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 4 derivative of x is calculated.
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ {(1 + \frac{1}{(2x)})}^{{x}^{\frac{1}{2}}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = (\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( (\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}\right)}{dx}\\=&((\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}((\frac{\frac{1}{2}}{x^{\frac{1}{2}}})ln(\frac{\frac{1}{2}}{x} + 1) + \frac{(x^{\frac{1}{2}})(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{(\frac{\frac{1}{2}}{x} + 1)}))\\=&\frac{(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln(\frac{\frac{1}{2}}{x} + 1)}{2x^{\frac{1}{2}}} - \frac{(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{2(\frac{\frac{1}{2}}{x} + 1)x^{\frac{3}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln(\frac{\frac{1}{2}}{x} + 1)}{2x^{\frac{1}{2}}} - \frac{(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{2(\frac{\frac{1}{2}}{x} + 1)x^{\frac{3}{2}}}\right)}{dx}\\=&\frac{\frac{-1}{2}(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln(\frac{\frac{1}{2}}{x} + 1)}{2x^{\frac{3}{2}}} + \frac{((\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}((\frac{\frac{1}{2}}{x^{\frac{1}{2}}})ln(\frac{\frac{1}{2}}{x} + 1) + \frac{(x^{\frac{1}{2}})(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{(\frac{\frac{1}{2}}{x} + 1)}))ln(\frac{\frac{1}{2}}{x} + 1)}{2x^{\frac{1}{2}}} + \frac{(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{2x^{\frac{1}{2}}(\frac{\frac{1}{2}}{x} + 1)} - \frac{(\frac{-(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{(\frac{\frac{1}{2}}{x} + 1)^{2}})(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{2x^{\frac{3}{2}}} - \frac{\frac{-3}{2}(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{2(\frac{\frac{1}{2}}{x} + 1)x^{\frac{5}{2}}} - \frac{((\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}((\frac{\frac{1}{2}}{x^{\frac{1}{2}}})ln(\frac{\frac{1}{2}}{x} + 1) + \frac{(x^{\frac{1}{2}})(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{(\frac{\frac{1}{2}}{x} + 1)}))}{2(\frac{\frac{1}{2}}{x} + 1)x^{\frac{3}{2}}}\\=&\frac{-(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln(\frac{\frac{1}{2}}{x} + 1)}{4x^{\frac{3}{2}}} + \frac{(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln^{2}(\frac{\frac{1}{2}}{x} + 1)}{4x} - \frac{(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln(\frac{\frac{1}{2}}{x} + 1)}{2(\frac{\frac{1}{2}}{x} + 1)x^{2}} + \frac{(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{2(\frac{\frac{1}{2}}{x} + 1)x^{\frac{5}{2}}} - \frac{(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{4(\frac{\frac{1}{2}}{x} + 1)^{2}x^{\frac{7}{2}}} + \frac{(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{4(\frac{\frac{1}{2}}{x} + 1)^{2}x^{3}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln(\frac{\frac{1}{2}}{x} + 1)}{4x^{\frac{3}{2}}} + \frac{(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln^{2}(\frac{\frac{1}{2}}{x} + 1)}{4x} - \frac{(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln(\frac{\frac{1}{2}}{x} + 1)}{2(\frac{\frac{1}{2}}{x} + 1)x^{2}} + \frac{(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{2(\frac{\frac{1}{2}}{x} + 1)x^{\frac{5}{2}}} - \frac{(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{4(\frac{\frac{1}{2}}{x} + 1)^{2}x^{\frac{7}{2}}} + \frac{(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{4(\frac{\frac{1}{2}}{x} + 1)^{2}x^{3}}\right)}{dx}\\=&\frac{-\frac{-3}{2}(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln(\frac{\frac{1}{2}}{x} + 