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