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.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ ({2}^{(x + 1)}){\frac{1}{x}}^{x}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = {2}^{(x + 1)}{\frac{1}{x}}^{x}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( {2}^{(x + 1)}{\frac{1}{x}}^{x}\right)}{dx}\\=&({2}^{(x + 1)}((1 + 0)ln(2) + \frac{(x + 1)(0)}{(2)})){\frac{1}{x}}^{x} + {2}^{(x + 1)}({\frac{1}{x}}^{x}((1)ln(\frac{1}{x}) + \frac{(x)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))\\=&{2}^{(x + 1)}{\frac{1}{x}}^{x}ln(2) + {\frac{1}{x}}^{x}{2}^{(x + 1)}ln(\frac{1}{x}) - {\frac{1}{x}}^{x}{2}^{(x + 1)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( {2}^{(x + 1)}{\frac{1}{x}}^{x}ln(2) + {\frac{1}{x}}^{x}{2}^{(x + 1)}ln(\frac{1}{x}) - {\frac{1}{x}}^{x}{2}^{(x + 1)}\right)}{dx}\\=&({2}^{(x + 1)}((1 + 0)ln(2) + \frac{(x + 1)(0)}{(2)})){\frac{1}{x}}^{x}ln(2) + {2}^{(x + 1)}({\frac{1}{x}}^{x}((1)ln(\frac{1}{x}) + \frac{(x)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))ln(2) + \frac{{2}^{(x + 1)}{\frac{1}{x}}^{x}*0}{(2)} + ({\frac{1}{x}}^{x}((1)ln(\frac{1}{x}) + \frac{(x)(\frac{-1}{x^{2}})}{(\frac{1}{x})})){2}^{(x + 1)}ln(\frac{1}{x}) + {\frac{1}{x}}^{x}({2}^{(x + 1)}((1 + 0)ln(2) + \frac{(x + 1)(0)}{(2)}))ln(\frac{1}{x}) + \frac{{\frac{1}{x}}^{x}{2}^{(x + 1)}*-1}{(\frac{1}{x})x^{2}} - ({\frac{1}{x}}^{x}((1)ln(\frac{1}{x}) + \frac{(x)(\frac{-1}{x^{2}})}{(\frac{1}{x})})){2}^{(x + 1)} - {\frac{1}{x}}^{x}({2}^{(x + 1)}((1 + 0)ln(2) + \frac{(x + 1)(0)}{(2)}))\\=&{\frac{1}{x}}^{x}{2}^{(x + 1)}ln(\frac{1}{x})ln(2) + {2}^{(x + 1)}{\frac{1}{x}}^{x}ln(2)ln(\frac{1}{x}) + {\frac{1}{x}}^{x}{2}^{(x + 1)}ln^{2}(\frac{1}{x}) - {\frac{1}{x}}^{x}{2}^{(x + 1)}ln(2) - 2{\frac{1}{x}}^{x}{2}^{(x + 1)}ln(\frac{1}{x}) + {2}^{(x + 1)}{\frac{1}{x}}^{x}ln^{2}(2) - \frac{{2}^{(x + 1)}{\frac{1}{x}}^{x}}{x} - {2}^{(x + 1)}{\frac{1}{x}}^{x}ln(2) + {\frac{1}{x}}^{x}{2}^{(x + 1)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( {\frac{1}{x}}^{x}{2}^{(x + 1)}ln(\frac{1}{x})ln(2) + {2}^{(x + 1)}{\frac{1}{x}}^{x}ln(2)ln(\frac{1}{x}) + {\frac{1}{x}}^{x}{2}^{(x + 1)}ln^{2}(\frac{1}{x}) - {\frac{1}{x}}^{x}{2}^{(x + 1)}ln(2) - 2{\frac{1}{x}}^{x}{2}^{(x + 1)}ln(\frac{1}{x}) + {2}^{(x + 1)}{\frac{1}{x}}^{x}ln^{2}(2) - \frac{{2}^{(x + 1)}{\frac{1}{x}}^{x}}{x} - {2}^{(x + 1)}{\frac{1}{x}}^{x}ln(2) + {\frac{1}{x}}^{x}{2}^{(x + 1)}\right)}{dx}\\=&({\frac{1}{x}}^{x}((1)ln(\frac{1}{x}) + \frac{(x)(\frac{-1}{x^{2}})}{(\frac{1}{x})})){2}^{(x + 1)}ln(\frac{1}{x})ln(2) + {\frac{1}{x}}^{x}({2}^{(x + 1)}((1 + 0)ln(2) + \frac{(x + 1)(0)}{(2)}))ln(\frac{1}{x})ln(2) + \frac{{\frac{1}{x}}^{x}{2}^{(x + 1)}*-ln(2)}{(\frac{1}{x})x^{2}} + \frac{{\frac{1}{x}}^{x}{2}^{(x + 1)}ln(\frac{1}{x})*0}{(2)} + ({2}^{(x + 1)}((1 + 0)ln(2) + \frac{(x + 1)(0)}{(2)})){\frac{1}{x}}^{x}ln(2)ln(\frac{1}{x}) + {2}^{(x + 1)}({\frac{1}{x}}^{x}((1)ln(\frac{1}{x}) + \frac{(x)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))ln(2)ln(\frac{1}{x}) + \frac{{2}^{(x + 1)}{\frac{1}{x}}^{x}*0ln(\frac{1}{x})}{(2)} + \frac{{2}^{(x + 1)}{\frac{1}{x}}^{x}ln(2)*-1}{(\frac{1}{x})x^{2}} + ({\frac{1}{x}}^{x}((1)ln(\frac{1}{x}) + \frac{(x)(\frac{-1}{x^{2}})}{(\frac{1}{x})})){2}^{(x + 1)}ln^{2}(\frac{1}{x}) + {\frac{1}{x}}^{x}({2}^{(x + 1)}((1 + 0)ln(2) + \frac{(x + 1)(0)}{(2)}))ln^{2}(\frac{1}{x}) + \frac{{\frac{1}{x}}^{x}{2}^{(x + 1)}*2ln(\frac{1}{x})*-1}{(\frac{1}{x})x^{2}} - ({\frac{1}{x}}^{x}((1)ln(\frac{1}{x}) + \frac{(x)(\frac{-1}{x^{2}})}{(\frac{1}{x})})){2}^{(x + 1)}ln(2) - {\frac{1}{x}}^{x}({2}^{(x + 1)}((1 + 0)ln(2) + \frac{(x + 1)(0)}{(2)}))ln(2) - \frac{{\frac{1}{x}}^{x}{2}^{(x + 1)}*0}{(2)} - 2({\frac{1}{x}}^{x}((1)ln(\frac{1}{x}) + \frac{(x)(\frac{-1}{x^{2}})}{(\frac{1}{x})})){2}^{(x + 1)}ln(\frac{1}{x}) - 2{\frac{1}{x}}^{x}({2}^{(x + 1)}((1 + 0)ln(2) + \frac{(x + 1)(0)}{(2)}))ln(\frac{1}{x}) - \frac{2{\frac{1}{x}}^{x}{2}^{(x + 1)}*-1}{(\frac{1}{x})x^{2}} + ({2}^{(x + 1)}((1 + 0)ln(2) + \frac{(x + 1)(0)}{(2)})){\frac{1}{x}}^{x}ln^{2}(2) + {2}^{(x + 