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.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ log_{log_{a}^{x}}^{log_{b}^{x}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( log_{log_{a}^{x}}^{log_{b}^{x}}\right)}{dx}\\=&(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{(ln(b))}))}{(log_{b}^{x})} - \frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))}))log_{log_{a}^{x}}^{log_{b}^{x}}}{(log_{a}^{x})})}{(ln(log_{a}^{x}))})\\=&\frac{1}{xlog(b, x)ln(b)ln(log_{a}^{x})} - \frac{log_{log_{a}^{x}}^{log_{b}^{x}}}{xlog(a, x)ln(a)ln(log_{a}^{x})}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{1}{xlog(b, x)ln(b)ln(log_{a}^{x})} - \frac{log_{log_{a}^{x}}^{log_{b}^{x}}}{xlog(a, x)ln(a)ln(log_{a}^{x})}\right)}{dx}\\=&\frac{-1}{x^{2}log(b, x)ln(b)ln(log_{a}^{x})} + \frac{(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{{\left(log(b, x)^{2}(ln(b))})}{xln(b)ln(log_{a}^{x})} + \frac{-0}{xlog(b, x)ln^{2}(b)(b)ln(log_{a}^{x})} + \frac{-(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{xlog(b, x)ln(b)ln^{2}(log_{a}^{x})(log_{a}^{x})} - \frac{-log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{2}log(a, x)ln(a)ln(log_{a}^{x})} - \frac{(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{2}(ln(a))})log_{log_{a}^{x}}^{log_{b}^{x}}}{xln(a)ln(log_{a}^{x})} - \frac{(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{(ln(b))}))}{(log_{b}^{x})} - \frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))}))log_{log_{a}^{x}}^{log_{b}^{x}}}{(log_{a}^{x})})}{(ln(log_{a}^{x}))})}{xlog(a, x)ln(a)ln(log_{a}^{x})} - \frac{log_{log_{a}^{x}}^{log_{b}^{x}}*-0}{xlog(a, x)ln^{2}(a)(a)ln(log_{a}^{x})} - \frac{log_{log_{a}^{x}}^{log_{b}^{x}}*-(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{xlog(a, x)ln(a)ln^{2}(log_{a}^{x})(log_{a}^{x})}\\=&\frac{-1}{x^{2}log(b, x)ln(b)ln(log_{a}^{x})} - \frac{1}{x^{2}{\left(log(b, x)^{2}ln^{2}(b)ln(log_{a}^{x})} - \frac{1}{x^{2}log(b, x)log(a, x)ln(b)ln(a)ln^{2}(log_{a}^{x})} - \frac{1}{x^{2}log(b, x)log(a, x)ln(b)ln^{2}(log_{a}^{x})ln(a)} + \frac{log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{2}{\left(log(a, x)^{2}ln^{2}(a)ln(log_{a}^{x})} + \frac{log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{2}log(a, x)ln(a)ln(log_{a}^{x})} + \frac{log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{2}{\left(log(a, x)^{2}ln^{2}(a)ln^{2}(log_{a}^{x})} + \frac{log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{2}{\left(log(a, x)^{2}ln^{2}(a)ln^{2}(log_{a}^{x})}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-1}{x^{2}log(b, x)ln(b)ln(log_{a}^{x})} - \frac{1}{x^{2}{\left(log(b, x)^{2}ln^{2}(b)ln(log_{a}^{x})} - \frac{1}{x^{2}log(b, x)log(a, x)ln(b)ln(a)ln^{2}(log_{a}^{x})} - \frac{1}{x^{2}log(b, x)log(a, x)ln(b)ln^{2}(log_{a}^{x})ln(a)} + \frac{log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{2}{\left(log(a, x)^{2}ln^{2}(a)ln(log_{a}^{x})} + \frac{log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{2}log(a, x)ln(a)ln(log_{a}^{x})} + \frac{log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{2}{\left(log(a, x)^{2}ln^{2}(a)ln^{2}(log_{a}^{x})} + \frac{log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{2}{\left(log(a, x)^{2}ln^{2}(a)ln^{2}(log_{a}^{x})}\right)}{dx}\\=&\frac{--2}{x^{3}log(b, x)ln(b)ln(log_{a}^{x})} - \frac{(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{{\left(log(b, x)^{2}(ln(b))})}{x^{2}ln(b)ln(log_{a}^{x})} - \frac{-0}{x^{2}log(b, x)ln^{2}(b)(b)ln(log_{a}^{x})} - \frac{-(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{2}log(b, x)ln(b)ln^{2}(log_{a}^{x})(log_{a}^{x})} - \frac{-2}{x^{3}{\left(log(b, x)^{2}ln^{2}(b)ln(log_{a}^{x})} - \frac{(\frac{-2(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{{\left(log(b, x)^{3}(ln(b))})}{x^{2}ln^{2}(b)ln(log_{a}^{x})} - \frac{-2*0}{x^{2}{\left(log(b, x)^{2}ln^{3}(b)(b)ln(log_{a}^{x})} - \frac{-(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{2}{\left(log(b, x)^{2}ln^{2}(b)ln^{2}(log_{a}^{x})(log_{a}^{x})} - \frac{-2}{x^{3}log(b, x)log(a, x)ln(b)ln(a)ln^{2}(log_{a}^{x})} - \frac{(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{{\left(log(b, x)^{2}(ln(b))})}{x^{2}log(a, x)ln(b)ln(a)ln^{2}(log_{a}^{x})} - \frac{(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{2}(ln(a))})}{x^{2}log(b, x)ln(b)ln(a)ln^{2}(log_{a}^{x})} - \frac{-0}{x^{2}log(b, x)log(a, x)ln^{2}(b)(b)ln(a)ln^{2}(log_{a}^{x})} - \frac{-0}{x^{2}log(b, x)log(a, x)ln(b)ln^{2}(a)(a)ln^{2}(log_{a}^{x})} - \frac{-2(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{2}log(b, x)log(a, x)ln(b)ln(a)ln^{3}(log_{a}^{x})(log_{a}^{x})} - \frac{-2}{x^{3}log(b, x)log(a, x)ln(b)ln^{2}(log_{a}^{x})ln(a)} - \frac{(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{{\left(log(b, x)^{2}(ln(b))})}{x^{2}log(a, x)ln(b)ln^{2}(log_{a}^{x})ln(a)} - \frac{(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{2}(ln(a))})}{x^{2}log(b, x)ln(b)ln^{2}(log_{a}^{x})ln(a)} - \frac{-0}{x^{2}log(b, x)log(a, x)ln^{2}(b)(b)ln^{2}(log_{a}^{x})ln(a)} - \frac{-2(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{2}log(b, x)log(a, x)ln(b)ln^{3}(log_{a}^{x})(log_{a}^{x})ln(a)} - \frac{-0}{x^{2}log(b, x)log(a, x)ln(b)ln^{2}(log_{a}^{x})ln^{2}(a)(a)} + \frac{-2log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}{\left(log(a, x)^{2}ln^{2}(a)ln(log_{a}^{x})} + \frac{(\frac{-2(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{3}(ln(a))})log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{2}ln^{2}(a)ln(log_{a}^{x})} + \frac{(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{(ln(b))}))}{(log_{b}^{x})} - \frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))}))log_{log_{a}^{x}}^{log_{b}^{x}}}{(log_{a}^{x})})}{(ln(log_{a}^{x}))})}{x^{2}{\left(log(a, x)^{2}ln^{2}(a)ln(log_{a}^{x})} + \frac{log_{log_{a}^{x}}^{log_{b}^{x}}*-2*0}{x^{2}{\left(log(a, x)^{2}ln^{3}(a)(a)ln(log_{a}^{x})} + \frac{log_{log_{a}^{x}}^{log_{b}^{x}}*-(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{2}{\left(log(a, x)^{2}ln^{2}(a)ln^{2}(log_{a}^{x})(log_{a}^{x})} + \frac{-2log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}log(a, x)ln(a)ln(log_{a}^{x})} + \frac{(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{2}(ln(a))})log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{2}ln(a)ln(log_{a}^{x})} + \frac{(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{(ln(b))}))}{(log_{b}^{x})} - \frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))}))log_{log_{a}^{x}}^{log_{b}^{x}}}{(log_{a}^{x})})}{(ln(log_{a}^{x}))})}{x^{2}log(a, x)ln(a)ln(log_{a}^{x})} + \frac{log_{log_{a}^{x}}^{log_{b}^{x}}*-0}{x^{2}log(a, x)ln^{2}(a)(a)ln(log_{a}^{x})} + \frac{log_{log_{a}^{x}}^{log_{b}^{x}}*-(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{2}log(a, x)ln(a)ln^{2}(log_{a}^{x})(log_{a}^{x})} + \frac{-2log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}{\left(log(a, x)^{2}ln^{2}(a)ln^{2}(log_{a}^{x})} + \frac{(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{(ln(b))}))}{(log_{b}^{x})} - \frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))}))log_{log_{a}^{x}}^{log_{b}^{x}}}{(log_{a}^{x})})}{(ln(log_{a}^{x}))})}{x^{2}{\left(log(a, x)^{2}ln^{2}(a)ln^{2}(log_{a}^{x})} + \frac{log_{log_{a}^{x}}^{log_{b}^{x}}(\frac{-2(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{3}(ln(a))})}{x^{2}ln^{2}(a)ln^{2}(log_{a}^{x})} + \frac{log_{log_{a}^{x}}^{log_{b}^{x}}*-2*0}{x^{2}{\left(log(a, x)^{2}ln^{3}(a)(a)ln^{2}(log_{a}^{x})} + \frac{log_{log_{a}^{x}}^{log_{b}^{x}}*-2(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{2}{\left(log(a, x)^{2}ln^{2}(a)ln^{3}(log_{a}^{x})(log_{a}^{x})} + \frac{-2log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