1)}{4x^{\frac{5}{2}}} - \frac{((\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}((\frac{\frac{1}{2}}{x^{\frac{1}{2}}})ln(\frac{\frac{1}{2}}{x} + 1) + \frac{(x^{\frac{1}{2}})(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{(\frac{\frac{1}{2}}{x} + 1)}))ln(\frac{\frac{1}{2}}{x} + 1)}{4x^{\frac{3}{2}}} - \frac{(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{4x^{\frac{3}{2}}(\frac{\frac{1}{2}}{x} + 1)} + \frac{-(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln^{2}(\frac{\frac{1}{2}}{x} + 1)}{4x^{2}} + \frac{((\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}((\frac{\frac{1}{2}}{x^{\frac{1}{2}}})ln(\frac{\frac{1}{2}}{x} + 1) + \frac{(x^{\frac{1}{2}})(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{(\frac{\frac{1}{2}}{x} + 1)}))ln^{2}(\frac{\frac{1}{2}}{x} + 1)}{4x} + \frac{(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}*2ln(\frac{\frac{1}{2}}{x} + 1)(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{4x(\frac{\frac{1}{2}}{x} + 1)} - \frac{(\frac{-(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{(\frac{\frac{1}{2}}{x} + 1)^{2}})(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln(\frac{\frac{1}{2}}{x} + 1)}{2x^{2}} - \frac{-2(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln(\frac{\frac{1}{2}}{x} + 1)}{2(\frac{\frac{1}{2}}{x} + 1)x^{3}} - \frac{((\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}((\frac{\frac{1}{2}}{x^{\frac{1}{2}}})ln(\frac{\frac{1}{2}}{x} + 1) + \frac{(x^{\frac{1}{2}})(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{(\frac{\frac{1}{2}}{x} + 1)}))ln(\frac{\frac{1}{2}}{x} + 1)}{2(\frac{\frac{1}{2}}{x} + 1)x^{2}} - \frac{(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{2(\frac{\frac{1}{2}}{x} + 1)x^{2}(\frac{\frac{1}{2}}{x} + 1)} + \frac{(\frac{-(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{(\frac{\frac{1}{2}}{x} + 1)^{2}})(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{2x^{\frac{5}{2}}} + \frac{\frac{-5}{2}(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{2(\frac{\frac{1}{2}}{x} + 1)x^{\frac{7}{2}}} + \frac{((\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}((\frac{\frac{1}{2}}{x^{\frac{1}{2}}})ln(\frac{\frac{1}{2}}{x} + 1) + \frac{(x^{\frac{1}{2}})(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{(\frac{\frac{1}{2}}{x} + 1)}))}{2(\frac{\frac{1}{2}}{x} + 1)x^{\frac{5}{2}}} - \frac{(\frac{-2(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{(\frac{\frac{1}{2}}{x} + 1)^{3}})(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{4x^{\frac{7}{2}}} - \frac{\frac{-7}{2}(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{4(\frac{\frac{1}{2}}{x} + 1)^{2}x^{\frac{9}{2}}} - \frac{((\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}((\frac{\frac{1}{2}}{x^{\frac{1}{2}}})ln(\frac{\frac{1}{2}}{x} + 1) + \frac{(x^{\frac{1}{2}})(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{(\frac{\frac{1}{2}}{x} + 1)}))}{4(\frac{\frac{1}{2}}{x} + 1)^{2}x^{\frac{7}{2}}} + \frac{(\frac{-2(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{(\frac{\frac{1}{2}}{x} + 1)^{3}})(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{4x^{3}} + \frac{-3(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{4(\frac{\frac{1}{2}}{x} + 1)^{2}x^{4}} + \frac{((\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}((\frac{\frac{1}{2}}{x^{\frac{1}{2}}})ln(\frac{\frac{1}{2}}{x} + 1) + \frac{(x^{\frac{1}{2}})(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{(\frac{\frac{1}{2}}{x} + 1)}))}{4(\frac{\frac{1}{2}}{x} + 1)^{2}x^{3}}\\=&\frac{3(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln(\frac{\frac{1}{2}}{x} + 