1)}({\frac{1}{x}}^{x}((1)ln(\frac{1}{x}) + \frac{(x)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))ln^{2}(2) + \frac{{2}^{(x + 1)}{\frac{1}{x}}^{x}*2ln(2)*0}{(2)} - \frac{-{2}^{(x + 1)}{\frac{1}{x}}^{x}}{x^{2}} - \frac{({2}^{(x + 1)}((1 + 0)ln(2) + \frac{(x + 1)(0)}{(2)})){\frac{1}{x}}^{x}}{x} - \frac{{2}^{(x + 1)}({\frac{1}{x}}^{x}((1)ln(\frac{1}{x}) + \frac{(x)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))}{x} - ({2}^{(x + 1)}((1 + 0)ln(2) + \frac{(x + 1)(0)}{(2)})){\frac{1}{x}}^{x}ln(2) - {2}^{(x + 1)}({\frac{1}{x}}^{x}((1)ln(\frac{1}{x}) + \frac{(x)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))ln(2) - \frac{{2}^{(x + 1)}{\frac{1}{x}}^{x}*0}{(2)} + ({\frac{1}{x}}^{x}((1)ln(\frac{1}{x}) + \frac{(x)(\frac{-1}{x^{2}})}{(\frac{1}{x})})){2}^{(x + 1)} + {\frac{1}{x}}^{x}({2}^{(x + 1)}((1 + 0)ln(2) + \frac{(x + 1)(0)}{(2)}))\\=&2{\frac{1}{x}}^{x}{2}^{(x + 1)}ln^{2}(\frac{1}{x})ln(2) + 2 * {2}^{(x + 1)}{\frac{1}{x}}^{x}ln^{2}(2)ln(\frac{1}{x}) - \frac{2 * {2}^{(x + 1)}{\frac{1}{x}}^{x}ln(2)}{x} - 3{\frac{1}{x}}^{x}{2}^{(x + 1)}ln(\frac{1}{x})ln(2) - {\frac{1}{x}}^{x}{2}^{(x + 1)}ln(2)ln(\frac{1}{x}) - \frac{{\frac{1}{x}}^{x}{2}^{(x + 1)}ln(2)}{x} + {2}^{(x + 1)}{\frac{1}{x}}^{x}ln(2)ln^{2}(\frac{1}{x}) + {\frac{1}{x}}^{x}{2}^{(x + 1)}ln(\frac{1}{x})ln^{2}(2) - \frac{3{\frac{1}{x}}^{x}{2}^{(x + 1)}ln(\frac{1}{x})}{x} - 2 * {2}^{(x + 1)}{\frac{1}{x}}^{x}ln(2)ln(\frac{1}{x}) + {\frac{1}{x}}^{x}{2}^{(x + 1)}ln^{3}(\frac{1}{x}) - 2 * {2}^{(x + 1)}{\frac{1}{x}}^{x}ln^{2}(2) + 3{\frac{1}{x}}^{x}{2}^{(x + 1)}ln(\frac{1}{x}) - 3{\frac{1}{x}}^{x}{2}^{(x + 1)}ln^{2}(\frac{1}{x}) + \frac{2 * {2}^{(x + 1)}{\frac{1}{x}}^{x}}{x} + {2}^{(x + 1)}{\frac{1}{x}}^{x}ln^{3}(2) + 2{\frac{1}{x}}^{x}{2}^{(x + 1)}ln(2) - {\frac{1}{x}}^{x}{2}^{(x + 1)}ln^{2}(2) + \frac{{2}^{(x + 1)}{\frac{1}{x}}^{x}}{x^{2}} + \frac{{\frac{1}{x}}^{x}{2}^{(x + 1)}}{x} + {2}^{(x + 1)}{\frac{1}{x}}^{x}ln(2) - {\frac{1}{x}}^{x}{2}^{(x + 1)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 2{\frac{1}{x}}^{x}{2}^{(x + 1)}ln^{2}(\frac{1}{x})ln(2) + 2 * {2}^{(x + 