}{\left(log(a, x)^{2}ln^{2}(a)ln^{2}(log_{a}^{x})} + \frac{(\frac{-2(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{3}(ln(a))})log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{2}ln^{2}(a)ln^{2}(log_{a}^{x})} + \frac{(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{(ln(b))}))}{(log_{b}^{x})} - \frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))}))log_{log_{a}^{x}}^{log_{b}^{x}}}{(log_{a}^{x})})}{(ln(log_{a}^{x}))})}{x^{2}{\left(log(a, x)^{2}ln^{2}(a)ln^{2}(log_{a}^{x})} + \frac{log_{log_{a}^{x}}^{log_{b}^{x}}*-2*0}{x^{2}{\left(log(a, x)^{2}ln^{3}(a)(a)ln^{2}(log_{a}^{x})} + \frac{log_{log_{a}^{x}}^{log_{b}^{x}}*-2(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{2}{\left(log(a, x)^{2}ln^{2}(a)ln^{3}(log_{a}^{x})(log_{a}^{x})}\\=&\frac{2}{x^{3}log(b, x)ln(b)ln(log_{a}^{x})} + \frac{3}{x^{3}{\left(log(b, x)^{2}ln^{2}(b)ln(log_{a}^{x})} + \frac{3}{x^{3}log(b, x)log(a, x)ln(b)ln(a)ln^{2}(log_{a}^{x})} + \frac{2}{x^{3}{\left(log(b, x)^{3}ln^{3}(b)ln(log_{a}^{x})} + \frac{2}{x^{3}{\left(log(b, x)^{2}log(a, x)ln^{2}(b)ln(a)ln^{2}(log_{a}^{x})} + \frac{2}{x^{3}{\left(log(a, x)^{2}log(b, x)ln^{2}(a)ln(b)ln^{2}(log_{a}^{x})} + \frac{2}{x^{3}{\left(log(a, x)^{2}log(b, x)ln(b)ln^{3}(log_{a}^{x})ln^{2}(a)} + \frac{3}{x^{3}log(b, x)log(a, x)ln(b)ln^{2}(log_{a}^{x})ln(a)} + \frac{1}{x^{3}{\left(log(b, x)^{2}log(a, x)ln^{2}(b)ln^{2}(log_{a}^{x})ln(a)} + \frac{2}{x^{3}{\left(log(a, x)^{2}log(b, x)ln(b)ln^{2}(a)ln^{3}(log_{a}^{x})} + \frac{1}{x^{3}log(b, x){\left(log(a, x)^{2}ln(b)ln^{2}(log_{a}^{x})ln^{2}(a)} + \frac{2}{x^{3}log(b, x){\left(log(a, x)^{2}ln(b)ln^{3}(log_{a}^{x})ln^{2}(a)} - \frac{3log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}{\left(log(a, x)^{2}ln^{2}(a)ln(log_{a}^{x})} - \frac{log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln^{2}(log_{a}^{x})} - \frac{5log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln^{2}(log_{a}^{x})} - \frac{2log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}log(a, x)ln(a)ln(log_{a}^{x})} - \frac{3log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}{\left(log(a, x)^{2}ln^{2}(a)ln^{2}(log_{a}^{x})} - \frac{3log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}{\left(log(a, x)^{2}ln^{2}(a)ln^{2}(log_{a}^{x})} - \frac{2log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln(log_{a}^{x})} - \frac{2log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln^{3}(log_{a}^{x})} - \frac{4log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln^{3}(log_{a}^{x})}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{2}{x^{3}log(b, x)ln(b)ln(log_{a}^{x})} + \frac{3}{x^{3}{\left(log(b, x)^{2}ln^{2}(b)ln(log_{a}^{x})} + \frac{3}{x^{3}log(b, x)log(a, x)ln(b)ln(a)ln^{2}(log_{a}^{x})} + \frac{2}{x^{3}{\left(log(b, x)^{3}ln^{3}(b)ln(log_{a}^{x})} + \frac{2}{x^{3}{\left(log(b, x)^{2}log(a, x)ln^{2}(b)ln(a)ln^{2}(log_{a}^{x})} + \frac{2}{x^{3}{\left(log(a, x)^{2}log(b, x)ln^{2}(a)ln(b)ln^{2}(log_{a}^{x})} + \frac{2}{x^{3}{\left(log(a, x)^{2}log(b, x)ln(b)ln^{3}(log_{a}^{x})ln^{2}(a)} + \frac{3}{x^{3}log(b, x)log(a, x)ln(b)ln^{2}(log_{a}^{x})ln(a)} + \frac{1}{x^{3}{\left(log(b, x)^{2}log(a, x)ln^{2}(b)ln^{2}(log_{a}^{x})ln(a)} + \frac{2}{x^{3}{\left(log(a, x)^{2}log(b, x)ln(b)ln^{2}(a)ln^{3}(log_{a}^{x})} + \frac{1}{x^{3}log(b, x){\left(log(a, x)^{2}ln(b)ln^{2}(log_{a}^{x})ln^{2}(a)} + \frac{2}{x^{3}log(b, x){\left(log(a, x)^{2}ln(b)ln^{3}(log_{a}^{x})ln^{2}(a)} - \frac{3log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}{\left(log(a, x)^{2}ln^{2}(a)ln(log_{a}^{x})} - \frac{log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln^{2}(log_{a}^{x})} - \frac{5log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln^{2}(log_{a}^{x})} - \frac{2log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}log(a, x)ln(a)ln(log_{a}^{x})} - \frac{3log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}{\left(log(a, x)^{2}ln^{2}(a)ln^{2}(log_{a}^{x})} - \frac{3log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}{\left(log(a, x)^{2}ln^{2}(a)ln^{2}(log_{a}^{x})} - \frac{2log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln(log_{a}^{x})} - \frac{2log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln^{3}(log_{a}^{x})} - \frac{4log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln^{3}(log_{a}^{x})}\right)}{dx}\\=&\frac{2*-3}{x^{4}log(b, x)ln(b)ln(log_{a}^{x})} + \frac{2(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{{\left(log(b, x)^{2}(ln(b))})}{x^{3}ln(b)ln(log_{a}^{x})} + \frac{2*-0}{x^{3}log(b, x)ln^{2}(b)(b)ln(log_{a}^{x})} + \frac{2*-(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{3}log(b, x)ln(b)ln^{2}(log_{a}^{x})(log_{a}^{x})} + \frac{3*-3}{x^{4}{\left(log(b, x)^{2}ln^{2}(b)ln(log_{a}^{x})} + \frac{3(\frac{-2(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{{\left(log(b, x)^{3}(ln(b))})}{x^{3}ln^{2}(b)ln(log_{a}^{x})} + \frac{3*-2*0}{x^{3}{\left(log(b, x)^{2}ln^{3}(b)(b)ln(log_{a}^{x})} + \frac{3*-(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{3}{\left(log(b, x)^{2}ln^{2}(b)ln^{2}(log_{a}^{x})(log_{a}^{x})} + \frac{3*-3}{x^{4}log(b, x)log(a, x)ln(b)ln(a)ln^{2}(log_{a}^{x})} + \frac{3(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{{\left(log(b, x)^{2}(ln(b))})}{x^{3}log(a, x)ln(b)ln(a)ln^{2}(log_{a}^{x})} + \frac{3(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{2}(ln(a))})}{x^{3}log(b, x)ln(b)ln(a)ln^{2}(log_{a}^{x})} + \frac{3*-0}{x^{3}log(b, x)log(a, x)ln^{2}(b)(b)ln(a)ln^{2}(log_{a}^{x})} + \frac{3*-0}{x^{3}log(b, x)log(a, x)ln(b)ln^{2}(a)(a)ln^{2}(log_{a}^{x})} + \frac{3*-2(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{3}log(b, x)log(a, x)ln(b)ln(a)ln^{3}(log_{a}^{x})(log_{a}^{x})} + \frac{2*-3}{x^{4}{\left(log(b, x)^{3}ln^{3}(b)ln(log_{a}^{x})} + \frac{2(\frac{-3(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{{\left(log(b, x)^{4}(ln(b))})}{x^{3}ln^{3}(b)ln(log_{a}^{x})} + \frac{2*-3*0}{x^{3}{\left(log(b, x)^{3}ln^{4}(b)(b)ln(log_{a}^{x})} + \frac{2*-(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{3}{\left(log(b, x)^{3}ln^{3}(b)ln^{2}(log_{a}^{x})(log_{a}^{x})} + \frac{2*-3}{x^{4}{\left(log(b, x)^{2}log(a, x)ln^{2}(b)ln(a)ln^{2}(log_{a}^{x})} + \frac{2(\frac{-2(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{{\left(log(b, x)^{3}(ln(b))})}{x^{3}log(a, x)ln^{2}(b)ln(a)ln^{2}(log_{a}^{x})} + \frac{2(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{2}(ln(a))})}{x^{3}{\left(log(b, x)^{2}ln^{2}(b)ln(a)ln^{2}(log_{a}^{x})} + \frac{2*-2*0}{x^{3}{\left(log(b, x)^{2}log(a, x)ln^{3}(b)(b)ln(a)ln^{2}(log_{a}^{x})} + \frac{2*-0}{x^{3}{\left(log(b, x)^{2}log(a, x)ln^{2}(b)ln^{2}(a)(a)ln^{2}(log_{a}^{x})} + \frac{2*-2(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{3}{\left(log(b, x)^{2}log(a, x)ln^{2}(b)ln(a)ln^{3}(log_{a}^{x})(log_{a}^{x})} + \frac{2*-3}{x^{4}{\left(log(a, x)^{2}log(b, x)ln^{2}(a)ln(b)ln^{2}(log_{a}^{x})} + \frac{2(\frac{-2(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{3}(ln(a))})}{x^{3}log(b, x)ln^{2}(a)ln(b)ln^{2}(log_{a}^{x})} + \frac{2(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{{\left(log(b, x)^{2}(ln(b))})}{x^{3}{\left(log(a, x)^{2}ln^{2}(a)ln(b)ln^{2}(log_{a}^{x})} + \frac{2*-2*0}{x^{3}{\left(log(a, x)^{2}log(b, x)ln^{3}(a)(a)ln(b)ln^{2}(log_{a}^{x})} + \frac{2*-0}{x^{3}{\left(log(a, x)^{2}log(b, x)ln^{2}(a)ln^{2}(b)(b)ln^{2}(log_{a}^{x})} + \frac{2*-2(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{3}{\left(log(a, x)^{2}log(b, x)ln^{2}(a)ln(b)ln^{3}(log_{a}^{x})(log_{a}^{x})} + \frac{2*-3}{x^{4}{\left(log(a, x)^{2}log(b, x)ln(b)ln^{3}(log_{a}^{x})ln^{2}(a)} + \frac{2(\frac{-2(