1)}{8x^{\frac{5}{2}}} - \frac{3(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln^{2}(\frac{\frac{1}{2}}{x} + 1)}{8x^{2}} - \frac{3(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln^{2}(\frac{\frac{1}{2}}{x} + 1)}{8(\frac{\frac{1}{2}}{x} + 1)x^{\frac{5}{2}}} + \frac{9(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln(\frac{\frac{1}{2}}{x} + 1)}{8(\frac{\frac{1}{2}}{x} + 1)x^{3}} + \frac{(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln^{3}(\frac{\frac{1}{2}}{x} + 1)}{8x^{\frac{3}{2}}} - \frac{3(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln(\frac{\frac{1}{2}}{x} + 1)}{8(\frac{\frac{1}{2}}{x} + 1)^{2}x^{4}} + \frac{3(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln(\frac{\frac{1}{2}}{x} + 1)}{8(\frac{\frac{1}{2}}{x} + 1)^{2}x^{\frac{7}{2}}} - \frac{9(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{8(\frac{\frac{1}{2}}{x} + 1)x^{\frac{7}{2}}} - \frac{3(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{4(\frac{\frac{1}{2}}{x} + 1)^{2}x^{4}} + \frac{9(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{8(\frac{\frac{1}{2}}{x} + 1)^{2}x^{\frac{9}{2}}} - \frac{(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{4(\frac{\frac{1}{2}}{x} + 1)^{3}x^{\frac{11}{2}}} + \frac{3(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{8(\frac{\frac{1}{2}}{x} + 1)^{3}x^{5}} - \frac{(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{8(\frac{\frac{1}{2}}{x} + 1)^{3}x^{\frac{9}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{3(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln(\frac{\frac{1}{2}}{x} + 1)}{8x^{\frac{5}{2}}} - \frac{3(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln^{2}(\frac{\frac{1}{2}}{x} + 1)}{8x^{2}} - \frac{3(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln^{2}(\frac{\frac{1}{2}}{x} + 1)}{8(\frac{\frac{1}{2}}{x} + 1)x^{\frac{5}{2}}} + \frac{9(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln(\frac{\frac{1}{2}}{x} + 1)}{8(\frac{\frac{1}{2}}{x} + 1)x^{3}} + \frac{(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln^{3}(\frac{\frac{1}{2}}{x} + 1)}{8x^{\frac{3}{2}}} - \frac{3(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln(\frac{\frac{1}{2}}{x} + 1)}{8(\frac{\frac{1}{2}}{x} + 1)^{2}x^{4}} + \frac{3(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln(\frac{\frac{1}{2}}{x} + 1)}{8(\frac{\frac{1}{2}}{x} + 1)^{2}x^{\frac{7}{2}}} - \frac{9(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{8(\frac{\frac{1}{2}}{x} + 1)x^{\frac{7}{2}}} - \frac{3(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{4(\frac{\frac{1}{2}}{x} + 1)^{2}x^{4}} + \frac{9(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{8(\frac{\frac{1}{2}}{x} + 1)^{2}x^{\frac{9}{2}}} - \frac{(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{4(\frac{\frac{1}{2}}{x} + 1)^{3}x^{\frac{11}{2}}} + \frac{3(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{8(\frac{\frac{1}{2}}{x} + 1)^{3}x^{5}} - \frac{(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{8(\frac{\frac{1}{2}}{x} + 1)^{3}x^{\frac{9}{2}}}\right)}{dx}\\=&\frac{3*\frac{-5}{2}(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln(\frac{\frac{1}{2}}{x} + 1)}{8x^{\frac{7}{2}}} + \frac{3((\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}((\frac{\frac{1}{2}}{x^{\frac{1}{2}}})ln(\frac{\frac{1}{2}}{x} + 1) + \frac{(x^{\frac{1}{2}})(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{(\frac{\frac{1}{2}}{x} + 1)}))ln(\frac{\frac{1}{2}}{x} + 1)}{8x^{\frac{5}{2}}} + \frac{3(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{8x^{\frac{5}{2}}(\frac{\frac{1}{2}}{x} + 1)} - \frac{3*-2(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln^{2}(\frac{\frac{1}{2}}{x} + 