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 3 derivative of x is calculated.
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ third\ derivative\ of\ function\ ln(\frac{(1 - x)}{(1 + x)})\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = ln(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( ln(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})\right)}{dx}\\=&\frac{(-(\frac{-(1 + 0)}{(x + 1)^{2}})x - \frac{1}{(x + 1)} + (\frac{-(1 + 0)}{(x + 1)^{2}}))}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})}\\=&\frac{x}{(x + 1)^{2}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})} - \frac{1}{(x + 1)^{2}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})} - \frac{1}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})(x + 1)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{x}{(x + 1)^{2}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})} - \frac{1}{(x + 1)^{2}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})} - \frac{1}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})(x + 1)}\right)}{dx}\\=&\frac{(\frac{-2(1 + 0)}{(x + 1)^{3}})x}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})} + \frac{(\frac{-(-(\frac{-(1 + 0)}{(x + 1)^{2}})x - \frac{1}{(x + 1)} + (\frac{-(1 + 0)}{(x + 1)^{2}}))}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{2}})x}{(x + 1)^{2}} + \frac{1}{(x + 1)^{2}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})} - \frac{(\frac{-2(1 + 0)}{(x + 1)^{3}})}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})} - \frac{(\frac{-(-(\frac{-(1 + 0)}{(x + 1)^{2}})x - \frac{1}{(x + 1)} + (\frac{-(1 + 0)}{(x + 1)^{2}}))}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{2}})}{(x + 1)^{2}} - \frac{(\frac{-(-(\frac{-(1 + 0)}{(x + 1)^{2}})x - \frac{1}{(x + 1)} + (\frac{-(1 + 0)}{(x + 1)^{2}}))}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{2}})}{(x + 1)} - \frac{(\frac{-(1 + 0)}{(x + 1)^{2}})}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})}\\=&\frac{-2x}{(x + 1)^{3}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})} - \frac{x^{2}}{(x + 1)^{4}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{2}} + \frac{x}{(x + 1)^{3}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{2}} + \frac{2x}{(x + 1)^{4}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{2}} + \frac{x}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{2}(x + 1)^{3}} + \frac{2}{(x + 1)^{2}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})} - \frac{1}{(x + 1)^{4}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{2}} + \frac{2}{(x + 1)^{3}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})} - \frac{1}{(x + 1)^{3}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{2}} - \frac{1}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{2}(x + 1)^{3}} - \frac{1}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{2}(x + 1)^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2x}{(x + 1)^{3}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})} - \frac{x^{2}}{(x + 1)^{4}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{2}} + \frac{x}{(x + 1)^{3}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{2}} + \frac{2x}{(x + 1)^{4}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{2}} + \frac{x}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{2}(x + 1)^{3}} + \frac{2}{(x + 1)^{2}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})} - \frac{1}{(x + 1)^{4}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{2}} + \frac{2}{(x + 1)^{3}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})} - \frac{1}{(x + 1)^{3}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{2}} - \frac{1}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{2}(x + 1)^{3}} - \frac{1}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{2}(x + 1)^{2}}\right)}{dx}\\=&\frac{-2(\frac{-3(1 + 0)}{(x + 1)^{4}})x}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})} - \frac{2(\frac{-(-(\frac{-(1 + 0)}{(x + 1)^{2}})x - \frac{1}{(x + 1)} + (\frac{-(1 + 0)}{(x + 1)^{2}}))}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{2}})x}{(x + 1)^{3}} - \frac{2}{(x + 1)^{3}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})} - \frac{(\frac{-4(1 + 0)}{(x + 1)^{5}})x^{2}}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{2}} - \frac{(\frac{-2(-(\frac{-(1 + 0)}{(x + 1)^{2}})x - \frac{1}{(x + 1)} + (\frac{-(1 + 0)}{(x + 1)^{2}}))}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{3}})x^{2}}{(x + 1)^{4}} - \frac{2x}{(x + 1)^{4}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{2}} + \frac{(\frac{-3(1 + 0)}{(x + 1)^{4}})x}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{2}} + \frac{(\frac{-2(-(\frac{-(1 + 0)}{(x + 1)^{2}})x - \frac{1}{(x + 1)} + (\frac{-(1 + 0)}{(x + 1)^{2}}))}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{3}})x}{(x + 1)^{3}} + \frac{1}{(x + 1)^{3}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{2}} + \frac{2(\frac{-4(1 + 0)}{(x + 1)^{5}})x}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{2}} + \frac{2(\frac{-2(-(\frac{-(1 + 0)}{(x + 1)^{2}})x - \frac{1}{(x + 1)} + (\frac{-(1 + 0)}{(x + 1)^{2}}))}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{3}})x}{(x + 1)^{4}} + \frac{2}{(x + 1)^{4}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{2}} + \frac{(\frac{-2(-(\frac{-(1 + 0)}{(x + 1)^{2}})x - \frac{1}{(x + 1)} + (\frac{-(1 + 0)}{(x + 1)^{2}}))}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{3}})x}{(x + 1)^{3}} + \frac{(\frac{-3(1 + 0)}{(x + 1)^{4}})x}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{2}} + \frac{1}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{2}(x + 1)^{3}} + \frac{2(\frac{-2(1 + 0)}{(x + 1)^{3}})}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})} + \frac{2(\frac{-(-(\frac{-(1 + 0)}{(x + 1)^{2}})x - \frac{1}{(x + 1)} + (\frac{-(1 + 0)}{(x + 1)^{2}}))}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{2}})}{(x + 1)^{2}} - \frac{(\frac{-4(1 + 0)}{(x + 1)^{5}})}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{2}} - \frac{(\frac{-2(-(\frac{-(1 + 0)}{(x + 1)^{2}})x - \frac{1}{(x + 1)} + (\frac{-(1 + 0)}{(x + 1)^{2}}))}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{3}})}{(x + 1)^{4}} + \frac{2(\frac{-3(1 + 0)}{(x + 1)^{4}})}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})} + \frac{2(\frac{-(-(\frac{-(1 + 0)}{(x + 1)^{2}})x - \frac{1}{(x + 1)} + (\frac{-(1 + 0)}{(x + 1)^{2}}))}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{2}})}{(x + 1)^{3}} - \frac{(\frac{-3(1 + 0)}{(x + 1)^{4}})}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{2}} - \frac{(\frac{-2(-(\frac{-(1 + 0)}{(x + 1)^{2}})x - \frac{1}{(x + 1)} + (\frac{-(1 + 0)}{(x + 1)^{2}}))}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{3}})}{(x + 1)^{3}} - \frac{(\frac{-2(-(\frac{-(1 + 0)}{(x + 1)^{2}})x - \frac{1}{(x + 1)} + (\frac{-(1 + 0)}{(x + 1)^{2}}))}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{3}})}{(x + 1)^{3}} - \frac{(\frac{-3(1 + 0)}{(x + 1)^{4}})}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{2}} - \frac{(\frac{-2(-(\frac{-(1 + 0)}{(x + 1)^{2}})x - \frac{1}{(x + 1)} + (\frac{-(1 + 0)}{(x + 1)^{2}}))}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{3}})}{(x + 1)^{2}} - \frac{(\frac{-2(1 + 0)}{(x + 1)^{3}})}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{2}}\\=&\frac{6x}{(x + 1)^{4}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})} + \frac{6x^{2}}{(x + 1)^{5}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{2}} - \frac{10x}{(x + 1)^{4}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{2}} - \frac{12x}{(x + 1)^{5}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{2}} - \frac{2x}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{2}(x + 1)^{4}} + \frac{2x^{3}}{(x + 1)^{6}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{3}} - \frac{4x^{2}}{(x + 1)^{5}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{3}} - \frac{6x^{2}}{(x + 1)^{6}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{3}} - \frac{2x^{2}}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{3}(x + 1)^{5}} + \frac{2x}{(x + 1)^{4}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{3}} + \frac{8x}{(x + 1)^{5}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{3}} + \frac{6x}{(x + 1)^{6}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{3}} + \frac{4x}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{3}(x + 1)^{4}} + \frac{4x}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{3}(x + 1)^{5}} + \frac{1}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{2}(x + 1)^{3}} + \frac{12}{(x + 1)^{4}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{2}} + \frac{6}{(x + 1)^{5}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{2}} + \frac{5}{(x + 1)^{3}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{2}} - \frac{2}{(x + 1)^{6}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{3}} - \frac{6}{(x + 1)^{4}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})} - \frac{6}{(x + 1)^{3}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})} - \frac{4}{(x + 1)^{5}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{3}} - \frac{2}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{3}(x + 1)^{5}} - \frac{4}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{3}(x + 1)^{4}} - \frac{2}{(x + 1)^{4}(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{3}} - \frac{2}{(\frac{-x}{(x + 1)} + \frac{1}{(x + 1)})^{3}(x + 1)^{3}}\\ \end{split}\end{equation} \]