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{3}(ln(a))})}{x^{3}log(b, x)ln(b)ln^{3}(log_{a}^{x})ln^{2}(a)} + \frac{2(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{{\left(log(b, x)^{2}(ln(b))})}{x^{3}{\left(log(a, x)^{2}ln(b)ln^{3}(log_{a}^{x})ln^{2}(a)} + \frac{2*-0}{x^{3}{\left(log(a, x)^{2}log(b, x)ln^{2}(b)(b)ln^{3}(log_{a}^{x})ln^{2}(a)} + \frac{2*-3(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{3}{\left(log(a, x)^{2}log(b, x)ln(b)ln^{4}(log_{a}^{x})(log_{a}^{x})ln^{2}(a)} + \frac{2*-2*0}{x^{3}{\left(log(a, x)^{2}log(b, x)ln(b)ln^{3}(log_{a}^{x})ln^{3}(a)(a)} + \frac{3*-3}{x^{4}log(b, x)log(a, x)ln(b)ln^{2}(log_{a}^{x})ln(a)} + \frac{3(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{{\left(log(b, x)^{2}(ln(b))})}{x^{3}log(a, x)ln(b)ln^{2}(log_{a}^{x})ln(a)} + \frac{3(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{2}(ln(a))})}{x^{3}log(b, x)ln(b)ln^{2}(log_{a}^{x})ln(a)} + \frac{3*-0}{x^{3}log(b, x)log(a, x)ln^{2}(b)(b)ln^{2}(log_{a}^{x})ln(a)} + \frac{3*-2(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{3}log(b, x)log(a, x)ln(b)ln^{3}(log_{a}^{x})(log_{a}^{x})ln(a)} + \frac{3*-0}{x^{3}log(b, x)log(a, x)ln(b)ln^{2}(log_{a}^{x})ln^{2}(a)(a)} + \frac{-3}{x^{4}{\left(log(b, x)^{2}log(a, x)ln^{2}(b)ln^{2}(log_{a}^{x})ln(a)} + \frac{(\frac{-2(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{{\left(log(b, x)^{3}(ln(b))})}{x^{3}log(a, x)ln^{2}(b)ln^{2}(log_{a}^{x})ln(a)} + \frac{(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{2}(ln(a))})}{x^{3}{\left(log(b, x)^{2}ln^{2}(b)ln^{2}(log_{a}^{x})ln(a)} + \frac{-2*0}{x^{3}{\left(log(b, x)^{2}log(a, x)ln^{3}(b)(b)ln^{2}(log_{a}^{x})ln(a)} + \frac{-2(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{3}{\left(log(b, x)^{2}log(a, x)ln^{2}(b)ln^{3}(log_{a}^{x})(log_{a}^{x})ln(a)} + \frac{-0}{x^{3}{\left(log(b, x)^{2}log(a, x)ln^{2}(b)ln^{2}(log_{a}^{x})ln^{2}(a)(a)} + \frac{2*-3}{x^{4}{\left(log(a, x)^{2}log(b, x)ln(b)ln^{2}(a)ln^{3}(log_{a}^{x})} + \frac{2(\frac{-2(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{3}(ln(a))})}{x^{3}log(b, x)ln(b)ln^{2}(a)ln^{3}(log_{a}^{x})} + \frac{2(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{{\left(log(b, x)^{2}(ln(b))})}{x^{3}{\left(log(a, x)^{2}ln(b)ln^{2}(a)ln^{3}(log_{a}^{x})} + \frac{2*-0}{x^{3}{\left(log(a, x)^{2}log(b, x)ln^{2}(b)(b)ln^{2}(a)ln^{3}(log_{a}^{x})} + \frac{2*-2*0}{x^{3}{\left(log(a, x)^{2}log(b, x)ln(b)ln^{3}(a)(a)ln^{3}(log_{a}^{x})} + \frac{2*-3(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{3}{\left(log(a, x)^{2}log(b, x)ln(b)ln^{2}(a)ln^{4}(log_{a}^{x})(log_{a}^{x})} + \frac{-3}{x^{4}log(b, x){\left(log(a, x)^{2}ln(b)ln^{2}(log_{a}^{x})ln^{2}(a)} + \frac{(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{{\left(log(b, x)^{2}(ln(b))})}{x^{3}{\left(log(a, x)^{2}ln(b)ln^{2}(log_{a}^{x})ln^{2}(a)} + \frac{(\frac{-2(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{3}(ln(a))})}{x^{3}log(b, x)ln(b)ln^{2}(log_{a}^{x})ln^{2}(a)} + \frac{-0}{x^{3}log(b, x){\left(log(a, x)^{2}ln^{2}(b)(b)ln^{2}(log_{a}^{x})ln^{2}(a)} + \frac{-2(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{3}log(b, x){\left(log(a, x)^{2}ln(b)ln^{3}(log_{a}^{x})(log_{a}^{x})ln^{2}(a)} + \frac{-2*0}{x^{3}log(b, x){\left(log(a, x)^{2}ln(b)ln^{2}(log_{a}^{x})ln^{3}(a)(a)} + \frac{2*-3}{x^{4}log(b, x){\left(log(a, x)^{2}ln(b)ln^{3}(log_{a}^{x})ln^{2}(a)} + \frac{2(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{{\left(log(b, x)^{2}(ln(b))})}{x^{3}{\left(log(a, x)^{2}ln(b)ln^{3}(log_{a}^{x})ln^{2}(a)} + \frac{2(\frac{-2(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{3}(ln(a))})}{x^{3}log(b, x)ln(b)ln^{3}(log_{a}^{x})ln^{2}(a)} + \frac{2*-0}{x^{3}log(b, x){\left(log(a, x)^{2}ln^{2}(b)(b)ln^{3}(log_{a}^{x})ln^{2}(a)} + \frac{2*-3(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{3}log(b, x){\left(log(a, x)^{2}ln(b)ln^{4}(log_{a}^{x})(log_{a}^{x})ln^{2}(a)} + \frac{2*-2*0}{x^{3}log(b, x){\left(log(a, x)^{2}ln(b)ln^{3}(log_{a}^{x})ln^{3}(a)(a)} - \frac{3*-3log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{2}ln^{2}(a)ln(log_{a}^{x})} - \frac{3(\frac{-2(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{3}(ln(a))})log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}ln^{2}(a)ln(log_{a}^{x})} - \frac{3(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{(ln(b))}))}{(log_{b}^{x})} - \frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))}))log_{log_{a}^{x}}^{log_{b}^{x}}}{(log_{a}^{x})})}{(ln(log_{a}^{x}))})}{x^{3}{\left(log(a, x)^{2}ln^{2}(a)ln(log_{a}^{x})} - \frac{3log_{log_{a}^{x}}^{log_{b}^{x}}*-2*0}{x^{3}{\left(log(a, x)^{2}ln^{3}(a)(a)ln(log_{a}^{x})} - \frac{3log_{log_{a}^{x}}^{log_{b}^{x}}*-(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{3}{\left(log(a, x)^{2}ln^{2}(a)ln^{2}(log_{a}^{x})(log_{a}^{x})} - \frac{-3log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{3}ln^{3}(a)ln^{2}(log_{a}^{x})} - \frac{(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{(ln(b))}))}{(log_{b}^{x})} - \frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))}))log_{log_{a}^{x}}^{log_{b}^{x}}}{(log_{a}^{x})})}{(ln(log_{a}^{x}))})}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln^{2}(log_{a}^{x})} - \frac{log_{log_{a}^{x}}^{log_{b}^{x}}(\frac{-3(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{4}(ln(a))})}{x^{3}ln^{3}(a)ln^{2}(log_{a}^{x})} - \frac{log_{log_{a}^{x}}^{log_{b}^{x}}*-3*0}{x^{3}{\left(log(a, x)^{3}ln^{4}(a)(a)ln^{2}(log_{a}^{x})} - \frac{log_{log_{a}^{x}}^{log_{b}^{x}}*-2(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln^{3}(log_{a}^{x})(log_{a}^{x})} - \frac{5*-3log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{3}ln^{3}(a)ln^{2}(log_{a}^{x})} - \frac{5(\frac{-3(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{4}(ln(a))})log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}ln^{3}(a)ln^{2}(log_{a}^{x})} - \frac{5(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{(ln(b))}))}{(log_{b}^{x})} - \frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))}))log_{log_{a}^{x}}^{log_{b}^{x}}}{(log_{a}^{x})})}{(ln(log_{a}^{x}))})}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln^{2}(log_{a}^{x})} - \frac{5log_{log_{a}^{x}}^{log_{b}^{x}}*-3*0}{x^{3}{\left(log(a, x)^{3}ln^{4}(a)(a)ln^{2}(log_{a}^{x})} - \frac{5log_{log_{a}^{x}}^{log_{b}^{x}}*-2(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln^{3}(log_{a}^{x})(log_{a}^{x})} - \frac{2*-3log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}log(a, x)ln(a)ln(log_{a}^{x})} - \frac{2(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{2}(ln(a))})log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}ln(a)ln(log_{a}^{x})} - \frac{2(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{(ln(b))}))}{(log_{b}^{x})} - \frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))}))log_{log_{a}^{x}}^{log_{b}^{x}}}{(log_{a}^{x})})}{(ln(log_{a}^{x}))})}{x^{3}log(a, x)ln(a)ln(log_{a}^{x})} - \frac{2log_{log_{a}^{x}}^{log_{b}^{x}}*-0}{x^{3}log(a, x)ln^{2}(a)(a)ln(log_{a}^{x})} - \frac{2log_{log_{a}^{x}}^{log_{b}^{x}}*-(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{3}log(a, x)ln(a)ln^{2}(log_{a}^{x})(log_{a}^{x})} - \frac{3*-3log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{2}ln^{2}(a)ln^{2}(log_{a}^{x})} - \frac{3(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{(ln(b))}))}{(log_{b}^{x})} - \frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))}))log_{log_{a}^{x}}^{log_{b}^{x}}}{(log_{a}^{x})})}{(ln(log_{a}^{x}))})}{x^{3}{\left(log(a, x)^{2}ln^{2}(a)ln^{2}(log_{a}^{x})} - \frac{3log_{log_{a}^{x}}^{log_{b}^{x}}(\frac{-2(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