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 3 questions in this calculation: for each question, the 4 derivative of x is calculated.
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/3]Find\ the\ 4th\ derivative\ of\ function\ log_{x}^{sin(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( log_{x}^{sin(x)}\right)}{dx}\\=&(\frac{(\frac{(cos(x))}{(sin(x))} - \frac{(1)log_{x}^{sin(x)}}{(x)})}{(ln(x))})\\=&\frac{cos(x)}{ln(x)sin(x)} - \frac{log_{x}^{sin(x)}}{xln(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{cos(x)}{ln(x)sin(x)} - \frac{log_{x}^{sin(x)}}{xln(x)}\right)}{dx}\\=&\frac{-cos(x)}{ln^{2}(x)(x)sin(x)} + \frac{-cos(x)cos(x)}{ln(x)sin^{2}(x)} + \frac{-sin(x)}{ln(x)sin(x)} - \frac{-log_{x}^{sin(x)}}{x^{2}ln(x)} - \frac{(\frac{(\frac{(cos(x))}{(sin(x))} - \frac{(1)log_{x}^{sin(x)}}{(x)})}{(ln(x))})}{xln(x)} - \frac{log_{x}^{sin(x)}*-1}{xln^{2}(x)(x)}\\=&\frac{-2cos(x)}{xln^{2}(x)sin(x)} - \frac{cos^{2}(x)}{ln(x)sin^{2}(x)} - \frac{1}{ln(x)} + \frac{log_{x}^{sin(x)}}{x^{2}ln(x)} + \frac{2log_{x}^{sin(x)}}{x^{2}ln^{2}(x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2cos(x)}{xln^{2}(x)sin(x)} - \frac{cos^{2}(x)}{ln(x)sin^{2}(x)} - \frac{1}{ln(x)} + \frac{log_{x}^{sin(x)}}{x^{2}ln(x)} + \frac{2log_{x}^{sin(x)}}{x^{2}ln^{2}(x)}\right)}{dx}\\=&\frac{-2*-cos(x)}{x^{2}ln^{2}(x)sin(x)} - \frac{2*-2cos(x)}{xln^{3}(x)(x)sin(x)} - \frac{2*-cos(x)cos(x)}{xln^{2}(x)sin^{2}(x)} - \frac{2*-sin(x)}{xln^{2}(x)sin(x)} - \frac{-cos^{2}(x)}{ln^{2}(x)(x)sin^{2}(x)} - \frac{-2cos(x)cos^{2}(x)}{ln(x)sin^{3}(x)} - \frac{-2cos(x)sin(x)}{ln(x)sin^{2}(x)} - \frac{-1}{ln^{2}(x)(x)} + \frac{-2log_{x}^{sin(x)}}{x^{3}ln(x)} + \frac{(\frac{(\frac{(cos(x))}{(sin(x))} - \frac{(1)log_{x}^{sin(x)}}{(x)})}{(ln(x))})}{x^{2}ln(x)} + \frac{log_{x}^{sin(x)}*-1}{x^{2}ln^{2}(x)(x)} + \frac{2*-2log_{x}^{sin(x)}}{x^{3}ln^{2}(x)} + \frac{2(\frac{(\frac{(cos(x))}{(sin(x))} - \frac{(1)log_{x}^{sin(x)}}{(x)})}{(ln(x))})}{x^{2}ln^{2}(x)} + \frac{2log_{x}^{sin(x)}*-2}{x^{2}ln^{3}(x)(x)}\\=&\frac{3cos(x)}{x^{2}ln^{2}(x)sin(x)} + \frac{6cos(x)}{x^{2}ln^{3}(x)sin(x)} + \frac{3cos^{2}(x)}{xln^{2}(x)sin^{2}(x)} + \frac{3}{xln^{2}(x)} + \frac{2cos^{3}(x)}{ln(x)sin^{3}(x)} + \frac{2cos(x)}{ln(x)sin(x)} - \frac{2log_{x}^{sin(x)}}{x^{3}ln(x)} - \frac{6log_{x}^{sin(x)}}{x^{3}ln^{2}(x)} - \frac{6log_{x}^{sin(x)}}{x^{3}ln^{3}(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{3cos(x)}{x^{2}ln^{2}(x)sin(x)} + \frac{6cos(x)}{x^{2}ln^{3}(x)sin(x)} + \frac{3cos^{2}(x)}{xln^{2}(x)sin^{2}(x)} + \frac{3}{xln^{2}(x)} + \frac{2cos^{3}(x)}{ln(x)sin^{3}(x)} + \frac{2cos(x)}{ln(x)sin(x)} - \frac{2log_{x}^{sin(x)}}{x^{3}ln(x)} - \frac{6log_{x}^{sin(x)}}{x^{3}ln^{2}(x)} - \frac{6log_{x}^{sin(x)}}{x^{3}ln^{3}(x)}\right)}{dx}\\=&\frac{3*-2cos(x)}{x^{3}ln