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 m is calculated.
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ first\ derivative\ of\ function\ \frac{({k}^{2} - 2rkm - 2(1 - r)k + r{m}^{2})}{({k}^{2} - 2rkm - (1 - r)k + r{m}^{2})} + r + \frac{(1 - r)k(k - rm - 2(1 - r))}{(k - rm)(k - rm - (1 - r))} - rm\ with\ respect\ to\ m：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = - \frac{2krm}{(-2krm + kr - k + k^{2} + rm^{2})} + \frac{2kr}{(-2krm + kr - k + k^{2} + rm^{2})} - \frac{2k}{(-2krm + kr - k + k^{2} + rm^{2})} + \frac{k^{2}}{(-2krm + kr - k + k^{2} + rm^{2})} + \frac{rm^{2}}{(-2krm + kr - k + k^{2} + rm^{2})} - rm - \frac{krm}{(k - rm)(k - rm + r - 1)} + \frac{kr^{2}m}{(k - rm)(k - rm + r - 1)} + \frac{4kr}{(k - rm)(k - rm + r - 1)} - \frac{k^{2}r}{(k - rm)(k - rm + r - 1)} - \frac{2kr^{2}}{(k - rm)(k - rm + r - 1)} - \frac{2k}{(k - rm)(k - rm + r - 1)} + \frac{k^{2}}{(k - rm)(k - rm + r - 1)} + r\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( - \frac{2krm}{(-2krm + kr - k + k^{2} + rm^{2})} + \frac{2kr}{(-2krm + kr - k + k^{2} + rm^{2})} - \frac{2k}{(-2krm + kr - k + k^{2} + rm^{2})} + \frac{k^{2}}{(-2krm + kr - k + k^{2} + rm^{2})} + \frac{rm^{2}}{(-2krm + kr - k + k^{2} + rm^{2})} - rm - \frac{krm}{(k - rm)(k - rm + r - 1)} + \frac{kr^{2}m}{(k - rm)(k - rm + r - 1)} + \frac{4kr}{(k - rm)(k - rm + r - 1)} - \frac{k^{2}r}{(k - rm)(k - rm + r - 1)} - \frac{2kr^{2}}{(k - rm)(k - rm + r - 1)} - \frac{2k}{(k - rm)(k - rm + r - 1)} + \frac{k^{2}}{(k - rm)(k - rm + r - 1)} + r\right)}{dm}\\=& - 2(\frac{-(-2kr + 0 + 0 + 0 + r*2m)}{(-2krm + kr - k + k^{2} + rm^{2})^{2}})krm - \frac{2kr}{(-2krm + kr - k + k^{2} + rm^{2})} + 2(\frac{-(-2kr + 0 + 0 + 0 + r*2m)}{(-2krm + kr - k + k^{2} + rm^{2})^{2}})kr + 0 - 2(\frac{-(-2kr + 0 + 0 + 0 + r*2m)}{(-2krm + kr - k + k^{2} + rm^{2})^{2}})k + 0 + (\frac{-(-2kr + 0 + 0 + 0 + r*2m)}{(-2krm + kr - k + k^{2} + rm^{2})^{2}})k^{2} + 0 + (\frac{-(-2kr + 0 + 0 + 0 + r*2m)}{(-2krm + kr - k + k^{2} + rm^{2})^{2}})rm^{2} + \frac{r*2m}{(-2krm + kr - k + k^{2} + rm^{2})} - r - \frac{(\frac{-(0 - r)}{(k - rm)^{2}})krm}{(k - rm + r - 1)} - \frac{(\frac{-(0 - r + 0 + 0)}{(k - rm + r - 1)^{2}})krm}{(k - rm)} - \frac{kr}{(k - rm)(k - rm + r - 1)} + \frac{(\frac{-(0 - r)}{(k - rm)^{2}})kr^{2}m}{(k - rm + r - 1)} + \frac{(\frac{-(0 - r + 0 + 0)}{(k - rm + r - 1)^{2}})kr^{2}m}{(k - rm)} + \frac{kr^{2}}{(k - rm)(k - rm + r - 1)} + \frac{4(\frac{-(0 - r)}{(k - rm)^{2}})kr}{(k - rm + r - 1)} + \frac{4(\frac{-(0 - r + 0 + 0)}{(k - rm + r - 1)^{2}})kr}{(k - rm)} + 0 - \frac{(\frac{-(0 - r)}{(k - rm)^{2}})k^{2}r}{(k - rm + r - 1)} - \frac{(\frac{-(0 - r + 0 + 0)}{(k - rm + r - 1)^{2}})k^{2}r}{(k - rm)} + 0 - \frac{2(\frac{-(0 - r)}{(k - rm)^{2}})kr^{2}}{(k - rm + r - 1)} - \frac{2(\frac{-(0 - r + 0 + 0)}{(k - rm + r - 1)^{2}})kr^{2}}{(k - rm)} + 0 - \frac{2(\frac{-(0 - r)}{(k - rm)^{2}})k}{(k - rm + r - 1)} - \frac{2(\frac{-(0 - r + 0 + 0)}{(k - rm + r - 1)^{2}})k}{(k - rm)} + 0 + \frac{(\frac{-(0 - r)}{(k - rm)^{2}})k^{2}}{(k - rm + r - 1)} + \frac{(\frac{-(0 - r + 0 + 0)}{(k - rm + r - 1)^{2}})k^{2}}{(k - rm)} + 0 + 0\\=&\frac{-4k^{2}r^{2}m}{(-2krm + kr - k + k^{2} + rm^{2})^{2}} + \frac{6kr^{2}m^{2}}{(-2krm + kr - k + k^{2} + rm^{2})^{2}} - \frac{4kr^{2}m}{(-2krm + kr - k + k^{2} + rm^{2})^{2}} + \frac{4krm}{(-2krm + kr - k + k^{2} + rm^{2})^{2}} - \frac{2k^{2}rm}{(-2krm + kr - k + k^{2} + rm^{2})^{2}} - \frac{4k^{2}r}{(-2krm + kr - k + k^{2} + rm^{2})^{2}} + \frac{4k^{2}r^{2}}{(-2krm + kr - k + k^{2} + rm^{2})^{2}} + \frac{2k^{3}r}{(-2krm + kr - k + k^{2} + rm^{2})^{2}} - \frac{2kr}{(-2krm + kr - k + k^{2} + rm^{2})} - \frac{2r^{2}m^{3}}{(-2krm + kr - k + k^{2} + rm^{2})^{2}} + \frac{2rm}{(-2krm + kr - k + k^{2} + rm^{2})} - r - \frac{kr^{2}m}{(k - rm)^{2}(k - rm + r - 1)} - \frac{kr^{2}m}{(k - rm + r - 1)^{2}(k - rm)} + \frac{kr^{3}m}{(k - rm)^{2}(k - rm + r - 1)} + \frac{kr^{3}m}{(k - rm + r - 1)^{2}(k - rm)} - \frac{kr}{(k - rm)(k - rm + r - 1)} + \frac{kr^{2}}{(k - rm)(k - rm + r - 1)} + \frac{4kr^{2}}{(k - rm)^{2}(k - rm + r - 1)} + \frac{4kr^{2}}{(k - rm + r - 1)^{2}(k - rm)} - \frac{k^{2}r^{2}}{(k - rm)^{2}(k - rm + r - 1)} - \frac{k^{2}r^{2}}{(k - rm + r - 1)^{2}(k - rm)} - \frac{2kr^{3}}{(k - rm)^{2}(k - rm + r - 1)} - \frac{2kr^{3}}{(k - rm + r - 1)^{2}(k - rm)} - \frac{2kr}{(k - rm)^{2}(k - rm + r - 1)} - \frac{2kr}{(k - rm + r - 1)^{2}(k - rm)} + \frac{k^{2}r}{(k - rm)^{2}(k - rm + r - 1)} + \frac{k^{2}r}{(k - rm + r - 1)^{2}(k - rm)}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ first\ derivative\ of\ function\ \frac{({k}^{2} - 2rkm - 2(1 - r)k + r{m}^{2})}{({k}^{2} - 2rkm - (1 - r)k + r{m}^{2})} + r + \frac{(1 - r)k(k - rm - 2(1 - r))}{(k - rm)(k - rm - (1 - r))} - rm\ with\ respect\ to\ m：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = - \frac{2krm}{(-2krm + kr - k + k^{2} + rm^{2})} + \frac{2kr}{(-2krm + kr - k + k^{2} + rm^{2})} - \frac{2k}{(-2krm + kr - k + k^{2} + rm^{2})} + \frac{k^{2}}{(-2krm + kr - k + k^{2} + rm^{2})} + \frac{rm^{2}}{(-2krm + kr - k + k^{2} + rm^{2})} - rm - \frac{krm}{(k - rm)(k - rm + r - 1)} + \frac{kr^{2}m}{(k - rm)(k - rm + r - 1)} + \frac{4kr}{(k - rm)(k - rm + r - 1)} - \frac{k^{2}r}{(k - rm)(k - rm + r - 1)} - \frac{2kr^{2}}{(k - rm)(k - rm + r - 1)} - \frac{2k}{(k - rm)(k - rm + r - 1)} + \frac{k^{2}}{(k - rm)(k - rm + r - 1)} + r\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( - \frac{2krm}{(-2krm + kr - k + k^{2} + rm^{2})} + \frac{2kr}{(-2krm + kr - k + k^{2} + rm^{2})} - \frac{2k}{(-2krm + kr - k + k^{2} + rm^{2})} + \frac{k^{2}}{(-2krm + kr - k + k^{2} + rm^{2})} + \frac{rm^{2}}{(-2krm + kr - k + k^{2} + rm^{2})} - rm - \frac{krm}{(k - rm)(k - rm + r - 1)} + \frac{kr^{2}m}{(k - rm)(k - rm + r - 1)} + \frac{4kr}{(k - rm)(k - rm + r - 1)} - \frac{k^{2}r}{(k - rm)(k - rm + r - 1)} - \frac{2kr^{2}}{(k - rm)(k - rm + r - 1)} - \frac{2k}{(k - rm)(k - rm + r - 1)} + \frac{k^{2}}{(k - rm)(k - rm + r - 1)} + r\right)}{dm}\\=& - 2(\frac{-(-2kr + 0 + 0 + 0 + r*2m)}{(-2krm + kr - k + k^{2} + rm^{2})^{2}})krm - \frac{2kr}{(-2krm + kr - k + k^{2} + rm^{2})} + 2(\frac{-(-2kr + 0 + 0 + 0 + r*2m)}{(-2krm + kr - k + k^{2} + rm^{2})^{2}})kr + 0 - 2(\frac{-(-2kr + 0 + 0 + 0 + r*2m)}{(-2krm + kr - k + k^{2} + rm^{2})^{2}})k + 0 + (\frac{-(-2kr + 0 + 0 + 0 + r*2m)}{(-2krm + kr - k + k^{2} + rm^{2})^{2}})k^{2} + 0 + (\frac{-(-2kr + 0 + 0 + 0 + r*2m)}{(-2krm + kr - k + k^{2} + rm^{2})^{2}})rm^{2} + \frac{r*2m}{(-2krm + kr - k + k^{2} + rm^{2})} - r - \frac{(\frac{-(0 - r)}{(k - rm)^{2}})krm}{(k - rm + r - 1)} - \frac{(\frac{-(0 - r + 0 + 0)}{(k - rm + r - 1)^{2}})krm}{(k - rm)} - \frac{kr}{(k - rm)(k - rm + r - 1)} + \frac{(\frac{-(0 - r)}{(k - rm)^{2}})kr^{2}m}{(k - rm + r - 1)} + \frac{(\frac{-(0 - r + 0 + 0)}{(k - rm + r - 1)^{2}})kr^{2}m}{(k - rm)} + \frac{kr^{2}}{(k - rm)(k - rm + r - 1)} + \frac{4(\frac{-(0 - r)}{(k - rm)^{2}})kr}{(k - rm + r - 1)} + \frac{4(\frac{-(0 - r + 0 + 0)}{(k - rm + r - 1)^{2}})kr}{(k - rm)} + 0 - \frac{(\frac{-(0 - r)}{(k - rm)^{2}})k^{2}r}{(k - rm + r - 1)} - \frac{(\frac{-(0 - r + 0 + 0)}{(k - rm + r - 1)^{2}})k^{2}r}{(k - rm)} + 0 - \frac{2(\frac{-(0 - r)}{(k - rm)^{2}})kr^{2}}{(k - rm + r - 1)} - \frac{2(\frac{-(0 - r + 0 + 0)}{(k - rm + r - 1)^{2}})kr^{2}}{(k - rm)} + 0 - \frac{2(\frac{-(0 - r)}{(k - rm)^{2}})k}{(k - rm + r - 1)} - \frac{2(\frac{-(0 - r + 0 + 0)}{(k - rm + r - 1)^{2}})k}{(k - rm)} + 0 + \frac{(\frac{-(0 - r)}{(k - rm)^{2}})k^{2}}{(k - rm + r - 1)} + \frac{(\frac{-(0 - r + 0 + 0)}{(k - rm + r - 1)^{2}})k^{2}}{(k - rm)} + 0 + 0\\=&\frac{-4k^{2}r^{2}m}{(-2krm + kr - k + k^{2} + rm^{2})^{2}} + \frac{6kr^{2}m^{2}}{(-2krm + kr - k + k^{2} + rm^{2})^{2}} - \frac{4kr^{2}m}{(-2krm + kr - k + k^{2} + rm^{2})^{2}} + \frac{4krm}{(-2krm + kr - k + k^{2} + rm^{2})^{2}} - \frac{2k^{2}rm}{(-2krm + kr - k + k^{2} + rm^{2})^{2}} - \frac{4k^{2}r}{(-2krm + kr - k + k^{2} + rm^{2})^{2}} + \frac{4k^{2}r^{2}}{(-2krm + kr - k + k^{2} + rm^{2})^{2}} + \frac{2k^{3}r}{(-2krm + kr - k + k^{2} + rm^{2})^{2}} - \frac{2kr}{(-2krm + kr - k + k^{2} + rm^{2})} - \frac{2r^{2}m^{3}}{(-2krm + kr - k + k^{2} + rm^{2})^{2}} + \frac{2rm}{(-2krm + kr - k + k^{2} + rm^{2})} - r - \frac{kr^{2}m}{(k - rm)^{2}(k - rm + r - 1)} - \frac{kr^{2}m}{(k - rm + r - 1)^{2}(k - rm)} + \frac{kr^{3}m}{(k - rm)^{2}(k - rm + r - 1)} + \frac{kr^{3}m}{(k - rm + r - 1)^{2}(k - rm)} - \frac{kr}{(k - rm)(k - rm + r - 1)} + \frac{kr^{2}}{(k - rm)(k - rm + r - 1)} + \frac{4kr^{2}}{(k - rm)^{2}(k - rm + r - 1)} + \frac{4kr^{2}}{(k - rm + r - 1)^{2}(k - rm)} - \frac{k^{2}r^{2}}{(k - rm)^{2}(k - rm + r - 1)} - \frac{k^{2}r^{2}}{(k - rm + r - 1)^{2}(k - rm)} - \frac{2kr^{3}}{(k - rm)^{2}(k - rm + r - 1)} - \frac{2kr^{3}}{(k - rm + r - 1)^{2}(k - rm)} - \frac{2kr}{(k - rm)^{2}(k - rm + r - 1)} - \frac{2kr}{(k - rm + r - 1)^{2}(k - rm)} + \frac{k^{2}r}{(k - rm)^{2}(k - rm + r - 1)} + \frac{k^{2}r}{(k - rm + r - 1)^{2}(k - rm)}\\ \end{split}\end{equation} \]