{3}(ln(a))})}{x^{3}ln^{2}(a)ln^{2}(log_{a}^{x})} - \frac{3log_{log_{a}^{x}}^{log_{b}^{x}}*-2*0}{x^{3}{\left(log(a, x)^{2}ln^{3}(a)(a)ln^{2}(log_{a}^{x})} - \frac{3log_{log_{a}^{x}}^{log_{b}^{x}}*-2(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{3}{\left(log(a, x)^{2}ln^{2}(a)ln^{3}(log_{a}^{x})(log_{a}^{x})} - \frac{3*-3log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{2}ln^{2}(a)ln^{2}(log_{a}^{x})} - \frac{3(\frac{-2(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{3}(ln(a))})log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}ln^{2}(a)ln^{2}(log_{a}^{x})} - \frac{3(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{(ln(b))}))}{(log_{b}^{x})} - \frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))}))log_{log_{a}^{x}}^{log_{b}^{x}}}{(log_{a}^{x})})}{(ln(log_{a}^{x}))})}{x^{3}{\left(log(a, x)^{2}ln^{2}(a)ln^{2}(log_{a}^{x})} - \frac{3log_{log_{a}^{x}}^{log_{b}^{x}}*-2*0}{x^{3}{\left(log(a, x)^{2}ln^{3}(a)(a)ln^{2}(log_{a}^{x})} - \frac{3log_{log_{a}^{x}}^{log_{b}^{x}}*-2(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{3}{\left(log(a, x)^{2}ln^{2}(a)ln^{3}(log_{a}^{x})(log_{a}^{x})} - \frac{2*-3log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{3}ln^{3}(a)ln(log_{a}^{x})} - \frac{2(\frac{-3(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{4}(ln(a))})log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}ln^{3}(a)ln(log_{a}^{x})} - \frac{2(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{(ln(b))}))}{(log_{b}^{x})} - \frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))}))log_{log_{a}^{x}}^{log_{b}^{x}}}{(log_{a}^{x})})}{(ln(log_{a}^{x}))})}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln(log_{a}^{x})} - \frac{2log_{log_{a}^{x}}^{log_{b}^{x}}*-3*0}{x^{3}{\left(log(a, x)^{3}ln^{4}(a)(a)ln(log_{a}^{x})} - \frac{2log_{log_{a}^{x}}^{log_{b}^{x}}*-(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln^{2}(log_{a}^{x})(log_{a}^{x})} - \frac{2*-3log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{3}ln^{3}(a)ln^{3}(log_{a}^{x})} - \frac{2(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{(ln(b))}))}{(log_{b}^{x})} - \frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))}))log_{log_{a}^{x}}^{log_{b}^{x}}}{(log_{a}^{x})})}{(ln(log_{a}^{x}))})}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln^{3}(log_{a}^{x})} - \frac{2log_{log_{a}^{x}}^{log_{b}^{x}}(\frac{-3(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{4}(ln(a))})}{x^{3}ln^{3}(a)ln^{3}(log_{a}^{x})} - \frac{2log_{log_{a}^{x}}^{log_{b}^{x}}*-3*0}{x^{3}{\left(log(a, x)^{3}ln^{4}(a)(a)ln^{3}(log_{a}^{x})} - \frac{2log_{log_{a}^{x}}^{log_{b}^{x}}*-3(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln^{4}(log_{a}^{x})(log_{a}^{x})} - \frac{4*-3log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{3}ln^{3}(a)ln^{3}(log_{a}^{x})} - \frac{4(\frac{-3(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{4}(ln(a))})log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}ln^{3}(a)ln^{3}(log_{a}^{x})} - \frac{4(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{(ln(b))}))}{(log_{b}^{x})} - \frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))}))log_{log_{a}^{x}}^{log_{b}^{x}}}{(log_{a}^{x})})}{(ln(log_{a}^{x}))})}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln^{3}(log_{a}^{x})} - \frac{4log_{log_{a}^{x}}^{log_{b}^{x}}*-3*0}{x^{3}{\left(log(a, x)^{3}ln^{4}(a)(a)ln^{3}(log_{a}^{x})} - \frac{4log_{log_{a}^{x}}^{log_{b}^{x}}*-3(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln^{4}(log_{a}^{x})(log_{a}^{x})}\\=&\frac{-6}{x^{4}log(b, x)ln(b)ln(log_{a}^{x})} - \frac{11}{x^{4}{\left(log(b, x)^{2}ln^{2}(b)ln(log_{a}^{x})} - \frac{11}{x^{4}log(b, x)log(a, x)ln(b)ln(a)ln^{2}(log_{a}^{x})} - \frac{12}{x^{4}{\left(log(b, x)^{3}ln^{3}(b)ln(log_{a}^{x})} - \frac{12}{x^{4}{\left(log(b, x)^{2}log(a, x)ln^{2}(b)ln(a)ln^{2}(log_{a}^{x})} - \frac{12}{x^{4}{\left(log(a, x)^{2}log(b, x)ln^{2}(a)ln(b)ln^{2}(log_{a}^{x})} - \frac{12}{x^{4}{\left(log(a, x)^{2}log(b, x)ln(b)ln^{3}(log_{a}^{x})ln^{2}(a)} - \frac{6}{x^{4}{\left(log(b, x)^{4}ln^{4}(b)ln(log_{a}^{x})} - \frac{6}{x^{4}{\left(log(b, x)^{3}log(a, x)ln^{3}(b)ln(a)ln^{2}(log_{a}^{x})} - \frac{3}{x^{4}{\left(log(a, x)^{2}{\left(log(b, x)^{2}ln^{2}(a)ln^{2}(b)ln^{2}(log_{a}^{x})} - \frac{4}{x^{4}{\left(log(a, x)^{2}{\left(log(b, x)^{2}ln^{2}(b)ln^{3}(log_{a}^{x})ln^{2}(a)} - \frac{6}{x^{4}{\left(log(a, x)^{3}log(b, x)ln^{3}(a)ln(b)ln^{2}(log_{a}^{x})} - \frac{2}{x^{4}{\left(log(b, x)^{2}{\left(log(a, x)^{2}ln^{2}(b)ln^{2}(a)ln^{2}(log_{a}^{x})} - \frac{4}{x^{4}{\left(log(a, x)^{3}log(b, x)ln^{3}(a)ln^{3}(log_{a}^{x})ln(b)} - \frac{12}{x^{4}{\left(log(a, x)^{3}log(b, x)ln^{3}(a)ln(b)ln^{3}(log_{a}^{x})} - \frac{4}{x^{4}{\left(log(b, x)^{2}{\left(log(a, x)^{2}ln^{2}(b)ln^{3}(log_{a}^{x})ln^{2}(a)} - \frac{12}{x^{4}{\left(log(a, x)^{3}log(b, x)ln(b)ln^{3}(a)ln^{4}(log_{a}^{x})} - \frac{11}{x^{4}log(b, x)log(a, x)ln(b)ln^{2}(log_{a}^{x})ln(a)} - \frac{6}{x^{4}{\left(log(b, x)^{2}log(a, x)ln^{2}(b)ln^{2}(log_{a}^{x})ln(a)} - \frac{12}{x^{4}{\left(log(a, x)^{2}log(b, x)ln(b)ln^{2}(a)ln^{3}(log_{a}^{x})} - \frac{2}{x^{4}{\left(log(b, x)^{3}log(a, x)ln^{3}(b)ln^{2}(log_{a}^{x})ln(a)} - \frac{2}{x^{4}{\left(log(a, x)^{2}{\left(log(b, x)^{2}ln^{2}(b)ln^{2}(a)ln^{3}(log_{a}^{x})} - \frac{2}{x^{4}{\left(log(b, x)^{2}{\left(log(a, x)^{2}ln^{2}(b)ln^{2}(a)ln^{3}(log_{a}^{x})} - \frac{6}{x^{4}{\left(log(a, x)^{3}log(b, x)ln(b)ln^{4}(log_{a}^{x})ln^{3}(a)} - \frac{6}{x^{4}log(b, x){\left(log(a, x)^{2}ln(b)ln^{2}(log_{a}^{x})ln^{2}(a)} - \frac{1}{x^{4}{\left(log(b, x)^{2}{\left(log(a, x)^{2}ln^{2}(b)ln^{2}(log_{a}^{x})ln^{2}(a)} - \frac{2}{x^{4}{\left(log(a, x)^{3}log(b, x)ln(b)ln^{3}(a)ln^{3}(log_{a}^{x})} - \frac{12}{x^{4}log(b, x){\left(log(a, x)^{2}ln(b)ln^{3}(log_{a}^{x})ln^{2}(a)} - \frac{6}{x^{4}log(b, x){\left(log(a, x)^{3}ln(b)ln^{3}(log_{a}^{x})ln^{3}(a)} - \frac{2}{x^{4}log(b, x){\left(log(a, x)^{3}ln(b)ln^{2}(log_{a}^{x})ln^{3}(a)} - \frac{6}{x^{4}log(b, x){\left(log(a, x)^{3}ln(b)ln^{4}(log_{a}^{x})ln^{3}(a)} + \frac{30log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{3}ln^{3}(a)ln^{2}(log_{a}^{x})} + \frac{11log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{2}ln^{2}(a)ln(log_{a}^{x})} + \frac{6log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{4}ln^{4}(a)ln^{3}(log_{a}^{x})} + \frac{20log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{4}ln^{4}(a)ln^{2}(log_{a}^{x})} + \frac{30log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{4}ln^{4}(a)ln^{3}(log_{a}^{x})} + \frac{6log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}log(a, x)ln(a)ln(log_{a}^{x})} + \frac{11log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{2}ln^{2}(a)ln^{2}(log_{a}^{x})} + \frac{11log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{2}ln^{2}(a)ln^{2}(log_{a}^{x})} + \frac{12log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{3}ln^{3}(a)ln^{3}(log_{a}^{x})} + \frac{24log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{3}ln^{3}(a)ln^{3}(log_{a}^{x})} + \frac{6log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{4}ln^{4}(a)ln(log_{a}^{x})} + \frac{12log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{3}ln^{3}(a)ln(log_{a}^{x})} + \frac{2log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{4}ln^{4}(a)ln^{2}(log_{a}^{x})} + \frac{6log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{3}ln^{3}(a)ln^{2}(log_{a}^{x})} + \frac{6log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{4}ln^{4}(a)ln^{4}(log_{a}^{x})} + \frac{18log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{4}ln^{4}(a)ln^{4}(log_{a}^{x})}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ log_{log_{a}^{x}}^{log_{b}^{x}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( log_{log_{a}^{x}}^{log_{b}^{x}}\right)}{dx}\\=&(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{(ln(b))}))}{(log_{b}^{x})} - \frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))}))log_{log_{a}^{x}}^{log_{b}^{x}}}{(log_{a}^{x})})}{(ln(log_{a}^{x}))})\\=&\frac{1}{xlog(b, x)ln(b)ln(log_{a}^{x})} - \frac{log_{log_{a}^{x}}^{log_{b}^{x}}}{xlog(a, x)ln(a)ln(log_{a}^{x})}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{1}{xlog(b, x)ln(b)ln(log_{a}^{x})} - \frac{log_{log_{a}^{x}}^{log_{b}^{x}}}{xlog(a, x)ln(a)ln(log_{a}^{x})}\right)}{dx}\\=&\frac{-1}{x^{2}log(b, x)ln(b)ln(log_{a}^{x})} + \frac{(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{{\left(log(b, x)^{2}(ln(b))})}{xln(b)ln(log_{a}^{x})} + \frac{-0}{xlog(b, x)ln^{2}(b)(b)ln(log_{a}^{x})} + \frac{-(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{xlog(b, x)ln(b)ln^{2}(log_{a}^{x})(log_{a}^{x})} - \frac{-log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{2}log(a, x)ln(a)ln(log_{a}^{x})} - \frac{(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{2}(ln(a))})log_{log_{a}^{x}}^{log_{b}^{x}}}{xln(a)ln(log_{a}^{x})} - \frac{(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{(ln(b))}))}{(log_{b}^{x})} - \frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))}))log_{log_{a}^{x}}^{log_{b}^{x}}}{(log_{a}^{x})})}{(ln(log_{a}^{x}))})}{xlog(a, x)ln(a)ln(log_{a}^{x})} - \frac{log_{log_{a}^{x}}^{log_{b}^{x}}*-0}{xlog(a, x)ln^{2}(a)(a)ln(log_{a}^{x})} - \frac{log_{log_{a}^{x}}^{log_{b}^{x}}*-(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{xlog(a, x)ln(a)ln^{2}(log_{a}^{x})(log_{a}^{x})}\\=&\frac{-1}{x^{2}log(b, x)ln(b)ln(log_{a}^{x})} - \frac{1}{x^{2}{\left(log(b, x)^{2}ln^{2}(b)ln(log_{a}^{x})} - \frac{1}{x^{2}log(b, x)log(a, x)ln(b)ln(a)ln^{2}(log_{a}^{x})} - \frac{1}{x^{2}log(b, x)log(a, x)ln(b)ln^{2}(log_{a}^{x})ln(a)} + \frac{log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{2}{\left(log(a, x)^{2}ln^{2}(a)ln(log_{a}^{x})} + \frac{log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{2}log(a, x)ln(a)ln(log_{a}^{x})} + \frac{log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{2}{\left(log(a, x)^{2}ln^{2}(a)ln^{2}(log_{a}^{x})} + \frac{log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{2}{\left(log(a, x)^{2}ln^{2}(a)ln^{2}(log_{a}^{x})}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-1}{x^{2}log(b, x)ln(b)ln(log_{a}^{x})} - \frac{1}{x^{2}{\left(log(b, x)^{2}ln^{2}(b)ln(log_{a}^{x})} - \frac{1}{x^{2}log(b, x)log(a, x)ln(b)ln(a)ln^{2}(log_{a}^{x})} - \frac{1}{x^{2}log(b, x)log(a, x)ln(b)ln^{2}(log_{a}^{x})ln(a)} + \frac{log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{2}{\left(log(a, x)^{2}ln^{2}(a)ln(log_{a}^{x})} + \frac{log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{2}log(a, x)ln(a)ln(log_{a}^{x})} + \frac{log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{2}{\left(log(a, x)^{2}ln^{2}(a)ln^{2}(log_{a}^{x})} + \frac{log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{2}{\left(log(a, x)^{2}ln^{2}(a)ln^{2}(log_{a}^{x})}\right)}{dx}\\=&\frac{--2}{x^{3}log(b, x)ln(b)ln(log_{a}^{x})} - \frac{(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{{\left(log(b, x)^{2}(ln(b))})}{x^{2}ln(b)ln(log_{a}^{x})} - \frac{-0}{x^{2}log(b, x)ln^{2}(b)(b)ln(log_{a}^{x})} - \frac{-(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{2}log(b, x)ln(b)ln^{2}(log_{a}^{x})(log_{a}^{x})} - \frac{-2}{x^{3}{\left(log(b, x)^{2}ln^{2}(b)ln(log_{a}^{x})} - \frac{(\frac{-2(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{{\left(log(b, x)^{3}(ln(b))})}{x^{2}ln^{2}(b)ln(log_{a}^{x})} - \frac{-2*0}{x^{2}{\left(log(b, x)^{2}ln^{3}(b)(b)ln(log_{a}^{x})} - \frac{-(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{2}{\left(log(b, x)^{2}ln^{2}(b)ln^{2}(log_{a}^{x})(log_{a}^{x})} - \frac{-2}{x^{3}log(b, x)log(a, x)ln(b)ln(a)ln^{2}(log_{a}^{x})} - \frac{(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{{\left(log(b, x)^{2}(ln(b))})}{x^{2}log(a, x)ln(b)ln(a)ln^{2}(log_{a}^{x})} - \frac{(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{2}(ln(a))})}{x^{2}log(b, x)ln(b)ln(a)ln^{2}(log_{a}^{x})} - \frac{-0}{x^{2}log(b, x)log(a, x)ln^{2}(b)(b)ln(a)ln^{2}(log_{a}^{x})} - \frac{-0}{x^{2}log(b, x)log(a, x)ln(b)ln^{2}(a)(a)ln^{2}(log_{a}^{x})} - \frac{-2(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{2}log(b, x)log(a, x)ln(b)ln(a)ln^{3}(log_{a}^{x})(log_{a}^{x})} - \frac{-2}{x^{3}log(b, x)log(a, x)ln(b)ln^{2}(log_{a}^{x})ln(a)} - \frac{(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{{\left(log(b, x)^{2}(ln(b))})}{x^{2}log(a, x)ln(b)ln^{2}(log_{a}^{x})ln(a)} - \frac{(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{2}(ln(a))})}{x^{2}log(b, x)ln(b)ln^{2}(log_{a}^{x})ln(a)} - \frac{-0}{x^{2}log(b, x)log(a, x)ln^{2}(b)(b)ln^{2}(log_{a}^{x})ln(a)} - \frac{-2(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{2}log(b, x)log(a, x)ln(b)ln^{3}(log_{a}^{x})(log_{a}^{x})ln(a)} - \frac{-0}{x^{2}log(b, x)log(a, x)ln(b)ln^{2}(log_{a}^{x})ln^{2}(a)(a)} + \frac{-2log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}{\left(log(a, x)^{2}ln^{2}(a)ln(log_{a}^{x})} + \frac{(\frac{-2(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{3}(ln(a))})log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{2}ln^{2}(a)ln(log_{a}^{x})} + \frac{(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{(ln(b))}))}{(log_{b}^{x})} - \frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))}))log_{log_{a}^{x}}^{log_{b}^{x}}}{(log_{a}^{x})})}{(ln(log_{a}^{x}))})}{x^{2}{\left(log(a, x)^{2}ln^{2}(a)ln(log_{a}^{x})} + \frac{log_{log_{a}^{x}}^{log_{b}^{x}}*-2*0}{x^{2}{\left(log(a, x)^{2}ln^{3}(a)(a)ln(log_{a}^{x})} + \frac{log_{log_{a}^{x}}^{log_{b}^{x}}*-(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{2}{\left(log(a, x)^{2}ln^{2}(a)ln^{2}(log_{a}^{x})(log_{a}^{x})} + \frac{-2log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}log(a, x)ln(a)ln(log_{a}^{x})} + \frac{(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{2}(ln(a))})log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{2}ln(a)ln(log_{a}^{x})} + \frac{(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{(ln(b))}))}{(log_{b}^{x})} - \frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))}))log_{log_{a}^{x}}^{log_{b}^{x}}}{(log_{a}^{x})})}{(ln(log_{a}^{x}))})}{x^{2}log(a, x)ln(a)ln(log_{a}^{x})} + \frac{log_{log_{a}^{x}}^{log_{b}^{x}}*-0}{x^{2}log(a, x)ln^{2}(a)(a)ln(log_{a}^{x})} + \frac{log_{log_{a}^{x}}^{log_{b}^{x}}*-(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{2}log(a, x)ln(a)ln^{2}(log_{a}^{x})(log_{a}^{x})} + \frac{-2log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}{\left(log(a, x)^{2}ln^{2}(a)ln^{2}(log_{a}^{x})} + \frac{(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{(ln(b))}))}{(log_{b}^{x})} - \frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))}))log_{log_{a}^{x}}^{log_{b}^{x}}}{(log_{a}^{x})})}{(ln(log_{a}^{x}))})}{x^{2}{\left(log(a, x)^{2}ln^{2}(a)ln^{2}(log_{a}^{x})} + \frac{log_{log_{a}^{x}}^{log_{b}^{x}}(\frac{-2(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{3}(ln(a))})}{x^{2}ln^{2}(a)ln^{2}(log_{a}^{x})} + \frac{log_{log_{a}^{x}}^{log_{b}^{x}}*-2*0}{x^{2}{\left(log(a, x)^{2}ln^{3}(a)(a)ln^{2}(log_{a}^{x})} + \frac{log_{log_{a}^{x}}^{log_{b}^{x}}*-2(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{2}{\left(log(a, x)^{2}ln^{2}(a)ln^{3}(log_{a}^{x})(log_{a}^{x})} + \frac{-2log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}{\left(log(a, x)^{2}ln^{2}(a)ln^