^{2}(x)sin(x)} + \frac{3*-2cos(x)}{x^{2}ln^{3}(x)(x)sin(x)} + \frac{3*-cos(x)cos(x)}{x^{2}ln^{2}(x)sin^{2}(x)} + \frac{3*-sin(x)}{x^{2}ln^{2}(x)sin(x)} + \frac{6*-2cos(x)}{x^{3}ln^{3}(x)sin(x)} + \frac{6*-3cos(x)}{x^{2}ln^{4}(x)(x)sin(x)} + \frac{6*-cos(x)cos(x)}{x^{2}ln^{3}(x)sin^{2}(x)} + \frac{6*-sin(x)}{x^{2}ln^{3}(x)sin(x)} + \frac{3*-cos^{2}(x)}{x^{2}ln^{2}(x)sin^{2}(x)} + \frac{3*-2cos^{2}(x)}{xln^{3}(x)(x)sin^{2}(x)} + \frac{3*-2cos(x)cos^{2}(x)}{xln^{2}(x)sin^{3}(x)} + \frac{3*-2cos(x)sin(x)}{xln^{2}(x)sin^{2}(x)} + \frac{3*-1}{x^{2}ln^{2}(x)} + \frac{3*-2}{xln^{3}(x)(x)} + \frac{2*-cos^{3}(x)}{ln^{2}(x)(x)sin^{3}(x)} + \frac{2*-3cos(x)cos^{3}(x)}{ln(x)sin^{4}(x)} + \frac{2*-3cos^{2}(x)sin(x)}{ln(x)sin^{3}(x)} + \frac{2*-cos(x)}{ln^{2}(x)(x)sin(x)} + \frac{2*-cos(x)cos(x)}{ln(x)sin^{2}(x)} + \frac{2*-sin(x)}{ln(x)sin(x)} - \frac{2*-3log_{x}^{sin(x)}}{x^{4}ln(x)} - \frac{2(\frac{(\frac{(cos(x))}{(sin(x))} - \frac{(1)log_{x}^{sin(x)}}{(x)})}{(ln(x))})}{x^{3}ln(x)} - \frac{2log_{x}^{sin(x)}*-1}{x^{3}ln^{2}(x)(x)} - \frac{6*-3log_{x}^{sin(x)}}{x^{4}ln^{2}(x)} - \frac{6(\frac{(\frac{(cos(x))}{(sin(x))} - \frac{(1)log_{x}^{sin(x)}}{(x)})}{(ln(x))})}{x^{3}ln^{2}(x)} - \frac{6log_{x}^{sin(x)}*-2}{x^{3}ln^{3}(x)(x)} - \frac{6*-3log_{x}^{sin(x)}}{x^{4}ln^{3}(x)} - \frac{6(\frac{(\frac{(cos(x))}{(sin(x))} - \frac{(1)log_{x}^{sin(x)}}{(x)})}{(ln(x))})}{x^{3}ln^{3}(x)} - \frac{6log_{x}^{sin(x)}*-3}{x^{3}ln^{4}(x)(x)}\\=&\frac{-8cos(x)}{x^{3}ln^{2}(x)sin(x)} - \frac{24cos(x)}{x^{3}ln^{3}(x)sin(x)} - \frac{6cos^{2}(x)}{x^{2}ln^{2}(x)sin^{2}(x)} - \frac{8cos(x)}{xln^{2}(x)sin(x)} - \frac{24cos(x)}{x^{3}ln^{4}(x)sin(x)} - \frac{12cos^{2}(x)}{x^{2}ln^{3}(x)sin^{2}(x)} - \frac{8cos^{3}(x)}{xln^{2}(x)sin^{3}(x)} - \frac{12}{x^{2}ln^{3}(x)} - \frac{6}{x^{2}ln^{2}(x)} - \frac{6cos^{4}(x)}{ln(x)sin^{4}(x)} - \frac{8cos^{2}(x)}{ln(x)sin^{2}(x)} - \frac{2}{ln(x)} + \frac{6log_{x}^{sin(x)}}{x^{4}ln(x)} + \frac{22log_{x}^{sin(x)}}{x^{4}ln^{2}(x)} + \frac{36log_{x}^{sin(x)}}{x^{4}ln^{3}(x)} + \frac{24log_{x}^{sin(x)}}{x^{4}ln^{4}(x)}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/3]Find\ the\ 4th\ derivative\ of\ function\ log_{x}^{cos(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( log_{x}^{cos(x)}\right)}{dx}\\=&(\frac{(\frac{(-sin(x))}{(cos(x))} - \frac{(1)log_{x}^{cos(x)}}{(x)})}{(ln(x))})\\=&\frac{-sin(x)}{ln(x)cos(x)} - \frac{log_{x}^{cos(x)}}{xln(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-sin(x)}{ln(x)cos(x)} - \frac{log_{x}^{cos(x)}}{xln(x)}\right)}{dx}\\=&\frac{--sin(x)}{ln^{2}(x)(x)cos(x)} - \frac{cos(x)}{ln(x)cos(x)} - \frac{sin(x)sin(x)}{ln(x)cos^{2}(x)} - \frac{-log_{x}^{cos(x)}}{x^{2}ln(x)} - \frac{(\frac{(\frac{(-sin(x))}{(cos(x))} - \frac{(1)log_{x}^{cos(x)}}{(x)})}{(ln(x))})}{xln(x)} - \frac{log_{x}^{cos(x)}*-1}{xln^{2}(x)(x)}\\=&\frac{2sin(x)}{xln^{2}(x)cos(x)} - \frac{sin^{2}(x)}{ln(x)cos^{2}(x)} - \frac{1}{ln(x)} + \frac{log_{x}^{cos(x)}}{x^{2}ln(x)} + \frac{2log_{x}^{cos(x)}}{x^{2}ln^{2}(x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{2sin(x)}{xln^{2}(x)cos(x)} - \frac{sin^{2}(x)}{ln(x)cos^{2}(x)} - \frac{1}{ln(x)} + \frac{log_{x}^{cos(x)}}{x^{2}ln(x)} + \frac{2log_{x}^{cos(x)}}{x^{2}ln^{2}(x)}\right)}{dx}\\=&\frac{2*-sin(x)}{x^{2}ln^{2}(x)cos(x)} + \frac{2*-2sin(x)}{xln^{3}(x)(x)cos(x)} + \frac{2cos(x)}{xln^{2}(x)cos(x)} + \frac{2sin(x)sin(x)}{xln^{2}(x)cos^{2}(x)} - \frac{-sin^{2}(x)}{ln^{2}(x)(x)cos^{2}(x)} - \frac{2sin(x)cos(x)}{ln(x)cos^{2}(x)} - \frac{sin^{2}(x)*2sin(x)}{ln(x)cos^{3}(x)} - \frac{-1}{ln^{2}(x)(x)} + \frac{-2log_{x}^{cos(x)}}{x^{3}ln(x)} + \frac{(\frac{(\frac{(-sin(x))}{(cos(x))} - \frac{(1)log_{x}^{cos(x)}}{(x)})}{(ln(x))})}{x^{2}ln(x)} + \frac{log_{x}^{cos(x)}*-1}{x^{2}ln^{2}(x)(x)} + \frac{2*-2log_{x}^{cos(x)}}{x^{3}ln^{2}(x)} + \frac{2(\frac{(\frac{(-sin(x))}{(cos(x))} - \frac{(1)log_{x}^{cos(x)}}{(x)})}{(ln(x))})}{x^{2}ln^{2}(x)} + \frac{2log_{x}^{cos(x)}*-2}{x^{2}ln^{3}(x)(x)}\\=&\frac{-3sin(x)}{x^{2}ln^{2}(x)cos(x)} - \frac{6sin(x)}{x^{2}ln^{3}(x)cos(x)} + \frac{3sin^{2}(x)}{xln^{2}(x)cos^{2}(x)} + \frac{3}{xln^{2}(x)} - \frac{2sin(x)}{ln(x)cos(x)} - \frac{2sin^{3}(x)}{ln(x)cos^{3}(x)} - \frac{2log_{x}^{cos(x)}}{x^{3}ln(x)} - \frac{6log_{x}^{cos(x)}}{x^{3}ln^{2}(x)} - \frac{6log_{x}^{cos(x)}}{x^{3}ln^{3}(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-3sin(x)}{x^{2}ln^{2}(x)cos(x)} - \frac{6sin(x)}{x^{2}ln^{3}(x)cos(x)} + \frac{3sin^{2}(x)}{xln^{2}(x)cos^{2}(x)} + \frac{3}{xln^{2}(x)} - \frac{2sin(x)}{ln(x)cos(x)} - \frac{2sin^{3}(x)}{ln(x)cos^{3}(x)} - \frac{2log_{x}^{cos(x)}}{x^{3}ln(x)} - \frac{6log_{x}^{cos(x)}}{x^{3}ln^{2}(x)} - \frac{6log_{x}^{cos(x)}}{x^{3}ln^{3}(x)}\right)}{dx}\\=&\frac{-3*-2sin(x)}{x^{3}ln^{2}(x)cos(x)} - \frac{3*-2sin(x)}{x^{2}ln^{3}(x)(x)cos(x)} - \frac{3cos(x)}{x^{2}ln^{2}(x)cos(x)} - \frac{3sin(x)sin(x)}{x^{2}ln^{2}(x)cos^{2}(x)} - \frac{6*-2sin(x)}{x^{3}ln^{3}(x)cos(x)} - \frac{6*-3sin(x)}{x^{2}ln^{4}(x)(x)cos(x)} - \frac{6cos(x)}{x^{2}ln^{3}(x)cos(x)} - \frac{6sin(x)sin(x)}{x^{2}ln^{3}(x)cos^{2}(x)} + \frac{3*-sin^{2}(x)}{x^{2}ln^{2}(x)cos^{2}(x)} + \frac{3*-2sin^{2}(x)}{xln^{3}(x)(x)cos^{2}(x)} + \frac{3*2sin(x)cos(x)}{xln^{2}(x)cos^{2}(x)} + \frac{3sin^{2}(x)*2sin(x)}{xln^{2}(x)cos^{3}(x)} + \frac{3*-1}{x^{2}ln^{2}(x)} + \frac{3*-2}{xln^{3}(x)(x)} - \frac{2*-sin(x)}{ln^{2}(x)(x)cos(x)} - \frac{2cos(x)}{ln(x)cos(x)} - \frac{2sin(x)sin(x)}{ln(x)cos^{2}(x)} - \frac{2*-sin^{3}(x)}{ln^{2}(x)(x)cos^{3}(x)} - \frac{2*3sin^{2}(x)cos(x)}{ln(x)cos^{3}(x)} - \frac{2sin^{3}(x)*3sin(x)}{ln(x)cos^{4}(x)} - \frac{2*-3log_{x}^{cos