1)}{8x^{3}} - \frac{3((\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}((\frac{\frac{1}{2}}{x^{\frac{1}{2}}})ln(\frac{\frac{1}{2}}{x} + 1) + \frac{(x^{\frac{1}{2}})(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{(\frac{\frac{1}{2}}{x} + 1)}))ln^{2}(\frac{\frac{1}{2}}{x} + 1)}{8x^{2}} - \frac{3(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}*2ln(\frac{\frac{1}{2}}{x} + 1)(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{8x^{2}(\frac{\frac{1}{2}}{x} + 1)} - \frac{3(\frac{-(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{(\frac{\frac{1}{2}}{x} + 1)^{2}})(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln^{2}(\frac{\frac{1}{2}}{x} + 1)}{8x^{\frac{5}{2}}} - \frac{3*\frac{-5}{2}(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln^{2}(\frac{\frac{1}{2}}{x} + 1)}{8(\frac{\frac{1}{2}}{x} + 1)x^{\frac{7}{2}}} - \frac{3((\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}((\frac{\frac{1}{2}}{x^{\frac{1}{2}}})ln(\frac{\frac{1}{2}}{x} + 1) + \frac{(x^{\frac{1}{2}})(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{(\frac{\frac{1}{2}}{x} + 1)}))ln^{2}(\frac{\frac{1}{2}}{x} + 1)}{8(\frac{\frac{1}{2}}{x} + 1)x^{\frac{5}{2}}} - \frac{3(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}*2ln(\frac{\frac{1}{2}}{x} + 1)(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{8(\frac{\frac{1}{2}}{x} + 1)x^{\frac{5}{2}}(\frac{\frac{1}{2}}{x} + 1)} + \frac{9(\frac{-(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{(\frac{\frac{1}{2}}{x} + 1)^{2}})(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln(\frac{\frac{1}{2}}{x} + 1)}{8x^{3}} + \frac{9*-3(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln(\frac{\frac{1}{2}}{x} + 1)}{8(\frac{\frac{1}{2}}{x} + 1)x^{4}} + \frac{9((\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}((\frac{\frac{1}{2}}{x^{\frac{1}{2}}})ln(\frac{\frac{1}{2}}{x} + 1) + \frac{(x^{\frac{1}{2}})(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{(\frac{\frac{1}{2}}{x} + 1)}))ln(\frac{\frac{1}{2}}{x} + 1)}{8(\frac{\frac{1}{2}}{x} + 1)x^{3}} + \frac{9(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{8(\frac{\frac{1}{2}}{x} + 1)x^{3}(\frac{\frac{1}{2}}{x} + 1)} + \frac{\frac{-3}{2}(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln^{3}(\frac{\frac{1}{2}}{x} + 1)}{8x^{\frac{5}{2}}} + \frac{((\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}((\frac{\frac{1}{2}}{x^{\frac{1}{2}}})ln(\frac{\frac{1}{2}}{x} + 1) + \frac{(x^{\frac{1}{2}})(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{(\frac{\frac{1}{2}}{x} + 1)}))ln^{3}(\frac{\frac{1}{2}}{x} + 1)}{8x^{\frac{3}{2}}} + \frac{(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}*3ln^{2}(\frac{\frac{1}{2}}{x} + 1)(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{8x^{\frac{3}{2}}(\frac{\frac{1}{2}}{x} + 1)} - \frac{3(\frac{-2(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{(\frac{\frac{1}{2}}{x} + 1)^{3}})(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln(\frac{\frac{1}{2}}{x} + 1)}{8x^{4}} - \frac{3*-4(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln(\frac{\frac{1}{2}}{x} + 1)}{8(\frac{\frac{1}{2}}{x} + 1)^{2}x^{5}} - \frac{3((\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}((\frac{\frac{1}{2}}{x^{\frac{1}{2}}})ln(\frac{\frac{1}{2}}{x} + 1) + \frac{(x^{\frac{1}{2}})(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{(\frac{\frac{1}{2}}{x} + 1)}))ln(\frac{\frac{1}{2}}{x} + 1)}{8(\frac{\frac{1}{2}}{x} + 1)^{2}x^{4}} - \frac{3(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{8(\frac{\frac{1}{2}}{x} + 1)^{2}x^{4}(\frac{\frac{1}{2}}{x} + 