{2}(log_{a}^{x})} + \frac{(\frac{-2(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{3}(ln(a))})log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{2}ln^{2}(a)ln^{2}(log_{a}^{x})} + \frac{(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{(ln(b))}))}{(log_{b}^{x})} - \frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))}))log_{log_{a}^{x}}^{log_{b}^{x}}}{(log_{a}^{x})})}{(ln(log_{a}^{x}))})}{x^{2}{\left(log(a, x)^{2}ln^{2}(a)ln^{2}(log_{a}^{x})} + \frac{log_{log_{a}^{x}}^{log_{b}^{x}}*-2*0}{x^{2}{\left(log(a, x)^{2}ln^{3}(a)(a)ln^{2}(log_{a}^{x})} + \frac{log_{log_{a}^{x}}^{log_{b}^{x}}*-2(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{2}{\left(log(a, x)^{2}ln^{2}(a)ln^{3}(log_{a}^{x})(log_{a}^{x})}\\=&\frac{2}{x^{3}log(b, x)ln(b)ln(log_{a}^{x})} + \frac{3}{x^{3}{\left(log(b, x)^{2}ln^{2}(b)ln(log_{a}^{x})} + \frac{3}{x^{3}log(b, x)log(a, x)ln(b)ln(a)ln^{2}(log_{a}^{x})} + \frac{2}{x^{3}{\left(log(b, x)^{3}ln^{3}(b)ln(log_{a}^{x})} + \frac{2}{x^{3}{\left(log(b, x)^{2}log(a, x)ln^{2}(b)ln(a)ln^{2}(log_{a}^{x})} + \frac{2}{x^{3}{\left(log(a, x)^{2}log(b, x)ln^{2}(a)ln(b)ln^{2}(log_{a}^{x})} + \frac{2}{x^{3}{\left(log(a, x)^{2}log(b, x)ln(b)ln^{3}(log_{a}^{x})ln^{2}(a)} + \frac{3}{x^{3}log(b, x)log(a, x)ln(b)ln^{2}(log_{a}^{x})ln(a)} + \frac{1}{x^{3}{\left(log(b, x)^{2}log(a, x)ln^{2}(b)ln^{2}(log_{a}^{x})ln(a)} + \frac{2}{x^{3}{\left(log(a, x)^{2}log(b, x)ln(b)ln^{2}(a)ln^{3}(log_{a}^{x})} + \frac{1}{x^{3}log(b, x){\left(log(a, x)^{2}ln(b)ln^{2}(log_{a}^{x})ln^{2}(a)} + \frac{2}{x^{3}log(b, x){\left(log(a, x)^{2}ln(b)ln^{3}(log_{a}^{x})ln^{2}(a)} - \frac{3log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}{\left(log(a, x)^{2}ln^{2}(a)ln(log_{a}^{x})} - \frac{log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln^{2}(log_{a}^{x})} - \frac{5log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln^{2}(log_{a}^{x})} - \frac{2log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}log(a, x)ln(a)ln(log_{a}^{x})} - \frac{3log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}{\left(log(a, x)^{2}ln^{2}(a)ln^{2}(log_{a}^{x})} - \frac{3log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}{\left(log(a, x)^{2}ln^{2}(a)ln^{2}(log_{a}^{x})} - \frac{2log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln(log_{a}^{x})} - \frac{2log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln^{3}(log_{a}^{x})} - \frac{4log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln^{3}(log_{a}^{x})}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{2}{x^{3}log(b, x)ln(b)ln(log_{a}^{x})} + \frac{3}{x^{3}{\left(log(b, x)^{2}ln^{2}(b)ln(log_{a}^{x})} + \frac{3}{x^{3}log(b, x)log(a, x)ln(b)ln(a)ln^{2}(log_{a}^{x})} + \frac{2}{x^{3}{\left(log(b, x)^{3}ln^{3}(b)ln(log_{a}^{x})} + \frac{2}{x^{3}{\left(log(b, x)^{2}log(a, x)ln^{2}(b)ln(a)ln^{2}(log_{a}^{x})} + \frac{2}{x^{3}{\left(log(a, x)^{2}log(b, x)ln^{2}(a)ln(b)ln^{2}(log_{a}^{x})} + \frac{2}{x^{3}{\left(log(a, x)^{2}log(b, x)ln(b)ln^{3}(log_{a}^{x})ln^{2}(a)} + \frac{3}{x^{3}log(b, x)log(a, x)ln(b)ln^{2}(log_{a}^{x})ln(a)} + \frac{1}{x^{3}{\left(log(b, x)^{2}log(a, x)ln^{2}(b)ln^{2}(log_{a}^{x})ln(a)} + \frac{2}{x^{3}{\left(log(a, x)^{2}log(b, x)ln(b)ln^{2}(a)ln^{3}(log_{a}^{x})} + \frac{1}{x^{3}log(b, x){\left(log(a, x)^{2}ln(b)ln^{2}(log_{a}^{x})ln^{2}(a)} + \frac{2}{x^{3}log(b, x){\left(log(a, x)^{2}ln(b)ln^{3}(log_{a}^{x})ln^{2}(a)} - \frac{3log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}{\left(log(a, x)^{2}ln^{2}(a)ln(log_{a}^{x})} - \frac{log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln^{2}(log_{a}^{x})} - \frac{5log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln^{2}(log_{a}^{x})} - \frac{2log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}log(a, x)ln(a)ln(log_{a}^{x})} - \frac{3log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}{\left(log(a, x)^{2}ln^{2}(a)ln^{2}(log_{a}^{x})} - \frac{3log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}{\left(log(a, x)^{2}ln^{2}(a)ln^{2}(log_{a}^{x})} - \frac{2log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln(log_{a}^{x})} - \frac{2log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln^{3}(log_{a}^{x})} - \frac{4log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln^{3}(log_{a}^{x})}\right)}{dx}\\=&\frac{2*-3}{x^{4}log(b, x)ln(b)ln(log_{a}^{x})} + \frac{2(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{{\left(log(b, x)^{2}(ln(b))})}{x^{3}ln(b)ln(log_{a}^{x})} + \frac{2*-0}{x^{3}log(b, x)ln^{2}(b)(b)ln(log_{a}^{x})} + \frac{2*-(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{3}log(b, x)ln(b)ln^{2}(log_{a}^{x})(log_{a}^{x})} + \frac{3*-3}{x^{4}{\left(log(b, x)^{2}ln^{2}(b)ln(log_{a}^{x})} + \frac{3(\frac{-2(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{{\left(log(b, x)^{3}(ln(b))})}{x^{3}ln^{2}(b)ln(log_{a}^{x})} + \frac{3*-2*0}{x^{3}{\left(log(b, x)^{2}ln^{3}(b)(b)ln(log_{a}^{x})} + \frac{3*-(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{3}{\left(log(b, x)^{2}ln^{2}(b)ln^{2}(log_{a}^{x})(log_{a}^{x})} + \frac{3*-3}{x^{4}log(b, x)log(a, x)ln(b)ln(a)ln^{2}(log_{a}^{x})} + \frac{3(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{{\left(log(b, x)^{2}(ln(b))})}{x^{3}log(a, x)ln(b)ln(a)ln^{2}(log_{a}^{x})} + \frac{3(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{2}(ln(a))})}{x^{3}log(b, x)ln(b)ln(a)ln^{2}(log_{a}^{x})} + \frac{3*-0}{x^{3}log(b, x)log(a, x)ln^{2}(b)(b)ln(a)ln^{2}(log_{a}^{x})} + \frac{3*-0}{x^{3}log(b, x)log(a, x)ln(b)ln^{2}(a)(a)ln^{2}(log_{a}^{x})} + \frac{3*-2(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{3}log(b, x)log(a, x)ln(b)ln(a)ln^{3}(log_{a}^{x})(log_{a}^{x})} + \frac{2*-3}{x^{4}{\left(log(b, x)^{3}ln^{3}(b)ln(log_{a}^{x})} + \frac{2(\frac{-3(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{{\left(log(b, x)^{4}(ln(b))})}{x^{3}ln^{3}(b)ln(log_{a}^{x})} + \frac{2*-3*0}{x^{3}{\left(log(b, x)^{3}ln^{4}(b)(b)ln(log_{a}^{x})} + \frac{2*-(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{3}{\left(log(b, x)^{3}ln^{3}(b)ln^{2}(log_{a}^{x})(log_{a}^{x})} + \frac{2*-3}{x^{4}{\left(log(b, x)^{2}log(a, x)ln^{2}(b)ln(a)ln^{2}(log_{a}^{x})} + \frac{2(\frac{-2(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{{\left(log(b, x)^{3}(ln(b))})}{x^{3}log(a, x)ln^{2}(b)ln(a)ln^{2}(log_{a}^{x})} + \frac{2(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{2}(ln(a))})}{x^{3}{\left(log(b, x)^{2}ln^{2}(b)ln(a)ln^{2}(log_{a}^{x})} + \frac{2*-2*0}{x^{3}{\left(log(b, x)^{2}log(a, x)ln^{3}(b)(b)ln(a)ln^{2}(log_{a}^{x})} + \frac{2*-0}{x^{3}{\left(log(b, x)^{2}log(a, x)ln^{2}(b)ln^{2}(a)(a)ln^{2}(log_{a}^{x})} + \frac{2*-2(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{3}{\left(log(b, x)^{2}log(a, x)ln^{2}(b)ln(a)ln^{3}(log_{a}^{x})(log_{a}^{x})} + \frac{2*-3}{x^{4}{\left(log(a, x)^{2}log(b, x)ln^{2}(a)ln(b)ln^{2}(log_{a}^{x})} + \frac{2(\frac{-2(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{3}(ln(a))})}{x^{3}log(b, x)ln^{2}(a)ln(b)ln^{2}(log_{a}^{x})} + \frac{2(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{{\left(log(b, x)^{2}(ln(b))})}{x^{3}{\left(log(a, x)^{2}ln^{2}(a)ln(b)ln^{2}(log_{a}^{x})} + \frac{2*-2*0}{x^{3}{\left(log(a, x)^{2}log(b, x)ln^{3}(a)(a)ln(b)ln^{2}(log_{a}^{x})} + \frac{2*-0}{x^{3}{\left(log(a, x)^{2}log(b, x)ln^{2}(a)ln^{2}(b)(b)ln^{2}(log_{a}^{x})} + \frac{2*-2(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{3}{\left(log(a, x)^{2}log(b, x)ln^{2}(a)ln(b)ln^{3}(log_{a}^{x})(log_{a}^{x})} + \frac{2*-3}{x^{4}{\left(log(a, x)^{2}log(b, x)ln(b)ln^{3}(log_{a}^{x})ln^{2}(a)} + \frac{2(\frac{-2(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{3}(ln(a))})}{x^{3