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 s 求 1 阶导数。
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数\frac{(\frac{(\frac{25K(s + \frac{4}{5})}{s})}{(1 + \frac{Kts*25K}{(s(s + \frac{4}{5}))})})}{(1 + (\frac{(\frac{25K(s + \frac{4}{5})}{s})}{(1 + \frac{Kts*25K}{(s(s + \frac{4}{5}))})}))} 关于 s 的 1 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{20K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)(\frac{20K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)s} + \frac{25K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)} + 1)s} + \frac{25K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)(\frac{20K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)s} + \frac{25K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)} + 1)}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( \frac{20K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)(\frac{20K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)s} + \frac{25K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)} + 1)s} + \frac{25K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)(\frac{20K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)s} + \frac{25K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)} + 1)}\right)}{ds}\\=&\frac{20(\frac{-(25(\frac{-(1 + 0)}{(s + \frac{4}{5})^{2}})K^{2}t + 0 + 0)}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)^{2}})K}{(\frac{20K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)s} + \frac{25K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)} + 1)s} + \frac{20(\frac{-(\frac{20(\frac{-(25(\frac{-(1 + 0)}{(s + \frac{4}{5})^{2}})K^{2}t + 0 + 0)}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)^{2}})K}{s} + \frac{20K*-1}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)s^{2}} + 25(\frac{-(25(\frac{-(1 + 0)}{(s + \frac{4}{5})^{2}})K^{2}t + 0 + 0)}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)^{2}})K + 0 + 0)}{(\frac{20K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)s} + \frac{25K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)} + 1)^{2}})K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)s} + \frac{20K*-1}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)(\frac{20K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)s} + \frac{25K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)} + 1)s^{2}} + \frac{25(\frac{-(25(\frac{-(1 + 0)}{(s + \frac{4}{5})^{2}})K^{2}t + 0 + 0)}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)^{2}})K}{(\frac{20K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)s} + \frac{25K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)} + 1)} + \frac{25(\frac{-(\frac{20(\frac{-(25(\frac{-(1 + 0)}{(s + \frac{4}{5})^{2}})K^{2}t + 0 + 0)}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)^{2}})K}{s} + \frac{20K*-1}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)s^{2}} + 25(\frac{-(25(\frac{-(1 + 0)}{(s + \frac{4}{5})^{2}})K^{2}t + 0 + 0)}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)^{2}})K + 0 + 0)}{(\frac{20K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)s} + \frac{25K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)} + 1)^{2}})K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)} + 0\\=&\frac{500K^{3}t}{(s + \frac{4}{5})^{2}(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)^{2}(\frac{20K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)s} + \frac{25K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)} + 1)s} - \frac{10000K^{4}t}{(\frac{20K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)s} + \frac{25K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)} + 1)^{2}(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)^{3}(s + \frac{4}{5})^{2}s^{2}} + \frac{400K^{2}}{(\frac{20K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)s} + \frac{25K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)} + 1)^{2}(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)^{2}s^{3}} - \frac{25000K^{4}t}{(\frac{20K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)s} + \frac{25K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)} + 1)^{2}(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)^{3}(s + \frac{4}{5})^{2}s} - \frac{20K}{(\frac{20K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)s} + \frac{25K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)} + 1)(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)s^{2}} + \frac{625K^{3}t}{(s + \frac{4}{5})^{2}(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)^{2}(\frac{20K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)s} + \frac{25K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)} + 1)} + \frac{500K^{2}}{(\frac{20K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)s} + \frac{25K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)} + 1)^{2}(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)^{2}s^{2}} - \frac{15625K^{4}t}{(\frac{20K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)s} + \frac{25K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)} + 1)^{2}(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)^{3}(s + \frac{4}{5})^{2}}\\ \end{split}\end{equation} \]
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数\frac{(\frac{(\frac{25K(s + \frac{4}{5})}{s})}{(1 + \frac{Kts*25K}{(s(s + \frac{4}{5}))})})}{(1 + (\frac{(\frac{25K(s + \frac{4}{5})}{s})}{(1 + \frac{Kts*25K}{(s(s + \frac{4}{5}))})}))} 关于 s 的 1 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{20K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)(\frac{20K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)s} + \frac{25K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)} + 1)s} + \frac{25K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)(\frac{20K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)s} + \frac{25K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)} + 1)}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( \frac{20K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)(\frac{20K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)s} + \frac{25K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)} + 1)s} + \frac{25K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)(\frac{20K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)s} + \frac{25K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)} + 1)}\right)}{ds}\\=&\frac{20(\frac{-(25(\frac{-(1 + 0)}{(s + \frac{4}{5})^{2}})K^{2}t + 0 + 0)}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)^{2}})K}{(\frac{20K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)s} + \frac{25K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)} + 1)s} + \frac{20(\frac{-(\frac{20(\frac{-(25(\frac{-(1 + 0)}{(s + \frac{4}{5})^{2}})K^{2}t + 0 + 0)}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)^{2}})K}{s} + \frac{20K*-1}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)s^{2}} + 25(\frac{-(25(\frac{-(1 + 0)}{(s + \frac{4}{5})^{2}})K^{2}t + 0 + 0)}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)^{2}})K + 0 + 0)}{(\frac{20K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)s} + \frac{25K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)} + 1)^{2}})K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)s} + \frac{20K*-1}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)(\frac{20K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)s} + \frac{25K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)} + 1)s^{2}} + \frac{25(\frac{-(25(\frac{-(1 + 0)}{(s + \frac{4}{5})^{2}})K^{2}t + 0 + 0)}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)^{2}})K}{(\frac{20K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)s} + \frac{25K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)} + 1)} + \frac{25(\frac{-(\frac{20(\frac{-(25(\frac{-(1 + 0)}{(s + \frac{4}{5})^{2}})K^{2}t + 0 + 0)}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)^{2}})K}{s} + \frac{20K*-1}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)s^{2}} + 25(\frac{-(25(\frac{-(1 + 0)}{(s + \frac{4}{5})^{2}})K^{2}t + 0 + 0)}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)^{2}})K + 0 + 0)}{(\frac{20K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)s} + \frac{25K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)} + 1)^{2}})K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)} + 0\\=&\frac{500K^{3}t}{(s + \frac{4}{5})^{2}(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)^{2}(\frac{20K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)s} + \frac{25K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)} + 1)s} - \frac{10000K^{4}t}{(\frac{20K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)s} + \frac{25K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)} + 1)^{2}(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)^{3}(s + \frac{4}{5})^{2}s^{2}} + \frac{400K^{2}}{(\frac{20K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)s} + \frac{25K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)} + 1)^{2}(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)^{2}s^{3}} - \frac{25000K^{4}t}{(\frac{20K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)s} + \frac{25K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)} + 1)^{2}(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)^{3}(s + \frac{4}{5})^{2}s} - \frac{20K}{(\frac{20K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)s} + \frac{25K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)} + 1)(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)s^{2}} + \frac{625K^{3}t}{(s + \frac{4}{5})^{2}(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)^{2}(\frac{20K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)s} + \frac{25K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)} + 1)} + \frac{500K^{2}}{(\frac{20K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)s} + \frac{25K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)} + 1)^{2}(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)^{2}s^{2}} - \frac{15625K^{4}t}{(\frac{20K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)s} + \frac{25K}{(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)} + 1)^{2}(\frac{25K^{2}t}{(s + \frac{4}{5})} + 1)^{3}(s + \frac{4}{5})^{2}}\\ \end{split}\end{equation} \]
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