1)} + \frac{3(\frac{-2(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{(\frac{\frac{1}{2}}{x} + 1)^{3}})(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln(\frac{\frac{1}{2}}{x} + 1)}{8x^{\frac{7}{2}}} + \frac{3*\frac{-7}{2}(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln(\frac{\frac{1}{2}}{x} + 1)}{8(\frac{\frac{1}{2}}{x} + 1)^{2}x^{\frac{9}{2}}} + \frac{3((\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}((\frac{\frac{1}{2}}{x^{\frac{1}{2}}})ln(\frac{\frac{1}{2}}{x} + 1) + \frac{(x^{\frac{1}{2}})(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{(\frac{\frac{1}{2}}{x} + 1)}))ln(\frac{\frac{1}{2}}{x} + 1)}{8(\frac{\frac{1}{2}}{x} + 1)^{2}x^{\frac{7}{2}}} + \frac{3(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{8(\frac{\frac{1}{2}}{x} + 1)^{2}x^{\frac{7}{2}}(\frac{\frac{1}{2}}{x} + 1)} - \frac{9(\frac{-(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{(\frac{\frac{1}{2}}{x} + 1)^{2}})(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{8x^{\frac{7}{2}}} - \frac{9*\frac{-7}{2}(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{8(\frac{\frac{1}{2}}{x} + 1)x^{\frac{9}{2}}} - \frac{9((\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}((\frac{\frac{1}{2}}{x^{\frac{1}{2}}})ln(\frac{\frac{1}{2}}{x} + 1) + \frac{(x^{\frac{1}{2}})(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{(\frac{\frac{1}{2}}{x} + 1)}))}{8(\frac{\frac{1}{2}}{x} + 1)x^{\frac{7}{2}}} - \frac{3(\frac{-2(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{(\frac{\frac{1}{2}}{x} + 1)^{3}})(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{4x^{4}} - \frac{3*-4(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{4(\frac{\frac{1}{2}}{x} + 1)^{2}x^{5}} - \frac{3((\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}((\frac{\frac{1}{2}}{x^{\frac{1}{2}}})ln(\frac{\frac{1}{2}}{x} + 1) + \frac{(x^{\frac{1}{2}})(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{(\frac{\frac{1}{2}}{x} + 1)}))}{4(\frac{\frac{1}{2}}{x} + 1)^{2}x^{4}} + \frac{9(\frac{-2(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{(\frac{\frac{1}{2}}{x} + 1)^{3}})(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{8x^{\frac{9}{2}}} + \frac{9*\frac{-9}{2}(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{8(\frac{\frac{1}{2}}{x} + 1)^{2}x^{\frac{11}{2}}} + \frac{9((\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}((\frac{\frac{1}{2}}{x^{\frac{1}{2}}})ln(\frac{\frac{1}{2}}{x} + 1) + \frac{(x^{\frac{1}{2}})(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{(\frac{\frac{1}{2}}{x} + 1)}))}{8(\frac{\frac{1}{2}}{x} + 1)^{2}x^{\frac{9}{2}}} - \frac{(\frac{-3(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{(\frac{\frac{1}{2}}{x} + 1)^{4}})(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{4x^{\frac{11}{2}}} - \frac{\frac{-11}{2}(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{4(\frac{\frac{1}{2}}{x} + 1)^{3}x^{\frac{13}{2}}} - \frac{((\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}((\frac{\frac{1}{2}}{x^{\frac{1}{2}}})ln(\frac{\frac{1}{2}}{x} + 1) + \frac{(x^{\frac{1}{2}})(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{(\frac{\frac{1}{2}}{x} + 1)}))}{4(\frac{\frac{1}{2}}{x} + 1)^{3}x^{\frac{11}{2}}} + \frac{3(\frac{-3(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{(\frac{\frac{1}{2}}{x} + 1)^{4}})(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{8x^{5}} + \frac{3*-5(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{8(\frac{\frac{1}{2}}{x} + 1)^{3}x^{6}} + \frac{3((\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}((\frac{\frac{1}{2}}{x^{\frac{1}{2}}})ln(\frac{\frac{1}{2}}{x} + 1) + \frac{(x^{\frac{1}{2}})(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{(\frac{\frac{1}{2}}{x} + 