}((1)ln(\frac{x}{N} + 1) + \frac{(N)(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)}))}{(\frac{x}{N} + 1)^{2}N^{3}} + \frac{(\frac{-2(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)^{3}})x^{2}(\frac{x}{N} + 1)^{N}}{N^{2}} + \frac{x^{2}*-2(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{2}N^{3}} + \frac{x^{2}((\frac{x}{N} + 1)^{N}((1)ln(\frac{x}{N} + 1) + \frac{(N)(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)}))}{(\frac{x}{N} + 1)^{2}N^{2}} - e^{N}\\=&(\frac{x}{N} + 1)^{N}ln^{3}(\frac{x}{N} + 1) - \frac{3x(\frac{x}{N} + 1)^{N}ln^{2}(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)N} - \frac{3x^{2}(\frac{x}{N} + 1)^{N}ln(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)^{2}N^{3}} + \frac{3x^{2}(\frac{x}{N} + 1)^{N}ln(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)^{2}N^{2}} - \frac{2x^{3}(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{3}N^{5}} + \frac{3x^{2}(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{2}N^{4}} + \frac{3x^{3}(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{3}N^{4}} - \frac{x^{3}(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{3}N^{3}} - e^{N}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( (\frac{x}{N} + 1)^{N}ln^{3}(\frac{x}{N} + 1) - \frac{3x(\frac{x}{N} + 1)^{N}ln^{2}(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)N} - \frac{3x^{2}(\frac{x}{N} + 1)^{N}ln(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)^{2}N^{3}} + \frac{3x^{2}(\frac{x}{N} + 1)^{N}ln(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)^{2}N^{2}} - \frac{2x^{3}(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{3}N^{5}} + \frac{3x^{2}(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{2}N^{4}} + \frac{3x^{3}(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{3}N^{4}} - \frac{x^{3}(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{3}N^{3}} - e^{N}\right)}{dN}\\=&((\frac{x}{N} + 1)^{N}((1)ln(\frac{x}{N} + 1) + \frac{(N)(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)}))ln^{3}(\frac{x}{N} + 1) + \frac{(\frac{x}{N} + 1)^{N}*3ln^{2}(\frac{x}{N} + 1)(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)} - \frac{3(\frac{-(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)^{2}})x(\frac{x}{N} + 1)^{N}ln^{2}(\frac{x}{N} + 1)}{N} - \frac{3x*-(\frac{x}{N} + 1)^{N}ln^{2}(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)N^{2}} - \frac{3x((\frac{x}{N} + 1)^{N}((1)ln(\frac{x}{N} + 1) + \frac{(N)(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)}))ln^{2}(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)N} - \frac{3x(\frac{x}{N} + 1)^{N}*2ln(\frac{x}{N} + 1)(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)N(\frac{x}{N} + 1)} - \frac{3(\frac{-2(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)^{3}})x^{2}(\frac{x}{N} + 1)^{N}ln(\frac{x}{N} + 1)}{N^{3}} - \frac{3x^{2}*-3(\frac{x}{N} + 1)^{N}ln(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)^{2}N^{4}} - \frac{3x^{2}((\frac{x}{N} + 1)^{N}((1)ln(\frac{x}{N} + 1) + \frac{(N)(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)}))ln(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)^{2}N^{3}} - \frac{3x^{2}(\frac{x}{N} + 1)^{N}(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)^{2}N^{3}(\frac{x}{N} + 1)} + \frac{3(\frac{-2(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)^{3}})x^{2}(\frac{x}{N} + 1)^{N}ln(\frac{x}{N} + 1)}{N^{2}} + \frac{3x^{2}*-2(\frac{x}{N} + 1)^{N}ln(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)^{2}N^{3}} + \frac{3x^{2}((\frac{x}{N} + 1)^{N}((1)ln(\frac{x}{N} + 1) + \frac{(N)(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)}))ln(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)^{2}N^{2}} + \frac{3x^{2}(\frac{x}{N} + 1)^{N}(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)^{2}N^{2}(\frac{x}{N} + 1)} - \frac{2(\frac{-3(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)^{4}})x^{3}(\frac{x}{N} + 1)^{N}}{N^{5}} - \frac{2x^{3}*-5(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{3}N^{6}} - \frac{2x^{3}((\frac{x}{N} + 