}log(b, x)ln(b)ln^{3}(log_{a}^{x})ln^{2}(a)} + \frac{2(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{{\left(log(b, x)^{2}(ln(b))})}{x^{3}{\left(log(a, x)^{2}ln(b)ln^{3}(log_{a}^{x})ln^{2}(a)} + \frac{2*-0}{x^{3}{\left(log(a, x)^{2}log(b, x)ln^{2}(b)(b)ln^{3}(log_{a}^{x})ln^{2}(a)} + \frac{2*-3(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{3}{\left(log(a, x)^{2}log(b, x)ln(b)ln^{4}(log_{a}^{x})(log_{a}^{x})ln^{2}(a)} + \frac{2*-2*0}{x^{3}{\left(log(a, x)^{2}log(b, x)ln(b)ln^{3}(log_{a}^{x})ln^{3}(a)(a)} + \frac{3*-3}{x^{4}log(b, x)log(a, x)ln(b)ln^{2}(log_{a}^{x})ln(a)} + \frac{3(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{{\left(log(b, x)^{2}(ln(b))})}{x^{3}log(a, x)ln(b)ln^{2}(log_{a}^{x})ln(a)} + \frac{3(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{2}(ln(a))})}{x^{3}log(b, x)ln(b)ln^{2}(log_{a}^{x})ln(a)} + \frac{3*-0}{x^{3}log(b, x)log(a, x)ln^{2}(b)(b)ln^{2}(log_{a}^{x})ln(a)} + \frac{3*-2(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{3}log(b, x)log(a, x)ln(b)ln^{3}(log_{a}^{x})(log_{a}^{x})ln(a)} + \frac{3*-0}{x^{3}log(b, x)log(a, x)ln(b)ln^{2}(log_{a}^{x})ln^{2}(a)(a)} + \frac{-3}{x^{4}{\left(log(b, x)^{2}log(a, x)ln^{2}(b)ln^{2}(log_{a}^{x})ln(a)} + \frac{(\frac{-2(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{{\left(log(b, x)^{3}(ln(b))})}{x^{3}log(a, x)ln^{2}(b)ln^{2}(log_{a}^{x})ln(a)} + \frac{(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{2}(ln(a))})}{x^{3}{\left(log(b, x)^{2}ln^{2}(b)ln^{2}(log_{a}^{x})ln(a)} + \frac{-2*0}{x^{3}{\left(log(b, x)^{2}log(a, x)ln^{3}(b)(b)ln^{2}(log_{a}^{x})ln(a)} + \frac{-2(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{3}{\left(log(b, x)^{2}log(a, x)ln^{2}(b)ln^{3}(log_{a}^{x})(log_{a}^{x})ln(a)} + \frac{-0}{x^{3}{\left(log(b, x)^{2}log(a, x)ln^{2}(b)ln^{2}(log_{a}^{x})ln^{2}(a)(a)} + \frac{2*-3}{x^{4}{\left(log(a, x)^{2}log(b, x)ln(b)ln^{2}(a)ln^{3}(log_{a}^{x})} + \frac{2(\frac{-2(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{3}(ln(a))})}{x^{3}log(b, x)ln(b)ln^{2}(a)ln^{3}(log_{a}^{x})} + \frac{2(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{{\left(log(b, x)^{2}(ln(b))})}{x^{3}{\left(log(a, x)^{2}ln(b)ln^{2}(a)ln^{3}(log_{a}^{x})} + \frac{2*-0}{x^{3}{\left(log(a, x)^{2}log(b, x)ln^{2}(b)(b)ln^{2}(a)ln^{3}(log_{a}^{x})} + \frac{2*-2*0}{x^{3}{\left(log(a, x)^{2}log(b, x)ln(b)ln^{3}(a)(a)ln^{3}(log_{a}^{x})} + \frac{2*-3(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{3}{\left(log(a, x)^{2}log(b, x)ln(b)ln^{2}(a)ln^{4}(log_{a}^{x})(log_{a}^{x})} + \frac{-3}{x^{4}log(b, x){\left(log(a, x)^{2}ln(b)ln^{2}(log_{a}^{x})ln^{2}(a)} + \frac{(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{{\left(log(b, x)^{2}(ln(b))})}{x^{3}{\left(log(a, x)^{2}ln(b)ln^{2}(log_{a}^{x})ln^{2}(a)} + \frac{(\frac{-2(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{3}(ln(a))})}{x^{3}log(b, x)ln(b)ln^{2}(log_{a}^{x})ln^{2}(a)} + \frac{-0}{x^{3}log(b, x){\left(log(a, x)^{2}ln^{2}(b)(b)ln^{2}(log_{a}^{x})ln^{2}(a)} + \frac{-2(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{3}log(b, x){\left(log(a, x)^{2}ln(b)ln^{3}(log_{a}^{x})(log_{a}^{x})ln^{2}(a)} + \frac{-2*0}{x^{3}log(b, x){\left(log(a, x)^{2}ln(b)ln^{2}(log_{a}^{x})ln^{3}(a)(a)} + \frac{2*-3}{x^{4}log(b, x){\left(log(a, x)^{2}ln(b)ln^{3}(log_{a}^{x})ln^{2}(a)} + \frac{2(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{{\left(log(b, x)^{2}(ln(b))})}{x^{3}{\left(log(a, x)^{2}ln(b)ln^{3}(log_{a}^{x})ln^{2}(a)} + \frac{2(\frac{-2(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{3}(ln(a))})}{x^{3}log(b, x)ln(b)ln^{3}(log_{a}^{x})ln^{2}(a)} + \frac{2*-0}{x^{3}log(b, x){\left(log(a, x)^{2}ln^{2}(b)(b)ln^{3}(log_{a}^{x})ln^{2}(a)} + \frac{2*-3(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{3}log(b, x){\left(log(a, x)^{2}ln(b)ln^{4}(log_{a}^{x})(log_{a}^{x})ln^{2}(a)} + \frac{2*-2*0}{x^{3}log(b, x){\left(log(a, x)^{2}ln(b)ln^{3}(log_{a}^{x})ln^{3}(a)(a)} - \frac{3*-3log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{2}ln^{2}(a)ln(log_{a}^{x})} - \frac{3(\frac{-2(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{3}(ln(a))})log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}ln^{2}(a)ln(log_{a}^{x})} - \frac{3(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{(ln(b))}))}{(log_{b}^{x})} - \frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))}))log_{log_{a}^{x}}^{log_{b}^{x}}}{(log_{a}^{x})})}{(ln(log_{a}^{x}))})}{x^{3}{\left(log(a, x)^{2}ln^{2}(a)ln(log_{a}^{x})} - \frac{3log_{log_{a}^{x}}^{log_{b}^{x}}*-2*0}{x^{3}{\left(log(a, x)^{2}ln^{3}(a)(a)ln(log_{a}^{x})} - \frac{3log_{log_{a}^{x}}^{log_{b}^{x}}*-(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{3}{\left(log(a, x)^{2}ln^{2}(a)ln^{2}(log_{a}^{x})(log_{a}^{x})} - \frac{-3log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{3}ln^{3}(a)ln^{2}(log_{a}^{x})} - \frac{(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{(ln(b))}))}{(log_{b}^{x})} - \frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))}))log_{log_{a}^{x}}^{log_{b}^{x}}}{(log_{a}^{x})})}{(ln(log_{a}^{x}))})}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln^{2}(log_{a}^{x})} - \frac{log_{log_{a}^{x}}^{log_{b}^{x}}(\frac{-3(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{4}(ln(a))})}{x^{3}ln^{3}(a)ln^{2}(log_{a}^{x})} - \frac{log_{log_{a}^{x}}^{log_{b}^{x}}*-3*0}{x^{3}{\left(log(a, x)^{3}ln^{4}(a)(a)ln^{2}(log_{a}^{x})} - \frac{log_{log_{a}^{x}}^{log_{b}^{x}}*-2(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln^{3}(log_{a}^{x})(log_{a}^{x})} - \frac{5*-3log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{3}ln^{3}(a)ln^{2}(log_{a}^{x})} - \frac{5(\frac{-3(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{4}(ln(a))})log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}ln^{3}(a)ln^{2}(log_{a}^{x})} - \frac{5(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{(ln(b))}))}{(log_{b}^{x})} - \frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))}))log_{log_{a}^{x}}^{log_{b}^{x}}}{(log_{a}^{x})})}{(ln(log_{a}^{x}))})}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln^{2}(log_{a}^{x})} - \frac{5log_{log_{a}^{x}}^{log_{b}^{x}}*-3*0}{x^{3}{\left(log(a, x)^{3}ln^{4}(a)(a)ln^{2}(log_{a}^{x})} - \frac{5log_{log_{a}^{x}}^{log_{b}^{x}}*-2(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln^{3}(log_{a}^{x})(log_{a}^{x})} - \frac{2*-3log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}log(a, x)ln(a)ln(log_{a}^{x})} - \frac{2(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{2}(ln(a))})log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}ln(a)ln(log_{a}^{x})} - \frac{2(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{(ln(b))}))}{(log_{b}^{x})} - \frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))}))log_{log_{a}^{x}}^{log_{b}^{x}}}{(log_{a}^{x})})}{(ln(log_{a}^{x}))})}{x^{3}log(a, x)ln(a)ln(log_{a}^{x})} - \frac{2log_{log_{a}^{x}}^{log_{b}^{x}}*-0}{x^{3}log(a, x)ln^{2}(a)(a)ln(log_{a}^{x})} - \frac{2log_{log_{a}^{x}}^{log_{b}^{x}}*-(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{3}log(a, x)ln(a)ln^{2}(log_{a}^{x})(log_{a}^{x})} - \frac{3*-3log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{2}ln^{2}(a)ln^{2}(log_{a}^{x})} - \frac{3(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{(ln(b))}))}{(log_{b}^{x})} - \frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))}))log_{log_{a}^{x}}^{log_{b}^{x}}}{(log_{a}^{x})})}{(ln(log_{a}^{x}))})}{x^{3}{\left(log(a, x)^{2}ln^{2}(a)ln^{2}(log_{a}^{x})} - \frac{3log_{log_{a}^{x}}^{log_{b}^{x}}(\frac{-2(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{3}(ln(a))})}{x^{3}ln^{2}(a)ln^{2}(log_{a}^{x})