1)}))}{8(\frac{\frac{1}{2}}{x} + 1)^{3}x^{5}} - \frac{(\frac{-3(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{(\frac{\frac{1}{2}}{x} + 1)^{4}})(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{8x^{\frac{9}{2}}} - \frac{\frac{-9}{2}(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{8(\frac{\frac{1}{2}}{x} + 1)^{3}x^{\frac{11}{2}}} - \frac{((\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}((\frac{\frac{1}{2}}{x^{\frac{1}{2}}})ln(\frac{\frac{1}{2}}{x} + 1) + \frac{(x^{\frac{1}{2}})(\frac{\frac{1}{2}*-1}{x^{2}} + 0)}{(\frac{\frac{1}{2}}{x} + 1)}))}{8(\frac{\frac{1}{2}}{x} + 1)^{3}x^{\frac{9}{2}}}\\=&\frac{-15(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln(\frac{\frac{1}{2}}{x} + 1)}{16x^{\frac{7}{2}}} + \frac{15(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln^{2}(\frac{\frac{1}{2}}{x} + 1)}{16x^{3}} - \frac{(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln^{3}(\frac{\frac{1}{2}}{x} + 1)}{4(\frac{\frac{1}{2}}{x} + 1)x^{3}} - \frac{15(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln(\frac{\frac{1}{2}}{x} + 1)}{4(\frac{\frac{1}{2}}{x} + 1)x^{4}} - \frac{3(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln^{3}(\frac{\frac{1}{2}}{x} + 1)}{8x^{\frac{5}{2}}} + \frac{3(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln^{2}(\frac{\frac{1}{2}}{x} + 1)}{2(\frac{\frac{1}{2}}{x} + 1)x^{\frac{7}{2}}} - \frac{3(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln^{2}(\frac{\frac{1}{2}}{x} + 1)}{8(\frac{\frac{1}{2}}{x} + 1)^{2}x^{\frac{9}{2}}} - \frac{15(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln(\frac{\frac{1}{2}}{x} + 1)}{8(\frac{\frac{1}{2}}{x} + 1)^{2}x^{\frac{9}{2}}} + \frac{21(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln(\frac{\frac{1}{2}}{x} + 1)}{8(\frac{\frac{1}{2}}{x} + 1)^{2}x^{5}} + \frac{3(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln^{2}(\frac{\frac{1}{2}}{x} + 1)}{8(\frac{\frac{1}{2}}{x} + 1)^{2}x^{4}} - \frac{(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln(\frac{\frac{1}{2}}{x} + 1)}{2(\frac{\frac{1}{2}}{x} + 1)^{3}x^{6}} + \frac{3(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln(\frac{\frac{1}{2}}{x} + 1)}{4(\frac{\frac{1}{2}}{x} + 1)^{3}x^{\frac{11}{2}}} + \frac{(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln^{4}(\frac{\frac{1}{2}}{x} + 1)}{16x^{2}} - \frac{(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}ln(\frac{\frac{1}{2}}{x} + 1)}{4(\frac{\frac{1}{2}}{x} + 1)^{3}x^{5}} + \frac{3(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{(\frac{\frac{1}{2}}{x} + 1)^{2}x^{5}} - \frac{3(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{(\frac{\frac{1}{2}}{x} + 1)^{3}x^{6}} + \frac{15(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{4(\frac{\frac{1}{2}}{x} + 1)x^{\frac{9}{2}}} + \frac{3(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{4(\frac{\frac{1}{2}}{x} + 1)^{3}x^{\frac{11}{2}}} - \frac{45(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{8(\frac{\frac{1}{2}}{x} + 1)^{2}x^{\frac{11}{2}}} + \frac{5(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{2(\frac{\frac{1}{2}}{x} + 1)^{3}x^{\frac{13}{2}}} - \frac{3(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{8(\frac{\frac{1}{2}}{x} + 1)^{4}x^{\frac{15}{2}}} + \frac{11(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{16(\frac{\frac{1}{2}}{x} + 1)^{4}x^{7}} - \frac{3(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{8(\frac{\frac{1}{2}}{x} + 1)^{4}x^{\frac{13}{2}}} + \frac{(\frac{\frac{1}{2}}{x} + 1)^{x^{\frac{1}{2}}}}{16(\frac{\frac{1}{2}}{x} + 1)^{4}x^{6}}\\ \end{split}\end{equation} \]