1)^{N}((1)ln(\frac{x}{N} + 1) + \frac{(N)(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)}))}{(\frac{x}{N} + 1)^{3}N^{5}} + \frac{3(\frac{-2(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)^{3}})x^{2}(\frac{x}{N} + 1)^{N}}{N^{4}} + \frac{3x^{2}*-4(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{2}N^{5}} + \frac{3x^{2}((\frac{x}{N} + 1)^{N}((1)ln(\frac{x}{N} + 1) + \frac{(N)(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)}))}{(\frac{x}{N} + 1)^{2}N^{4}} + \frac{3(\frac{-3(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)^{4}})x^{3}(\frac{x}{N} + 1)^{N}}{N^{4}} + \frac{3x^{3}*-4(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{3}N^{5}} + \frac{3x^{3}((\frac{x}{N} + 1)^{N}((1)ln(\frac{x}{N} + 1) + \frac{(N)(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)}))}{(\frac{x}{N} + 1)^{3}N^{4}} - \frac{(\frac{-3(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)^{4}})x^{3}(\frac{x}{N} + 1)^{N}}{N^{3}} - \frac{x^{3}*-3(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{3}N^{4}} - \frac{x^{3}((\frac{x}{N} + 1)^{N}((1)ln(\frac{x}{N} + 1) + \frac{(N)(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)}))}{(\frac{x}{N} + 1)^{3}N^{3}} - e^{N}\\=&(\frac{x}{N} + 1)^{N}ln^{4}(\frac{x}{N} + 1) - \frac{4x(\frac{x}{N} + 1)^{N}ln^{3}(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)N} - \frac{6x^{2}(\frac{x}{N} + 1)^{N}ln^{2}(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)^{2}N^{3}} + \frac{6x^{2}(\frac{x}{N} + 1)^{N}ln^{2}(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)^{2}N^{2}} - \frac{8x^{3}(\frac{x}{N} + 1)^{N}ln(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)^{3}N^{5}} + \frac{12x^{2}(\frac{x}{N} + 1)^{N}ln(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)^{2}N^{4}} + \frac{12x^{3}(\frac{x}{N} + 1)^{N}ln(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)^{3}N^{4}} - \frac{4x^{3}(\frac{x}{N} + 1)^{N}ln(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)^{3}N^{3}} - \frac{12x^{3}(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{3}N^{5}} - \frac{6x^{4}(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{4}N^{7}} + \frac{16x^{3}(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{3}N^{6}} + \frac{11x^{4}(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{4}N^{6}} - \frac{12x^{2}(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{2}N^{5}} - \frac{6x^{4}(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{4}N^{5}} + \frac{x^{4}(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{4}N^{4}} - e^{N}\\ \end{split}\end{equation} \]
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数{(1 + \frac{x}{N})}^{N} - e^{N} 关于 N 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = (\frac{x}{N} + 1)^{N} - e^{N}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( (\frac{x}{N} + 1)^{N} - e^{N}\right)}{dN}\\=&((\frac{x}{N} + 1)^{N}((1)ln(\frac{x}{N} + 1) + \frac{(N)(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)})) - e^{N}\\=&(\frac{x}{N} + 1)^{N}ln(\frac{x}{N} + 1) - \frac{x(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)N} - e^{N}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( (\frac{x}{N} + 1)^{N}ln(\frac{x}{N} + 1) - \frac{x(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)N} - e^{N}\right)}{dN}\\=&((\frac{x}{N} + 1)^{N}((1)ln(\frac{x}{N} + 1) + \frac{(N)(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)}))ln(\frac{x}{N} + 1) + \frac{(\frac{x}{N} + 1)^{N}(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)} - \frac{(\frac{-(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)^{2}})x(\frac{x}{N} + 1)^{N}}{N} - \frac{x*-(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)N^{2}} - \frac{x((\frac{x}{N} + 1)^{N}((1)ln(\frac{x}{N} + 1) + \frac{(N)(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)}))}{(\frac{x}{N} + 1)N} - e^{N}\\=&(\frac{x}{N} + 1)^{N}ln^{2}(\frac{x}{N} + 1) - \frac{2x(\frac{x}{N} + 