} - \frac{3log_{log_{a}^{x}}^{log_{b}^{x}}*-2*0}{x^{3}{\left(log(a, x)^{2}ln^{3}(a)(a)ln^{2}(log_{a}^{x})} - \frac{3log_{log_{a}^{x}}^{log_{b}^{x}}*-2(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{3}{\left(log(a, x)^{2}ln^{2}(a)ln^{3}(log_{a}^{x})(log_{a}^{x})} - \frac{3*-3log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{2}ln^{2}(a)ln^{2}(log_{a}^{x})} - \frac{3(\frac{-2(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{3}(ln(a))})log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}ln^{2}(a)ln^{2}(log_{a}^{x})} - \frac{3(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{(ln(b))}))}{(log_{b}^{x})} - \frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))}))log_{log_{a}^{x}}^{log_{b}^{x}}}{(log_{a}^{x})})}{(ln(log_{a}^{x}))})}{x^{3}{\left(log(a, x)^{2}ln^{2}(a)ln^{2}(log_{a}^{x})} - \frac{3log_{log_{a}^{x}}^{log_{b}^{x}}*-2*0}{x^{3}{\left(log(a, x)^{2}ln^{3}(a)(a)ln^{2}(log_{a}^{x})} - \frac{3log_{log_{a}^{x}}^{log_{b}^{x}}*-2(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{3}{\left(log(a, x)^{2}ln^{2}(a)ln^{3}(log_{a}^{x})(log_{a}^{x})} - \frac{2*-3log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{3}ln^{3}(a)ln(log_{a}^{x})} - \frac{2(\frac{-3(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{4}(ln(a))})log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}ln^{3}(a)ln(log_{a}^{x})} - \frac{2(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{(ln(b))}))}{(log_{b}^{x})} - \frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))}))log_{log_{a}^{x}}^{log_{b}^{x}}}{(log_{a}^{x})})}{(ln(log_{a}^{x}))})}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln(log_{a}^{x})} - \frac{2log_{log_{a}^{x}}^{log_{b}^{x}}*-3*0}{x^{3}{\left(log(a, x)^{3}ln^{4}(a)(a)ln(log_{a}^{x})} - \frac{2log_{log_{a}^{x}}^{log_{b}^{x}}*-(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln^{2}(log_{a}^{x})(log_{a}^{x})} - \frac{2*-3log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{3}ln^{3}(a)ln^{3}(log_{a}^{x})} - \frac{2(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{(ln(b))}))}{(log_{b}^{x})} - \frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))}))log_{log_{a}^{x}}^{log_{b}^{x}}}{(log_{a}^{x})})}{(ln(log_{a}^{x}))})}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln^{3}(log_{a}^{x})} - \frac{2log_{log_{a}^{x}}^{log_{b}^{x}}(\frac{-3(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{4}(ln(a))})}{x^{3}ln^{3}(a)ln^{3}(log_{a}^{x})} - \frac{2log_{log_{a}^{x}}^{log_{b}^{x}}*-3*0}{x^{3}{\left(log(a, x)^{3}ln^{4}(a)(a)ln^{3}(log_{a}^{x})} - \frac{2log_{log_{a}^{x}}^{log_{b}^{x}}*-3(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln^{4}(log_{a}^{x})(log_{a}^{x})} - \frac{4*-3log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{3}ln^{3}(a)ln^{3}(log_{a}^{x})} - \frac{4(\frac{-3(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{4}(ln(a))})log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{3}ln^{3}(a)ln^{3}(log_{a}^{x})} - \frac{4(\frac{(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{b}^{x}}{(b)})}{(ln(b))}))}{(log_{b}^{x})} - \frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))}))log_{log_{a}^{x}}^{log_{b}^{x}}}{(log_{a}^{x})})}{(ln(log_{a}^{x}))})}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln^{3}(log_{a}^{x})} - \frac{4log_{log_{a}^{x}}^{log_{b}^{x}}*-3*0}{x^{3}{\left(log(a, x)^{3}ln^{4}(a)(a)ln^{3}(log_{a}^{x})} - \frac{4log_{log_{a}^{x}}^{log_{b}^{x}}*-3(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln^{4}(log_{a}^{x})(log_{a}^{x})}\\=&\frac{-6}{x^{4}log(b, x)ln(b)ln(log_{a}^{x})} - \frac{11}{x^{4}{\left(log(b, x)^{2}ln^{2}(b)ln(log_{a}^{x})} - \frac{11}{x^{4}log(b, x)log(a, x)ln(b)ln(a)ln^{2}(log_{a}^{x})} - \frac{12}{x^{4}{\left(log(b, x)^{3}ln^{3}(b)ln(log_{a}^{x})} - \frac{12}{x^{4}{\left(log(b, x)^{2}log(a, x)ln^{2}(b)ln(a)ln^{2}(log_{a}^{x})} - \frac{12}{x^{4}{\left(log(a, x)^{2}log(b, x)ln^{2}(a)ln(b)ln^{2}(log_{a}^{x})} - \frac{12}{x^{4}{\left(log(a, x)^{2}log(b, x)ln(b)ln^{3}(log_{a}^{x})ln^{2}(a)} - \frac{6}{x^{4}{\left(log(b, x)^{4}ln^{4}(b)ln(log_{a}^{x})} - \frac{6}{x^{4}{\left(log(b, x)^{3}log(a, x)ln^{3}(b)ln(a)ln^{2}(log_{a}^{x})} - \frac{3}{x^{4}{\left(log(a, x)^{2}{\left(log(b, x)^{2}ln^{2}(a)ln^{2}(b)ln^{2}(log_{a}^{x})} - \frac{4}{x^{4}{\left(log(a, x)^{2}{\left(log(b, x)^{2}ln^{2}(b)ln^{3}(log_{a}^{x})ln^{2}(a)} - \frac{6}{x^{4}{\left(log(a, x)^{3}log(b, x)ln^{3}(a)ln(b)ln^{2}(log_{a}^{x})} - \frac{2}{x^{4}{\left(log(b, x)^{2}{\left(log(a, x)^{2}ln^{2}(b)ln^{2}(a)ln^{2}(log_{a}^{x})} - \frac{4}{x^{4}{\left(log(a, x)^{3}log(b, x)ln^{3}(a)ln^{3}(log_{a}^{x})ln(b)} - \frac{12}{x^{4}{\left(log(a, x)^{3}log(b, x)ln^{3}(a)ln(b)ln^{3}(log_{a}^{x})} - \frac{4}{x^{4}{\left(log(b, x)^{2}{\left(log(a, x)^{2}ln^{2}(b)ln^{3}(log_{a}^{x})ln^{2}(a)} - \frac{12}{x^{4}{\left(log(a, x)^{3}log(b, x)ln(b)ln^{3}(a)ln^{4}(log_{a}^{x})} - \frac{11}{x^{4}log(b, x)log(a, x)ln(b)ln^{2}(log_{a}^{x})ln(a)} - \frac{6}{x^{4}{\left(log(b, x)^{2}log(a, x)ln^{2}(b)ln^{2}(log_{a}^{x})ln(a)} - \frac{12}{x^{4}{\left(log(a, x)^{2}log(b, x)ln(b)ln^{2}(a)ln^{3}(log_{a}^{x})} - \frac{2}{x^{4}{\left(log(b, x)^{3}log(a, x)ln^{3}(b)ln^{2}(log_{a}^{x})ln(a)} - \frac{2}{x^{4}{\left(log(a, x)^{2}{\left(log(b, x)^{2}ln^{2}(b)ln^{2}(a)ln^{3}(log_{a}^{x})} - \frac{2}{x^{4}{\left(log(b, x)^{2}{\left(log(a, x)^{2}ln^{2}(b)ln^{2}(a)ln^{3}(log_{a}^{x})} - \frac{6}{x^{4}{\left(log(a, x)^{3}log(b, x)ln(b)ln^{4}(log_{a}^{x})ln^{3}(a)} - \frac{6}{x^{4}log(b, x){\left(log(a, x)^{2}ln(b)ln^{2}(log_{a}^{x})ln^{2}(a)} - \frac{1}{x^{4}{\left(log(b, x)^{2}{\left(log(a, x)^{2}ln^{2}(b)ln^{2}(log_{a}^{x})ln^{2}(a)} - \frac{2}{x^{4}{\left(log(a, x)^{3}log(b, x)ln(b)ln^{3}(a)ln^{3}(log_{a}^{x})} - \frac{12}{x^{4}log(b, x){\left(log(a, x)^{2}ln(b)ln^{3}(log_{a}^{x})ln^{2}(a)} - \frac{6}{x^{4}log(b, x){\left(log(a, x)^{3}ln(b)ln^{3}(log_{a}^{x})ln^{3}(a)} - \frac{2}{x^{4}log(b, x){\left(log(a, x)^{3}ln(b)ln^{2}(log_{a}^{x})ln^{3}(a)} - \frac{6}{x^{4}log(b, x){\left(log(a, x)^{3}ln(b)ln^{4}(log_{a}^{x})ln^{3}(a)} + \frac{30log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{3}ln^{3}(a)ln^{2}(log_{a}^{x})} + \frac{11log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{2}ln^{2}(a)ln(log_{a}^{x})} + \frac{6log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{4}ln^{4}(a)ln^{3}(log_{a}^{x})} + \frac{20log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{4}ln^{4}(a)ln^{2}(log_{a}^{x})} + \frac{30log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{4}ln^{4}(a)ln^{3}(log_{a}^{x})} + \frac{6log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}log(a, x)ln(a)ln(log_{a}^{x})} + \frac{11log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{2}ln^{2}(a)ln^{2}(log_{a}^{x})} + \frac{11log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{2}ln^{2}(a)ln^{2}(log_{a}^{x})} + \frac{12log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{3}ln^{3}(a)ln^{3}(log_{a}^{x})} + \frac{24log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{3}ln^{3}(a)ln^{3}(log_{a}^{x})} + \frac{6log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{4}ln^{4}(a)ln(log_{a}^{x})} + \frac{12log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{3}ln^{3}(a)ln(log_{a}^{x})} + \frac{2log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{4}ln^{4}(a)ln^{2}(log_{a}^{x})} + \frac{6log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{3}ln^{3}(a)ln^{2}(log_{a}^{x})} + \frac{6log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{4}ln^{4}(a)ln^{4}(log_{a}^{x})} + \frac{18log_{log_{a}^{x}}^{log_{b}^{x}}}{x^{4}{\left(log(a, x)^{4}ln^{4}(a)ln^{4}(log_{a}^{x})}\\ \end{split}\end{equation} \]