(x)}}{x^{4}ln(x)} - \frac{2(\frac{(\frac{(-sin(x))}{(cos(x))} - \frac{(1)log_{x}^{cos(x)}}{(x)})}{(ln(x))})}{x^{3}ln(x)} - \frac{2log_{x}^{cos(x)}*-1}{x^{3}ln^{2}(x)(x)} - \frac{6*-3log_{x}^{cos(x)}}{x^{4}ln^{2}(x)} - \frac{6(\frac{(\frac{(-sin(x))}{(cos(x))} - \frac{(1)log_{x}^{cos(x)}}{(x)})}{(ln(x))})}{x^{3}ln^{2}(x)} - \frac{6log_{x}^{cos(x)}*-2}{x^{3}ln^{3}(x)(x)} - \frac{6*-3log_{x}^{cos(x)}}{x^{4}ln^{3}(x)} - \frac{6(\frac{(\frac{(-sin(x))}{(cos(x))} - \frac{(1)log_{x}^{cos(x)}}{(x)})}{(ln(x))})}{x^{3}ln^{3}(x)} - \frac{6log_{x}^{cos(x)}*-3}{x^{3}ln^{4}(x)(x)}\\=&\frac{8sin(x)}{x^{3}ln^{2}(x)cos(x)} + \frac{24sin(x)}{x^{3}ln^{3}(x)cos(x)} - \frac{6sin^{2}(x)}{x^{2}ln^{2}(x)cos^{2}(x)} + \frac{24sin(x)}{x^{3}ln^{4}(x)cos(x)} - \frac{12sin^{2}(x)}{x^{2}ln^{3}(x)cos^{2}(x)} + \frac{8sin(x)}{xln^{2}(x)cos(x)} + \frac{8sin^{3}(x)}{xln^{2}(x)cos^{3}(x)} - \frac{12}{x^{2}ln^{3}(x)} - \frac{6}{x^{2}ln^{2}(x)} - \frac{8sin^{2}(x)}{ln(x)cos^{2}(x)} - \frac{6sin^{4}(x)}{ln(x)cos^{4}(x)} - \frac{2}{ln(x)} + \frac{6log_{x}^{cos(x)}}{x^{4}ln(x)} + \frac{22log_{x}^{cos(x)}}{x^{4}ln^{2}(x)} + \frac{36log_{x}^{cos(x)}}{x^{4}ln^{3}(x)} + \frac{24log_{x}^{cos(x)}}{x^{4}ln^{4}(x)}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[3/3]Find\ the\ 4th\ derivative\ of\ function\ log_{x}^{tan(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( log_{x}^{tan(x)}\right)}{dx}\\=&(\frac{(\frac{(sec^{2}(x)(1))}{(tan(x))} - \frac{(1)log_{x}^{tan(x)}}{(x)})}{(ln(x))})\\=&\frac{sec^{2}(x)}{ln(x)tan(x)} - \frac{log_{x}^{tan(x)}}{xln(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{sec^{2}(x)}{ln(x)tan(x)} - \frac{log_{x}^{tan(x)}}{xln(x)}\right)}{dx}\\=&\frac{-sec^{2}(x)}{ln^{2}(x)(x)tan(x)} + \frac{-sec^{2}(x)(1)sec^{2}(x)}{ln(x)tan^{2}(x)} + \frac{2sec^{2}(x)tan(x)}{ln(x)tan(x)} - \frac{-log_{x}^{tan(x)}}{x^{2}ln(x)} - \frac{(\frac{(\frac{(sec^{2}(x)(1))}{(tan(x))} - \frac{(1)log_{x}^{tan(x)}}{(x)})}{(ln(x))})}{xln(x)} - \frac{log_{x}^{tan(x)}*-1}{xln^{2}(x)(x)}\\=&\frac{-2sec^{2}(x)}{xln^{2}(x)tan(x)} - \frac{sec^{4}(x)}{ln(x)tan^{2}(x)} + \frac{2sec^{2}(x)}{ln(x)} + \frac{log_{x}^{tan(x)}}{x^{2}ln(x)} + \frac{2log_{x}^{tan(x)}}{x^{2}ln^{2}(x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2sec^{2}(x)}{xln^{2}(x)tan(x)} - \frac{sec^{4}(x)}{ln(x)tan^{2}(x)} + \frac{2sec^{2}(x)}{ln(x)} + \frac{log_{x}^{tan(x)}}{x^{2}ln(x)} + \frac{2log_{x}^{tan(x)}}{x^{2}ln^{2}(x)}\right)}{dx}\\=&\frac{-2*-sec^{2}(x)}{x^{2}ln^{2}(x)tan(x)} - \frac{2*-2sec^{2}(x)}{xln^{3}(x)(x)tan(x)} - \frac{2*-sec^{2}(x)(1)sec^{2}(x)}{xln^{2}(x)tan^{2}(x)} - \frac{2*2sec^{2}(x)tan(x)}{xln^{2}(x)tan(x)} - \frac{-sec^{4}(x)}{ln^{2}(x)(x)tan^{2}(x)} - \frac{-2sec^{2}(x)(1)sec^{4}(x)}{ln(x)tan^{3}(x)} - \frac{4sec^{4}(x)tan(x)}{ln(x)tan^{2}(x)} + \frac{2*-