1)^{N}ln(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)N} - \frac{x^{2}(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{2}N^{3}} + \frac{x^{2}(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{2}N^{2}} - e^{N}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( (\frac{x}{N} + 1)^{N}ln^{2}(\frac{x}{N} + 1) - \frac{2x(\frac{x}{N} + 1)^{N}ln(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)N} - \frac{x^{2}(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{2}N^{3}} + \frac{x^{2}(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{2}N^{2}} - e^{N}\right)}{dN}\\=&((\frac{x}{N} + 1)^{N}((1)ln(\frac{x}{N} + 1) + \frac{(N)(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)}))ln^{2}(\frac{x}{N} + 1) + \frac{(\frac{x}{N} + 1)^{N}*2ln(\frac{x}{N} + 1)(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)} - \frac{2(\frac{-(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)^{2}})x(\frac{x}{N} + 1)^{N}ln(\frac{x}{N} + 1)}{N} - \frac{2x*-(\frac{x}{N} + 1)^{N}ln(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)N^{2}} - \frac{2x((\frac{x}{N} + 1)^{N}((1)ln(\frac{x}{N} + 1) + \frac{(N)(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)}))ln(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)N} - \frac{2x(\frac{x}{N} + 1)^{N}(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)N(\frac{x}{N} + 1)} - \frac{(\frac{-2(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)^{3}})x^{2}(\frac{x}{N} + 1)^{N}}{N^{3}} - \frac{x^{2}*-3(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{2}N^{4}} - \frac{x^{2}((\frac{x}{N} + 1)^{N}((1)ln(\frac{x}{N} + 1) + \frac{(N)(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)}))}{(\frac{x}{N} + 1)^{2}N^{3}} + \frac{(\frac{-2(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)^{3}})x^{2}(\frac{x}{N} + 1)^{N}}{N^{2}} + \frac{x^{2}*-2(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{2}N^{3}} + \frac{x^{2}((\frac{x}{N} + 1)^{N}((1)ln(\frac{x}{N} + 1) + \frac{(N)(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)}))}{(\frac{x}{N} + 1)^{2}N^{2}} - e^{N}\\=&(\frac{x}{N} + 1)^{N}ln^{3}(\frac{x}{N} + 1) - \frac{3x(\frac{x}{N} + 1)^{N}ln^{2}(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)N} - \frac{3x^{2}(\frac{x}{N} + 1)^{N}ln(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)^{2}N^{3}} + \frac{3x^{2}(\frac{x}{N} + 1)^{N}ln(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)^{2}N^{2}} - \frac{2x^{3}(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{3}N^{5}} + \frac{3x^{2}(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{2}N^{4}} + \frac{3x^{3}(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{3}N^{4}} - \frac{x^{3}(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{3}N^{3}} - e^{N}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( (\frac{x}{N} + 1)^{N}ln^{3}(\frac{x}{N} + 1) - \frac{3x(\frac{x}{N} + 1)^{N}ln^{2}(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)N} - \frac{3x^{2}(\frac{x}{N} + 1)^{N}ln(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)^{2}N^{3}} + \frac{3x^{2}(\frac{x}{N} + 1)^{N}ln(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)^{2}N^{2}} - \frac{2x^{3}(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{3}N^{5}} + \frac{3x^{2}(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{2}N^{4}} + \frac{3x^{3}(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{3}N^{4}} - \frac{x^{3}(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{3}N^{3}} - e^{N}\right)}{dN}\\=&((\frac{x}{N} + 1)^{N}((1)ln(\frac{x}{N} + 1) + \frac{(N)(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)}))ln^{3}(\frac{x}{N} + 1) + \frac{(\frac{x}{N} + 1)^{N}*3ln^{2}(\frac{x}{N} + 1)(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)} - \frac{3(\frac{-(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)^{2}})x(\frac{x}{N} + 1)^{N}ln^{2}(\frac{x}{N} + 1)}{N} - \frac{3x*-(\frac{x}{N} + 1)^{N}ln^{2}(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)N^{2}} - \frac{3x((\frac{x}{N} + 1)^{N}((1)ln(\frac{x}{N} + 1) + \frac{(N)(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)}))ln^{2}(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)N} - \frac{3x(\frac{x}{N} + 