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 2 derivative of m is calculated.
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ second\ derivative\ of\ function\ \frac{({k}^{2} - 2rkm - 2(1 - r)k + r{m}^{2})}{({k}^{2} - 2rkm - (1 - r)k + r{m}^{2})}\ with\ respect\ to\ m：\\\end{split}\end{equation} \]\[ \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})}\\&\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})}\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})}\\=&\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})}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \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})}\right)}{dm}\\=&-4(\frac{-2(-2kr + 0 + 0 + 0 + r*2m)}{(-2krm + kr - k + k^{2} + rm^{2})^{3}})k^{2}r^{2}m - \frac{4k^{2}r^{2}}{(-2krm + kr - k + k^{2} + rm^{2})^{2}} + 6(\frac{-2(-2kr + 0 + 0 + 0 + r*2m)}{(-2krm + kr - k + k^{2} + rm^{2})^{3}})kr^{2}m^{2} + \frac{6kr^{2}*2m}{(-2krm + kr - k + k^{2} + rm^{2})^{2}} - 4(\frac{-2(-2kr + 0 + 0 + 0 + r*2m)}{(-2krm + kr - k + k^{2} + rm^{2})^{3}})kr^{2}m - \frac{4kr^{2}}{(-2krm + kr - k + k^{2} + rm^{2})^{2}} + 4(\frac{-2(-2kr + 0 + 0 + 0 + r*2m)}{(-2krm + kr - k + k^{2} + rm^{2})^{3}})krm + \frac{4kr}{(-2krm + kr - k + k^{2} + rm^{2})^{2}} - 2(\frac{-2(-2kr + 0 + 0 + 0 + r*2m)}{(-2krm + kr - k + k^{2} + rm^{2})^{3}})k^{2}rm - \frac{2k^{2}r}{(-2krm + kr - k + k^{2} + rm^{2})^{2}} - 4(\frac{-2(-2kr + 0 + 0 + 0 + r*2m)}{(-2krm + kr - k + k^{2} + rm^{2})^{3}})k^{2}r + 0 + 4(\frac{-2(-2kr + 0 + 0 + 0 + r*2m)}{(-2krm + kr - k + k^{2} + rm^{2})^{3}})k^{2}r^{2} + 0 + 2(\frac{-2(-2kr + 0 + 0 + 0 + r*2m)}{(-2krm + kr - k + k^{2} + rm^{2})^{3}})k^{3}r + 0 - 2(\frac{-(-2kr + 0 + 0 + 0 + r*2m)}{(-2krm + kr - k + k^{2} + rm^{2})^{2}})kr + 0 - 2(\frac{-2(-2kr + 0 + 0 + 0 + r*2m)}{(-2krm + kr - k + k^{2} + rm^{2})^{3}})r^{2}m^{3} - \frac{2r^{2}*3m^{2}}{(-2krm + kr - k + k^{2} + rm^{2})^{2}} + 2(\frac{-(-2kr + 0 + 0 + 0 + r*2m)}{(-2krm + kr - k + k^{2} + rm^{2})^{2}})rm + \frac{2r}{(-2krm + kr - k + k^{2} + rm^{2})}\\=&\frac{-16k^{3}r^{3}m}{(-2krm + kr - k + k^{2} + rm^{2})^{3}} + \frac{40k^{2}r^{3}m^{2}}{(-2krm + kr - k + k^{2} + rm^{2})^{3}} + \frac{20kr^{2}m}{(-2krm + kr - k + k^{2} + rm^{2})^{2}} - \frac{32kr^{3}m^{3}}{(-2krm + kr - k + k^{2} + rm^{2})^{3}} - \frac{32k^{2}r^{3}m}{(-2krm + kr - k + k^{2} + rm^{2})^{3}} + \frac{16kr^{3}m^{2}}{(-2krm + kr - k + k^{2} + rm^{2})^{3}} + \frac{32k^{2}r^{2}m}{(-2krm + kr - k + k^{2} + rm^{2})^{3}} - \frac{16kr^{2}m^{2}}{(-2krm + kr - k + k^{2} + rm^{2})^{3}} - \frac{16k^{3}r^{2}m}{(-2krm + kr - k + k^{2} + rm^{2})^{3}} + \frac{8k^{2}r^{2}m^{2}}{(-2krm + kr - k + k^{2} + rm^{2})^{3}} + \frac{4kr}{(-2krm + kr - k + k^{2} + rm^{2})^{2}} - \frac{8k^{2}r^{2}}{(-2krm + kr - k + k^{2} + rm^{2})^{2}} - \frac{4kr^{2}}{(-2krm + kr - k + k^{2} + rm^{2})^{2}} - \frac{2k^{2}r}{(-2krm + kr - k + k^{2} + rm^{2})^{2}} - \frac{16k^{3}r^{2}}{(-2krm + kr - k + k^{2} + rm^{2})^{3}} + \frac{16k^{3}r^{3}}{(-2krm + kr - k + k^{2} + rm^{2})^{3}} + \frac{8k^{4}r^{2}}{(-2krm + kr - k + k^{2} + rm^{2})^{3}} + \frac{8r^{3}m^{4}}{(-2krm + kr - k + k^{2} + rm^{2})^{3}} - \frac{10r^{2}m^{2}}{(-2krm + kr - k + k^{2} + rm^{2})^{2}} + \frac{2r}{(-2krm + kr - k + k^{2} + rm^{2})}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ second\ derivative\ of\ function\ \frac{({k}^{2} - 2rkm - 2(1 - r)k + r{m}^{2})}{({k}^{2} - 2rkm - (1 - r)k + r{m}^{2})}\ with\ respect\ to\ m：\\\end{split}\end{equation} \]\[ \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})}\\&\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})}\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})}\\=&\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})}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \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})}\right)}{dm}\\=&-4(\frac{-2(-2kr + 0 + 0 + 0 + r*2m)}{(-2krm + kr - k + k^{2} + rm^{2})^{3}})k^{2}r^{2}m - \frac{4k^{2}r^{2}}{(-2krm + kr - k + k^{2} + rm^{2})^{2}} + 6(\frac{-2(-2kr + 0 + 0 + 0 + r*2m)}{(-2krm + kr - k + k^{2} + rm^{2})^{3}})kr^{2}m^{2} + \frac{6kr^{2}*2m}{(-2krm + kr - k + k^{2} + rm^{2})^{2}} - 4(\frac{-2(-2kr + 0 + 0 + 0 + r*2m)}{(-2krm + kr - k + k^{2} + rm^{2})^{3}})kr^{2}m - \frac{4kr^{2}}{(-2krm + kr - k + k^{2} + rm^{2})^{2}} + 4(\frac{-2(-2kr + 0 + 0 + 0 + r*2m)}{(-2krm + kr - k + k^{2} + rm^{2})^{3}})krm + \frac{4kr}{(-2krm + kr - k + k^{2} + rm^{2})^{2}} - 2(\frac{-2(-2kr + 0 + 0 + 0 + r*2m)}{(-2krm + kr - k + k^{2} + rm^{2})^{3}})k^{2}rm - \frac{2k^{2}r}{(-2krm + kr - k + k^{2} + rm^{2})^{2}} - 4(\frac{-2(-2kr + 0 + 0 + 0 + r*2m)}{(-2krm + kr - k + k^{2} + rm^{2})^{3}})k^{2}r + 0 + 4(\frac{-2(-2kr + 0 + 0 + 0 + r*2m)}{(-2krm + kr - k + k^{2} + rm^{2})^{3}})k^{2}r^{2} + 0 + 2(\frac{-2(-2kr + 0 + 0 + 0 + r*2m)}{(-2krm + kr - k + k^{2} + rm^{2})^{3}})k^{3}r + 0 - 2(\frac{-(-2kr + 0 + 0 + 0 + r*2m)}{(-2krm + kr - k + k^{2} + rm^{2})^{2}})kr + 0 - 2(\frac{-2(-2kr + 0 + 0 + 0 + r*2m)}{(-2krm + kr - k + k^{2} + rm^{2})^{3}})r^{2}m^{3} - \frac{2r^{2}*3m^{2}}{(-2krm + kr - k + k^{2} + rm^{2})^{2}} + 2(\frac{-(-2kr + 0 + 0 + 0 + r*2m)}{(-2krm + kr - k + k^{2} + rm^{2})^{2}})rm + \frac{2r}{(-2krm + kr - k + k^{2} + rm^{2})}\\=&\frac{-16k^{3}r^{3}m}{(-2krm + kr - k + k^{2} + rm^{2})^{3}} + \frac{40k^{2}r^{3}m^{2}}{(-2krm + kr - k + k^{2} + rm^{2})^{3}} + \frac{20kr^{2}m}{(-2krm + kr - k + k^{2} + rm^{2})^{2}} - \frac{32kr^{3}m^{3}}{(-2krm + kr - k + k^{2} + rm^{2})^{3}} - \frac{32k^{2}r^{3}m}{(-2krm + kr - k + k^{2} + rm^{2})^{3}} + \frac{16kr^{3}m^{2}}{(-2krm + kr - k + k^{2} + rm^{2})^{3}} + \frac{32k^{2}r^{2}m}{(-2krm + kr - k + k^{2} + rm^{2})^{3}} - \frac{16kr^{2}m^{2}}{(-2krm + kr - k + k^{2} + rm^{2})^{3}} - \frac{16k^{3}r^{2}m}{(-2krm + kr - k + k^{2} + rm^{2})^{3}} + \frac{8k^{2}r^{2}m^{2}}{(-2krm + kr - k + k^{2} + rm^{2})^{3}} + \frac{4kr}{(-2krm + kr - k + k^{2} + rm^{2})^{2}} - \frac{8k^{2}r^{2}}{(-2krm + kr - k + k^{2} + rm^{2})^{2}} - \frac{4kr^{2}}{(-2krm + kr - k + k^{2} + rm^{2})^{2}} - \frac{2k^{2}r}{(-2krm + kr - k + k^{2} + rm^{2})^{2}} - \frac{16k^{3}r^{2}}{(-2krm + kr - k + k^{2} + rm^{2})^{3}} + \frac{16k^{3}r^{3}}{(-2krm + kr - k + k^{2} + rm^{2})^{3}} + \frac{8k^{4}r^{2}}{(-2krm + kr - k + k^{2} + rm^{2})^{3}} + \frac{8r^{3}m^{4}}{(-2krm + kr - k + k^{2} + rm^{2})^{3}} - \frac{10r^{2}m^{2}}{(-2krm + kr - k + k^{2} + rm^{2})^{2}} + \frac{2r}{(-2krm + kr - k + k^{2} + rm^{2})}\\ \end{split}\end{equation} \]