sec^{2}(x)}{ln^{2}(x)(x)} + \frac{2*2sec^{2}(x)tan(x)}{ln(x)} + \frac{-2log_{x}^{tan(x)}}{x^{3}ln(x)} + \frac{(\frac{(\frac{(sec^{2}(x)(1))}{(tan(x))} - \frac{(1)log_{x}^{tan(x)}}{(x)})}{(ln(x))})}{x^{2}ln(x)} + \frac{log_{x}^{tan(x)}*-1}{x^{2}ln^{2}(x)(x)} + \frac{2*-2log_{x}^{tan(x)}}{x^{3}ln^{2}(x)} + \frac{2(\frac{(\frac{(sec^{2}(x)(1))}{(tan(x))} - \frac{(1)log_{x}^{tan(x)}}{(x)})}{(ln(x))})}{x^{2}ln^{2}(x)} + \frac{2log_{x}^{tan(x)}*-2}{x^{2}ln^{3}(x)(x)}\\=&\frac{3sec^{2}(x)}{x^{2}ln^{2}(x)tan(x)} + \frac{6sec^{2}(x)}{x^{2}ln^{3}(x)tan(x)} + \frac{3sec^{4}(x)}{xln^{2}(x)tan^{2}(x)} - \frac{6sec^{2}(x)}{xln^{2}(x)} + \frac{2sec^{6}(x)}{ln(x)tan^{3}(x)} - \frac{4sec^{4}(x)}{ln(x)tan(x)} + \frac{4tan(x)sec^{2}(x)}{ln(x)} - \frac{2log_{x}^{tan(x)}}{x^{3}ln(x)} - \frac{6log_{x}^{tan(x)}}{x^{3}ln^{2}(x)} - \frac{6log_{x}^{tan(x)}}{x^{3}ln^{3}(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{3sec^{2}(x)}{x^{2}ln^{2}(x)tan(x)} + \frac{6sec^{2}(x)}{x^{2}ln^{3}(x)tan(x)} + \frac{3sec^{4}(x)}{xln^{2}(x)tan^{2}(x)} - \frac{6sec^{2}(x)}{xln^{2}(x)} + \frac{2sec^{6}(x)}{ln(x)tan^{3}(x)} - \frac{4sec^{4}(x)}{ln(x)tan(x)} + \frac{4tan(x)sec^{2}(x)}{ln(x)} - \frac{2log_{x}^{tan(x)}}{x^{3}ln(x)} - \frac{6log_{x}^{tan(x)}}{x^{3}ln^{2}(x)} - \frac{6log_{x}^{tan(x)}}{x^{3}ln^{3}(x)}\right)}{dx}\\=&\frac{3*-2sec^{2}(x)}{x^{3}ln^{2}(x)tan(x)} + \frac{3*-2sec^{2}(x)}{x^{2}ln^{3}(x)(x)tan(x)} + \frac{3*-sec^{2}(x)(1)sec^{2}(x)}{x^{2}ln^{2}(x)tan^{2}(x)} + \frac{3*2sec^{2}(x)tan(x)}{x^{2}ln^{2}(x)tan(x)} + \frac{6*-2sec^{2}(x)}{x^{3}ln^{3}(x)tan(x)} + \frac{6*-3sec^{2}(x)}{x^{2}ln^{4}(x)(x)tan(x)} + \frac{6*-sec^{2}(x)(1)sec^{2}(x)}{x^{2}ln^{3}(x)tan^{2}(x)} + \frac{6*2sec^{2}(x)tan(x)}{x^{2}ln^{3}(x)tan(x)} + \frac{3*-sec^{4}(x)}{x^{2}ln^{2}(x)tan^{2}(x)} + \frac{3*-2sec^{4}(x)}{xln^{3}(x)(x)tan^{2}(x)} + \frac{3*-2sec^{2}(x)(1)sec^{4}(x)}{xln^{2}(x)tan^{3}(x)} + \frac{3*4sec^{4}(x)tan(x)}{xln^{2}(x)tan^{2}(x)} - \frac{6*-sec^{2}(x)}{x^{2}ln^{2}(x)} - \frac{6*-2sec^{2}(x)}{xln^{3}(x)(x)} - \frac{6*2sec^{2}(x)tan(x)}{xln^{2}(x)} + \frac{2*-sec^{6}(x)}{ln^{2}(x)(x)tan^{3}(x)} + \frac{2*-3sec^{2}(x)(1)sec^{6}(x)}{ln(x)tan^{4}(x)} + \frac{2*6sec^{6}(x)tan(x)}{ln(x)tan^{3}(x)} - \frac{4*-sec^{4}(x)}{ln^{2}(x)(x)tan(x)} - \frac{4*-sec^{2}(x)(1)sec^{4}(x)}{ln(x)tan^{2}(x)} - \frac{4*4sec^{4}(x)tan(x)}{ln(x)tan(x)} + \frac{4*-tan(x)sec^{2}(x)}{ln^{2}(x)(x)} + \frac{4sec^{2}(x)(1)sec^{2}(x)}{ln(x)} + \frac{4tan(x)*2sec^{2}(x)tan(x)}{ln(x)} - \frac{2*-3log_{x}^{tan(x)}}{x^{4}ln(x)} - \frac{2(\frac{(\frac{(sec^{2}(x)(1))}{(tan(x))} - \frac{(1)log_{x}^{tan(x)}}{(x)})}{(ln(x))})}{x^{3}ln(x)} - \frac{2log_{x}^{tan(x)}*-1}{x^{3}ln^{2}(x)(x)} - \frac{6*-3log_{x}^{tan(x)}}{x^{4}ln^{2}(x)} - \frac{6(\frac{(\frac{(sec^{2}(x)(1))}{(tan(x))} - \frac{(1)log_{x}^{tan(x)}}{(x)})}{(ln(x))})}{x^{3}ln^{2}(x)} - \frac{6log_{x}^{tan(x)}*-2}{x^{3}ln^{3}(x)(x)} - \frac{6*-3log_{x}^{tan(x)}}{x^{4}ln^{3}(x)} - \frac{6(\frac{(\frac{(sec^{2}(x)(1))}{(tan(x))} - \frac{(1)log_{x}^{tan(x)}}{(x)})}{(ln(x))})}{x^{3}ln^{3}(x)} - \frac{6log_{x}^{tan(x)}*-3}{x^{3}ln^{4}(x)(x)}\\=&\frac{-8sec^{2}(x)}{x^{3}ln^{2}(x)tan(x)} - \frac{24sec^{2}(x)}{x^{3}ln^{3}(x)tan(x)} - \frac{6sec^{4}(x)}{x^{2}ln^{2}(x)tan^{2}(x)} + \frac{12sec^{2}(x)}{x^{2}ln^{2}(x)} - \frac{24sec^{2}(x)}{x^{3}ln^{4}(x)tan(x)} - \frac{12sec^{4}(x)}{x^{2}ln^{3}(x)tan^{2}(x)} + \frac{24sec^{2}(x)}{x^{2}ln^{3}(x)} - \frac{8sec^{6}(x)}{xln^{2}(x)tan^{3}(x)} + \frac{16sec^{4}(x)}{xln^{2}(x)tan(x)} - \frac{16tan(x)sec^{2}(x)}{xln^{2}(x)} - \frac{6sec^{8}(x)}{ln(x)tan^{4}(x)} + \frac{16sec^{6}(x)}{ln(x)tan^{2}(x)} - \frac{12sec^{4}(x)}{ln(x)} + \frac{8tan^{2}(x)sec^{2}(x)}{ln(x)} + \frac{6log_{x}^{tan(x)}}{x^{4}ln(x)} + \frac{22log_{x}^{tan(x)}}{x^{4}ln^{2}(x)} + \frac{36log_{x}^{tan(x)}}{x^{4}ln^{3}(x)} + \frac{24log_{x}^{tan(x)}}{x^{4}ln^{4}(x)}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/3]Find\ the\ 4th\ derivative\ of\ function\ log_{x}^{sin(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( log_{x}^{sin(x)}\right)}{dx}\\=&(\frac{(\frac{(cos(x))}{(sin(x))} - \frac{(1)log_{x}^{sin(x)}}{(x)})}{(ln(x))})\\=&\frac{cos(x)}{ln(x)sin(x)} - \frac{log_{x}^{sin(x)}}{xln(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{cos(x)}{ln(x)sin(x)} - \frac{log_{x}^{sin(x)}}{xln(x)}\right)}{dx}\\=&\frac{-cos(x)}{ln^{2}(x)(x)sin(x)} + \frac{-cos(x)cos(x)}{ln(x)sin^{2}(x)} + \frac{-sin(x)}{ln(x)sin(x)} - \frac{-log_{x}^{sin(x)}}{x^{2}ln(x)} - \frac{(\frac{(\frac{(cos(x))}{(sin(x))} - \frac{(1)log_{x}^{sin(x)}}{(x)})}{(ln(x))})}{xln(x)} - \frac{log_{x}^{sin(x)}*-1}{xln^{2}(x)(x)}\\=&\frac{-2cos(x)}{xln^{2}(x)sin(x)} - \frac{cos^{2}(x)}{ln(x)sin^{2}(x)} - \frac{1}{ln(x)} + \frac{log_{x}^{sin(x)}}{x^{2}ln(x)} + \frac{2log_{x}^{sin(x)}}{x^{2}ln^{2}(x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2cos(x)}{xln^{2}(x)sin(x)} - \frac{cos^{2}(x)}{ln(x)sin^{2}(x)} - \frac{1}{ln(x)} + \frac{log_{x}^{sin(x)}}{x^{2}ln(x)} + \frac{2log_{x}^{sin(x)}}{x^{2}ln^{2}(x)}\right)}{dx}\\=&\frac{-2*-cos(x)}{x^{2}ln^{2}(x)sin(x)} - \frac{2*-2cos(x)}{xln^{3}(x)(x)sin(x)} - \frac{2*-cos(x)cos(x)}{xln^{2}(x)sin^{2}(x)} - \frac{2*-sin(x)}{xln^{2}(x)sin(x)} - \frac{-cos^{2}(x)}{ln^{2}(x)(x)sin^{2}(x)} - \frac{-2cos(x)cos^{2}(x)}{ln(x)sin^{3}(x)} - \frac{-2cos(x)sin(x)}{ln(x)sin^{2}(x)} - \frac{-1}{ln^{2}(x)(x)} + \frac{-2log_{x}^{sin(x)}}{x^{3}ln(x)} + \frac{(\frac{(\frac{(cos(x))}{(sin(x))} - \frac{(1)log_{x}^{sin(x)}}{(x)})}{(ln(x))})}{x^{2}ln(x)} + \frac{log_{x}^{sin(x)}*-1}{x^{2}ln^{2}(x)(x)} + \frac{2*-2log_{x}^{sin(x)}}{x^{3}ln^{2