1)}{\frac{1}{x}}^{x}ln^{2}(2)ln(\frac{1}{x}) - \frac{2 * {2}^{(x + 1)}{\frac{1}{x}}^{x}ln(2)}{x} - 3{\frac{1}{x}}^{x}{2}^{(x + 1)}ln(\frac{1}{x})ln(2) - {\frac{1}{x}}^{x}{2}^{(x + 1)}ln(2)ln(\frac{1}{x}) - \frac{{\frac{1}{x}}^{x}{2}^{(x + 1)}ln(2)}{x} + {2}^{(x + 1)}{\frac{1}{x}}^{x}ln(2)ln^{2}(\frac{1}{x}) + {\frac{1}{x}}^{x}{2}^{(x + 1)}ln(\frac{1}{x})ln^{2}(2) - \frac{3{\frac{1}{x}}^{x}{2}^{(x + 1)}ln(\frac{1}{x})}{x} - 2 * {2}^{(x + 1)}{\frac{1}{x}}^{x}ln(2)ln(\frac{1}{x}) + {\frac{1}{x}}^{x}{2}^{(x + 1)}ln^{3}(\frac{1}{x}) - 2 * {2}^{(x + 1)}{\frac{1}{x}}^{x}ln^{2}(2) + 3{\frac{1}{x}}^{x}{2}^{(x + 1)}ln(\frac{1}{x}) - 3{\frac{1}{x}}^{x}{2}^{(x + 1)}ln^{2}(\frac{1}{x}) + \frac{2 * {2}^{(x + 1)}{\frac{1}{x}}^{x}}{x} + {2}^{(x + 1)}{\frac{1}{x}}^{x}ln^{3}(2) + 2{\frac{1}{x}}^{x}{2}^{(x + 1)}ln(2) - {\frac{1}{x}}^{x}{2}^{(x + 1)}ln^{2}(2) + \frac{{2}^{(x + 1)}{\frac{1}{x}}^{x}}{x^{2}} + \frac{{\frac{1}{x}}^{x}{2}^{(x + 1)}}{x} + {2}^{(x + 1)}{\frac{1}{x}}^{x}ln(2) - {\frac{1}{x}}^{x}{2}^{(x + 1)}\right)}{dx}\\=&2({\frac{1}{x}}^{x}((1)ln(\frac{1}{x}) + \frac{(x)(\frac{-1}{x^{2}})}{(\frac{1}{x})})){2}^{(x + 1)}ln^{2}(\frac{1}{x})ln(2) + 2{\frac{1}{x}}^{x}({2}^{(x + 1)}((1 + 0)ln(2) + \frac{(x + 1)(0)}{(2)}))ln^{2}(\frac{1}{x})ln(2) + \frac{2{\frac{1}{x}}^{x}{2}^{(x + 1)}*2ln(\frac{1}{x})*-ln(2)}{(\frac{1}{x})x^{2}} + \frac{2{\frac{1}{x}}^{x}{2}^{(x + 1)}ln^{2}(\frac{1}{x})*0}{(2)} + 2({2}^{(x + 1)}((1 + 0)ln(2) + \frac{(x + 1)(0)}{(2)})){\frac{1}{x}}^{x}ln^{2}(2)ln(\frac{1}{x}) + 2 * {2}^{(x + 1)}({\frac{1}{x}}^{x}((1)ln(\frac{1}{x}) + \frac{(x)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))ln^{2}(2)ln(\frac{1}{x}) + \frac{2 * {2}^{(x + 1)}{\frac{1}{x}}^{x}*2ln(2)*0ln(\frac{1}{x})}{(2)} + \frac{2 * {2}^{(x + 1)}{\frac{1}{x}}^{x}ln^{2}(2)*-1}{(\frac{1}{x})x^{2}} - \frac{2*-{2}^{(x + 1)}{\frac{1}{x}}^{x}ln(2)}{x^{2}} - \frac{2({2}^{(x + 1)}((1 + 0)ln(2) + \frac{(x + 1)(0)}{(2)})){\frac{1}{x}}^{x}ln(2)}{x} - \frac{2 * {2}^{(x + 