1)^{N}*2ln(\frac{x}{N} + 1)(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)N(\frac{x}{N} + 1)} - \frac{3(\frac{-2(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)^{3}})x^{2}(\frac{x}{N} + 1)^{N}ln(\frac{x}{N} + 1)}{N^{3}} - \frac{3x^{2}*-3(\frac{x}{N} + 1)^{N}ln(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)^{2}N^{4}} - \frac{3x^{2}((\frac{x}{N} + 1)^{N}((1)ln(\frac{x}{N} + 1) + \frac{(N)(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)}))ln(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)^{2}N^{3}} - \frac{3x^{2}(\frac{x}{N} + 1)^{N}(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)^{2}N^{3}(\frac{x}{N} + 1)} + \frac{3(\frac{-2(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)^{3}})x^{2}(\frac{x}{N} + 1)^{N}ln(\frac{x}{N} + 1)}{N^{2}} + \frac{3x^{2}*-2(\frac{x}{N} + 1)^{N}ln(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)^{2}N^{3}} + \frac{3x^{2}((\frac{x}{N} + 1)^{N}((1)ln(\frac{x}{N} + 1) + \frac{(N)(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)}))ln(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)^{2}N^{2}} + \frac{3x^{2}(\frac{x}{N} + 1)^{N}(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)^{2}N^{2}(\frac{x}{N} + 1)} - \frac{2(\frac{-3(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)^{4}})x^{3}(\frac{x}{N} + 1)^{N}}{N^{5}} - \frac{2x^{3}*-5(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{3}N^{6}} - \frac{2x^{3}((\frac{x}{N} + 1)^{N}((1)ln(\frac{x}{N} + 1) + \frac{(N)(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)}))}{(\frac{x}{N} + 1)^{3}N^{5}} + \frac{3(\frac{-2(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)^{3}})x^{2}(\frac{x}{N} + 1)^{N}}{N^{4}} + \frac{3x^{2}*-4(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{2}N^{5}} + \frac{3x^{2}((\frac{x}{N} + 1)^{N}((1)ln(\frac{x}{N} + 1) + \frac{(N)(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)}))}{(\frac{x}{N} + 1)^{2}N^{4}} + \frac{3(\frac{-3(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)^{4}})x^{3}(\frac{x}{N} + 1)^{N}}{N^{4}} + \frac{3x^{3}*-4(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{3}N^{5}} + \frac{3x^{3}((\frac{x}{N} + 1)^{N}((1)ln(\frac{x}{N} + 1) + \frac{(N)(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)}))}{(\frac{x}{N} + 1)^{3}N^{4}} - \frac{(\frac{-3(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)^{4}})x^{3}(\frac{x}{N} + 1)^{N}}{N^{3}} - \frac{x^{3}*-3(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{3}N^{4}} - \frac{x^{3}((\frac{x}{N} + 1)^{N}((1)ln(\frac{x}{N} + 1) + \frac{(N)(\frac{x*-1}{N^{2}} + 0)}{(\frac{x}{N} + 1)}))}{(\frac{x}{N} + 1)^{3}N^{3}} - e^{N}\\=&(\frac{x}{N} + 1)^{N}ln^{4}(\frac{x}{N} + 1) - \frac{4x(\frac{x}{N} + 1)^{N}ln^{3}(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)N} - \frac{6x^{2}(\frac{x}{N} + 1)^{N}ln^{2}(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)^{2}N^{3}} + \frac{6x^{2}(\frac{x}{N} + 1)^{N}ln^{2}(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)^{2}N^{2}} - \frac{8x^{3}(\frac{x}{N} + 1)^{N}ln(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)^{3}N^{5}} + \frac{12x^{2}(\frac{x}{N} + 1)^{N}ln(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)^{2}N^{4}} + \frac{12x^{3}(\frac{x}{N} + 1)^{N}ln(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)^{3}N^{4}} - \frac{4x^{3}(\frac{x}{N} + 1)^{N}ln(\frac{x}{N} + 1)}{(\frac{x}{N} + 1)^{3}N^{3}} - \frac{12x^{3}(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{3}N^{5}} - \frac{6x^{4}(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{4}N^{7}} + \frac{16x^{3}(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{3}N^{6}} + \frac{11x^{4}(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{4}N^{6}} - \frac{12x^{2}(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{2}N^{5}} - \frac{6x^{4}(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{4}N^{5}} + \frac{x^{4}(\frac{x}{N} + 1)^{N}}{(\frac{x}{N} + 1)^{4}N^{4}} - e^{N}\\ \end{split}\end{equation} \]
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