Derivative function：
Enter an original function (that is, the function to be derived), then set the variable to be derived and the order of the derivative, and click the "Next" button to obtain the derivative function of the corresponding order of the function.
Note that the input function supports mathematical functions and other constants.
Enter an original function (that is, the function to be derived), then set the variable to be derived and the order of the derivative, and click the "Next" button to obtain the derivative function of the corresponding order of the function.
Note that the input function supports mathematical functions and other constants.
Current location：Derivative function > Derivative function calculation history > Answer
There are 1 questions in this calculation: for each question, the 1 derivative of s is calculated.
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ first\ derivative\ of\ function\ \frac{(\frac{(\frac{25K(s + \frac{4}{5})}{s})}{(1 + \frac{Kts*25K}{(s(s + \frac{4}{5}))})})}{(1 + (\frac{(\frac{25K(s + \frac{4}{5})}{s})}{(1 + \frac{Kts*25K}{(s(s + \frac{4}{5}))})}))}\ with\ respect\ to\ s：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \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}{The\ first\ derivative\ function：}\\&\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} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ first\ derivative\ of\ function\ \frac{(\frac{(\frac{25K(s + \frac{4}{5})}{s})}{(1 + \frac{Kts*25K}{(s(s + \frac{4}{5}))})})}{(1 + (\frac{(\frac{25K(s + \frac{4}{5})}{s})}{(1 + \frac{Kts*25K}{(s(s + \frac{4}{5}))})}))}\ with\ respect\ to\ s：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \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}{The\ first\ derivative\ function：}\\&\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} \]