1)}({\frac{1}{x}}^{x}((1)ln(\frac{1}{x}) + \frac{(x)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))ln(2)}{x} - \frac{2 * {2}^{(x + 1)}{\frac{1}{x}}^{x}*0}{x(2)} - 3({\frac{1}{x}}^{x}((1)ln(\frac{1}{x}) + \frac{(x)(\frac{-1}{x^{2}})}{(\frac{1}{x})})){2}^{(x + 1)}ln(\frac{1}{x})ln(2) - 3{\frac{1}{x}}^{x}({2}^{(x + 1)}((1 + 0)ln(2) + \frac{(x + 1)(0)}{(2)}))ln(\frac{1}{x})ln(2) - \frac{3{\frac{1}{x}}^{x}{2}^{(x + 1)}*-ln(2)}{(\frac{1}{x})x^{2}} - \frac{3{\frac{1}{x}}^{x}{2}^{(x + 1)}ln(\frac{1}{x})*0}{(2)} - ({\frac{1}{x}}^{x}((1)ln(\frac{1}{x}) + \frac{(x)(\frac{-1}{x^{2}})}{(\frac{1}{x})})){2}^{(x + 1)}ln(2)ln(\frac{1}{x}) - {\frac{1}{x}}^{x}({2}^{(x + 1)}((1 + 0)ln(2) + \frac{(x + 1)(0)}{(2)}))ln(2)ln(\frac{1}{x}) - \frac{{\frac{1}{x}}^{x}{2}^{(x + 1)}*0ln(\frac{1}{x})}{(2)} - \frac{{\frac{1}{x}}^{x}{2}^{(x + 1)}ln(2)*-1}{(\frac{1}{x})x^{2}} - \frac{-{\frac{1}{x}}^{x}{2}^{(x + 1)}ln(2)}{x^{2}} - \frac{({\frac{1}{x}}^{x}((1)ln(\frac{1}{x}) + \frac{(x)(\frac{-1}{x^{2}})}{(\frac{1}{x})})){2}^{(x + 1)}ln(2)}{x} - \frac{{\frac{1}{x}}^{x}({2}^{(x + 1)}((1 + 0)ln(2) + \frac{(x + 1)(0)}{(2)}))ln(2)}{x} - \frac{{\frac{1}{x}}^{x}{2}^{(x + 1)}*0}{x(2)} + ({2}^{(x + 1)}((1 + 0)ln(2) + \frac{(x + 1)(0)}{(2)})){\frac{1}{x}}^{x}ln(2)ln^{2}(\frac{1}{x}) + {2}^{(x + 1)}({\frac{1}{x}}^{x}((1)ln(\frac{1}{x}) + \frac{(x)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))ln(2)ln^{2}(\frac{1}{x}) + \frac{{2}^{(x + 1)}{\frac{1}{x}}^{x}*0ln^{2}(\frac{1}{x})}{(2)} + \frac{{2}^{(x + 1)}{\frac{1}{x}}^{x}ln(2)*2ln(\frac{1}{x})*-1}{(\frac{1}{x})x^{2}} + ({\frac{1}{x}}^{x}((1)ln(\frac{1}{x}) + \frac{(x)(\frac{-1}{x^{2}})}{(\frac{1}{x})})){2}^{(x + 1)}ln(\frac{1}{x})ln^{2}(2) + {\frac{1}{x}}^{x}({2}^{(x + 1)}((1 + 0)ln(2) + \frac{(x + 1)(0)}{(2)}))ln(\frac{1}{x})ln^{2}(2) + \frac{{\frac{1}{x}}^{x}{2}^{(x + 1)}*-ln^{2}(2)}{(\frac{1}{x})x^{2}} + \frac{{\frac{1}{x}}^{x}{2}^{(x + 1)}ln(\frac{1}{x})*2ln(2)*0}{(2)} - \frac{3*-{\frac{1}{x}}^{x}{2}^{(x + 1)}ln(\frac{1}{x})}{x^{2}} - \frac{3({\frac{1}{x}}^{x}((1)ln(\frac{1}{x}) + \frac{(x)(\frac{-1}{x^{2}})}{(\frac{1}{x})})){2}^{(x + 1)}ln(\frac{1}{x})}{x} - \frac{3{\frac{1}{x}}^{x}({2}^{(x + 1)}((1 + 0)ln(2) + \frac{(x + 1)(0)}{(2)}))ln(\frac{1}{x})}{x} - \frac{3{\frac{1}{x}}^{x}{2}^{(x + 1)}*-1}{x(\frac{1}{x})x^{2}} - 2({2}^{(x + 1)}((1 + 0)ln(2) + \frac{(x + 1)(0)}{(2)})){\frac{1}{x}}^{x}ln(2)ln(\frac{1}{x}) - 2 * {2}^{(x + 1)}({\frac{1}{x}}^{x}((1)ln(\frac{1}{x}) + \frac{(x)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))ln(2)ln(\frac{1}{x}) - \frac{2 * {2}^{(x + 1)}{\frac{1}{x}}^{x}*0ln(\frac{1}{x})}{(2)} - \frac{2 * {2}^{(x + 1)}{\frac{1}{x}}^{x}ln(2)*-1}{(\frac{1}{x})x^{2}} + ({\frac{1}{x}}^{x}((1)ln(\frac{1}{x}) + \frac{(x)(\frac{-1}{x^{2}})}{(\frac{1}{x})})){2}^{(x + 1)}ln^{3}(\frac{1}{x}) + {\frac{1}{x}}^{x}({2}^{(x + 1)}((1 + 0)ln(2) + \frac{(x + 1)(0)}{(2)}))ln^{3}(\frac{1}{x}) + \frac{{\frac{1}{x}}^{x}{2}^{(x + 1)}*3ln^{2}(\frac{1}{x})*-1}{(\frac{1}{x})x^{2}} - 2({2}^{(x + 1)}((1 + 0)ln(2) + \frac{(x + 1)(0)}{(2)})){\frac{1}{x}}^{x}ln^{2}(2) - 2 * {2}^{(x + 1)}({\frac{1}{x}}^{x}((1)ln(\frac{1}{x}) + \frac{(x)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))ln^{2}(2) - \frac{2 * {2}^{(x + 1)}{\frac{1}{x}}^{x}*2ln(2)*0}{(2)} + 3({\frac{1}{x}}^{x}((1)ln(\frac{1}{x}) + \frac{(x)(\frac{-1}{x^{2}})}{(\frac{1}{x})})){2}^{(x + 1)}ln(\frac{1}{x}) + 3{\frac{1}{x}}^{x}({2}^{(x + 1)}((1 + 0)ln(2) + \frac{(x + 1)(0)}{(2)}))ln(\frac{1}{x}) + \frac{3{\frac{1}{x}}^{x}{2}^{(x + 1)}*-1}{(\frac{1}{x})x^{2}} - 3({\frac{1}{x}}^{x}((1)ln(\frac{1}{x}) + \frac{(x)(\frac{-1}{x^{2}})}{(\frac{1}{x})})){2}^{(x + 1)}ln^{2}(\frac{1}{x}) - 3{\frac{1}{x}}^{x}({2}^{(x + 1)}((1 + 0)ln(2) + \frac{(x + 1)(0)}{(2)}))ln^{2}(\frac{1}{x}) - \frac{3{\frac{1}{x}}^{x}{2}^{(x + 1)}*2ln(\frac{1}{x})*-1}{(\frac{1}{x})x^{2}} + \frac{2*-{2}^{(x + 1)}{\frac{1}{x}}^{x}}{x^{2}} + \frac{2({2}^{(x + 1)}((1 + 0)ln(2) + \frac{(x + 1)(0)}{(2)})){\frac{1}{x}}^{x}}{x} + \frac{2 * {2}^{(x + 1)}({\frac{1}{x}}^{x}((1)ln(\frac{1}{x}) + \frac{(x)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))}{x} + ({2}^{(x + 1)}((1 + 0)ln(2) + \frac{(x + 1)(0)}{(2)})){\frac{1}{x}}^{x}ln^{3}(2) + {2}^{(x + 1)}({\frac{1}{x}}^{x}((1)ln(\frac{1}{x}) + \frac{(x)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))ln^{3}(2) + \frac{{2}^{(x + 1)}{\frac{1}{x}}^{x}*3ln^{2}(2)*0}{(2)} + 2({\frac{1}{x}}^{x}((1)ln(\frac{1}{x}) + \frac{(x)(\frac{-1}{x^{2}})}{(\frac{1}{x})})){2}^{(x + 1)}ln(2) + 2{\frac{1}{x}}^{x}({2}^{(x + 1)}((1 + 0)ln(2) + \frac{(x + 1)(0)}{(2)}))ln(2) + \frac{2{\frac{1}{x}}^{x}{2}^{(x + 1)}*0}{(2)} - ({\frac{1}{x}}^{x}((1)ln(\frac{1}{x}) + \frac{(x)(\frac{-1}{x^{2}})}{(\frac{1}{x})})){2}^{(x + 1)}ln^{2}(2) - {\frac{1}{x}}^{x}({2}^{(x + 1)}((1 + 0)ln(2) + \frac{(x + 1)(0)}{(2)}))ln^{2}(2) - \frac{{\frac{1}{x}}^{x}{2}^{(x + 1)}*2ln(2)*0}{(2)} + \frac{-2 * {2}^{(x + 1)}{\frac{1}{x}}^{x}}{x^{3}} + \frac{({2}^{(x + 1)}((1 + 0)ln(2) + \frac{(x + 1)(0)}{(2)})){\frac{1}{x}}^{x}}{x^{2}} + \frac{{2}^{(x + 1)}({\frac{1}{x}}^{x}((1)ln(\frac{1}{x}) + \frac{(x)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))}{x^{2}} + \frac{-{\frac{1}{x}}^{x}{2}^{(x + 1)}}{x^{2}} + \frac{({\frac{1}{x}}^{x}((1)ln(\frac{1}{x}) + \frac{(x)(\frac{-1}{x^{2}})}{(\frac{1}{x})})){2}^{(x + 1)}}{x} + \frac{{\frac{1}{x}}^{x}({2}^{(x + 1)}((1 + 0)ln(2) + \frac{(x + 1)(0)}{(2)}))}{x} + ({2}^{(x + 1)}((1 + 0)ln(2) + \frac{(x + 1)(0)}{(2)})){\frac{1}{x}}^{x}ln(2) + {2}^{(x + 1)}({\frac{1}{x}}^{x}((1)ln(\frac{1}{x}) + \frac{(x)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))ln(2) + \frac{{2}^{(x + 1)}{\frac{1}{x}}^{x}*0}{(2)} - ({\frac{1}{x}}^{x}((1)ln(\frac{1}{x}) + \frac{(x)(\frac{-1}{x^{2}})}{(\frac{1}{x})})){2}^{(x + 1)} - {\frac{1}{x}}^{x}({2}^{(x + 1)}((1 + 0)ln(2) + \frac{(x + 1)(0)}{(2)}))\\=&3{\frac{1}{x}}^{x}{2}^{(x + 1)}ln^{3}(\frac{1}{x})ln(2) + 3 * {2}^{(x + 1)}{\frac{1}{x}}^{x}ln^{2}(2)ln^{2}(\frac{1}{x}) + 3{\frac{1}{x}}^{x}{2}^{(x + 1)}ln^{2}(\frac{1}{x})ln^{2}(2) - \frac{7{\frac{1}{x}}^{x}{2}^{(x + 1)}ln(\frac{1}{x})ln(2)}{x} + 3 * {2}^{(x + 1)}{\frac{1}{x}}^{x}ln^{3}(2)ln(\frac{1}{x}) - 8{\frac{1}{x}}^{x}{2}^{(x + 1)}ln^{2}(\frac{1}{x})ln(2) - \frac{2 * {2}^{(x + 1)}{\frac{1}{x}}^{x}ln(\frac{1}{x})ln(2)}{x} - \frac{3 * {2}^{(x + 1)}{\frac{1}{x}}^{x}ln(2)ln(\frac{1}{x})}{x} - \frac{4 * {2}^{(x + 1)}{\frac{1}{x}}^{x}ln^{2}(2)}{x} - \frac{6{\frac{1}{x}}^{x}{2}^{(x + 1)}ln^{2}(\frac{1}{x})}{x} + \frac{7 * {2}^{(x + 1)}{\frac{1}{x}}^{x}ln(2)}{x} - 6 * {2}^{(x + 1)}{\frac{1}{x}}^{x}ln^{2}(2)ln(\frac{1}{x}) + \frac{{\frac{1}{x}}^{x}{2}^{(x + 1)}ln(2)}{x^{2}} - 2{\frac{1}{x}}^{x}{2}^{(x + 1)}ln^{2}(2)ln(\frac{1}{x}) + 3{\frac{1}{x}}^{x}{2}^{(x + 1)}ln(2)ln(\frac{1}{x}) - \frac{2{\frac{1}{x}}^{x}{2}^{(x + 1)}ln^{2}(2)}{x} - {\frac{1}{x}}^{x}{2}^{(x + 1)}ln(2)ln^{2}(\frac{1}{x}) + 6{\frac{1}{x}}^{x}{2}^{(x + 1)}ln(\frac{1}{x})ln(2) - 4{\frac{1}{x}}^{x}{2}^{(x + 1)}ln(\frac{1}{x})ln^{2}(2) + \frac{4{\frac{1}{x}}^{x}{2}^{(x + 1)}ln(\frac{1}{x})}{x^{2}} + \frac{5{\frac{1}{x}}^{x}{2}^{(x + 1)}ln(2)}{x} + \frac{12{\frac{1}{x}}^{x}{2}^{(x + 1)}ln(\frac{1}{x})}{x} + \frac{3 * {2}^{(x + 1)}{\frac{1}{x}}^{x}ln(2)}{x^{2}} + {2}^{(x + 1)}{\frac{1}{x}}^{x}ln(2)ln^{3}(\frac{1}{x}) + {\frac{1}{x}}^{x}{2}^{(x + 1)}ln(\frac{1}{x})ln^{3}(2) + \frac{{2}^{(x + 1)}{\frac{1}{x}}^{x}}{x^{2}} + 3 * {2}^{(x + 1)}{\frac{1}{x}}^{x}ln(2)ln(\frac{1}{x}) - 3 * {2}^{(x + 1)}{\frac{1}{x}}^{x}ln(2)ln^{2}(\frac{1}{x}) - 4{\frac{1}{x}}^{x}{2}^{(x + 1)}ln^{3}(\frac{1}{x}) + {\frac{1}{x}}^{x}{2}^{(x + 1)}ln^{4}(\frac{1}{x}) - 3 * {2}^{(x + 1)}{\frac{1}{x}}^{x}ln^{3}(2) + 3{\frac{1}{x}}^{x}{2}^{(x + 1)}ln^{2}(2) + 6{\frac{1}{x}}^{x}{2}^{(x + 1)}ln^{2}(\frac{1}{x}) - \frac{3 * {2}^{(x + 1)}{\frac{1}{x}}^{x}}{x} + {2}^{(x + 1)}{\frac{1}{x}}^{x}ln^{4}(2) - {\frac{1}{x}}^{x}{2}^{(x + 1)}ln^{3}(2) - 4{\frac{1}{x}}^{x}{2}^{(x + 1)}ln(\frac{1}{x}) - 3{\frac{1}{x}}^{x}{2}^{(x + 1)}ln(2) + 3 * {2}^{(x + 1)}{\frac{1}{x}}^{x}ln^{2}(2) - \frac{2 * {2}^{(x + 1)}{\frac{1}{x}}^{x}}{x^{3}} - \frac{3{\frac{1}{x}}^{x}{2}^{(x + 1)}}{x} - \frac{2{\frac{1}{x}}^{x}{2}^{(x + 1)}}{x^{2}} - {2}^{(x + 1)}{\frac{1}{x}}^{x}ln(2) + {\frac{1}{x}}^{x}{2}^{(x + 1)}\\ \end{split}\end{equation} \]