}(x)} + \frac{2(\frac{(\frac{(cos(x))}{(sin(x))} - \frac{(1)log_{x}^{sin(x)}}{(x)})}{(ln(x))})}{x^{2}ln^{2}(x)} + \frac{2log_{x}^{sin(x)}*-2}{x^{2}ln^{3}(x)(x)}\\=&\frac{3cos(x)}{x^{2}ln^{2}(x)sin(x)} + \frac{6cos(x)}{x^{2}ln^{3}(x)sin(x)} + \frac{3cos^{2}(x)}{xln^{2}(x)sin^{2}(x)} + \frac{3}{xln^{2}(x)} + \frac{2cos^{3}(x)}{ln(x)sin^{3}(x)} + \frac{2cos(x)}{ln(x)sin(x)} - \frac{2log_{x}^{sin(x)}}{x^{3}ln(x)} - \frac{6log_{x}^{sin(x)}}{x^{3}ln^{2}(x)} - \frac{6log_{x}^{sin(x)}}{x^{3}ln^{3}(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{3cos(x)}{x^{2}ln^{2}(x)sin(x)} + \frac{6cos(x)}{x^{2}ln^{3}(x)sin(x)} + \frac{3cos^{2}(x)}{xln^{2}(x)sin^{2}(x)} + \frac{3}{xln^{2}(x)} + \frac{2cos^{3}(x)}{ln(x)sin^{3}(x)} + \frac{2cos(x)}{ln(x)sin(x)} - \frac{2log_{x}^{sin(x)}}{x^{3}ln(x)} - \frac{6log_{x}^{sin(x)}}{x^{3}ln^{2}(x)} - \frac{6log_{x}^{sin(x)}}{x^{3}ln^{3}(x)}\right)}{dx}\\=&\frac{3*-2cos(x)}{x^{3}ln^{2}(x)sin(x)} + \frac{3*-2cos(x)}{x^{2}ln^{3}(x)(x)sin(x)} + \frac{3*-cos(x)cos(x)}{x^{2}ln^{2}(x)sin^{2}(x)} + \frac{3*-sin(x)}{x^{2}ln^{2}(x)sin(x)} + \frac{6*-2cos(x)}{x^{3}ln^{3}(x)sin(x)} + \frac{6*-3cos(x)}{x^{2}ln^{4}(x)(x)sin(x)} + \frac{6*-cos(x)cos(x)}{x^{2}ln^{3}(x)sin^{2}(x)} + \frac{6*-sin(x)}{x^{2}ln^{3}(x)sin(x)} + \frac{3*-cos^{2}(x)}{x^{2}ln^{2}(x)sin^{2}(x)} + \frac{3*-2cos^{2}(x)}{xln^{3}(x)(x)sin^{2}(x)} + \frac{3*-2cos(x)cos^{2}(x)}{xln^{2}(x)sin^{3}(x)} + \frac{3*-2cos(x)sin(x)}{xln^{2}(x)sin^{2}(x)} + \frac{3*-1}{x^{2}ln^{2}(x)} + \frac{3*-2}{xln^{3}(x)(x)} + \frac{2*-cos^{3}(x)}{ln^{2}(x)(x)sin^{3}(x)} + \frac{2*-3cos(x)cos^{3}(x)}{ln(x)sin^{4}(x)} + \frac{2*-3cos^{2}(x)sin(x)}{ln(x)sin^{3}(x)} + \frac{2*-cos(x)}{ln^{2}(x)(x)sin(x)} + \frac{2*-cos(x)cos(x)}{ln(x)sin^{2}(x)} + \frac{2*-sin(x)}{ln(x)sin(x)} - \frac{2*-3log_{x}^{sin(x)}}{x^{4}ln(x)} - \frac{2(\frac{(\frac{(cos(x))}{(sin(x))} - \frac{(1)log_{x}^{sin(x)}}{(x)})}{(ln(x))})}{x^{3}ln(x)} - \frac{2log_{x}^{sin(x)}*-1}{x^{3}ln^{2}(x)(x)} - \frac{6*-3log_{x}^{sin(x)}}{x^{4}ln^{2}(x)} - \frac{6(\frac{(\frac{(cos(x))}{(sin(x))} - \frac{(1)log_{x}^{sin(x)}}{(x)})}{(ln(x))})}{x^{3}ln^{2}(x)} - \frac{6log_{x}^{sin(x)}*-2}{x^{3}ln^{3}(x)(x)} - \frac{6*-3log_{x}^{sin(x)}}{x^{4}ln^{3}(x)} - \frac{6(\frac{(\frac{(cos(x))}{(sin(x))} - \frac{(1)log_{x}^{sin(x)}}{(x)})}{(ln(x))})}{x^{3}ln^{3}(x)} - \frac{6log_{x}^{sin(x)}*-3}{x^{3}ln^{4}(x)(x)}\\=&\frac{-8cos(x)}{x^{3}ln^{2}(x)sin(x)} - \frac{24cos(x)}{x^{3}ln^{3}(x)sin(x)} - \frac{6cos^{2}(x)}{x^{2}ln^{2}(x)sin^{2}(x)} - \frac{8cos(x)}{xln^{2}(x)sin(x)} - \frac{24cos(x)}{x^{3}ln^{4}(x)sin(x)} - \frac{12cos^{2}(x)}{x^{2}ln^{3}(x)sin^{2}(x)} - \frac{8cos^{3}(x)}{xln^{2}(x)sin^{3}(x)} - \frac{12}{x^{2}ln^{3}(x)} - \frac{6}{x^{2}ln^{2}(x)} - \frac{6cos^{4}(x)}{ln(x)sin^{4}(x)} - \frac{8cos^{2}(x)}{ln(x)sin^{2}(x)} - \frac{2}{ln(x)} + \frac{6log_{x}^{sin(x)}}{x^{4}ln(x)} + \frac{22log_{x}^{sin(x)}}{x^{4}ln^{2}(x)} + \frac{36log_{x}^{sin(x)}}{x^{4}ln^{3}(x)} + \frac{24log_{x}^{sin(x)}}{x^{4}ln^{4}(x)}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/3]Find\ the\ 4th\ derivative\ of\ function\ log_{x}^{cos(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( log_{x}^{cos(x)}\right)}{dx}\\=&(\frac{(\frac{(-sin(x))}{(cos(x))} - \frac{(1)log_{x}^{cos(x)}}{(x)})}{(ln(x))})\\=&\frac{-sin(x)}{ln(x)cos(x)} - \frac{log_{x}^{cos(x)}}{xln(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-sin(x)}{ln(x)cos(x)} - \frac{log_{x}^{cos(x)}}{xln(x)}\right)}{dx}\\=&\frac{--sin(x)}{ln^{2}(x)(x)cos(x)} - \frac{cos(x)}{ln(x)cos(x)} - \frac{sin(x)sin(x)}{ln(x)cos^{2}(x)} - \frac{-log_{x}^{cos(x)}}{x^{2}ln(x)} - \frac{(\frac{(\frac{(-sin(x))}{(cos(x))} - \frac{(1)log_{x}^{cos(x)}}{(x)})}{(ln(x))})}{xln(x)} - \frac{log_{x}^{cos(x)}*-1}{xln^{2}(x)(x)}\\=&\frac{2sin(x)}{xln^{2}(x)cos(x)} - \frac{sin^{2}(x)}{ln(x)cos^{2}(x)} - \frac{1}{ln(x)} + \frac{log_{x}^{cos(x)}}{x^{2}ln(x)} + \frac{2log_{x}^{cos(x)}}{x^{2}ln^{2}(x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{2sin(x)}{xln^{2}(x)cos(x)} - \frac{sin^{2}(x)}{ln(x)cos^{2}(x)} - \frac{1}{ln(x)} + \frac{log_{x}^{cos(x)}}{x^{2}ln(x)} + \frac{2log_{x}^{cos(x)}}{x^{2}ln^{2}(x)}\right)}{dx}\\=&\frac{2*-sin(x)}{x^{2}ln^{2}(x)cos(x)} + \frac{2*-2sin(x)}{xln^{3}(x)(x)cos(x)} + \frac{2cos(x)}{xln^{2}(x)cos(x)} + \frac{2sin(x)sin(x)}{xln^{2}(x)cos^{2}(x)} - \frac{-sin^{2}(x)}{ln^{2}(x)(x)cos^{2}(x)} - \frac{2sin(x)cos(x)}{ln(x)cos^{2}(x)} - \frac{sin^{2}(x)*2sin(x)}{ln(x)cos^{3}(x)} - \frac{-1}{ln^{2}(x)(x)} + \frac{-2log_{x}^{cos(x)}}{x^{3}ln(x)} + \frac{(\frac{(\frac{(-sin(x))}{(cos(x))} - \frac{(1)log_{x}^{cos(x)}}{(x)})}{(ln(x))})}{x^{2}ln(x)} + \frac{log_{x}^{cos(x)}*-1}{x^{2}ln^{2}(x)(x)} + \frac{2*-2log_{x}^{cos(x)}}{x^{3}ln^{2}(x)} + \frac{2(\frac{(\frac{(-sin(x))}{(cos(x))} - \frac{(1)log_{x}^{cos(x)}}{(x)})}{(ln(x))})}{x^{2}ln^{2}(x)} + \frac{2log_{x}^{cos(x)}*-2}{x^{2}ln^{3}(x)(x)}\\=&\frac{-3sin(x)}{x^{2}ln^{2}(x)cos(x)} - \frac{6sin(x)}{x^{2}ln^{3}(x)cos(x)} + \frac{3sin^{2}(x)}{xln^{2}(x)cos^{2}(x)} + \frac{3}{xln^{2}(x)} - \frac{2sin(x)}{ln(x)cos(x)} - \frac{2sin^{3}(x)}{ln(x)cos^{3}(x)} - \frac{2log_{x}^{cos(x)}}{x^{3}ln(x)} - \frac{6log_{x}^{cos(x)}}{x^{3}ln^{2}(x)} - \frac{6log_{x}^{cos(x)}}{x^{3}ln^{3}(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-3sin(x)}{x^{2}ln^{2}(x)cos(x)} - \frac{6sin(x)}{x^{2}ln^{3}(x)cos(x)} + \frac{3sin^{2}(x)}{xln^{2}(x)cos^{2}(x)} + \frac{3}{xln^{2}(x)} - \frac{2sin(x)}{ln(x)cos(x)} - \frac{2sin^{3}(x)}{ln(x)cos^{3}(x)} - \frac{2log_{x}^{cos(x)}}{x^{3}ln(x)} - \frac{6log_{x}^{cos(x)}}{x^{3}ln^{2}(x)} - \frac{6log_{x}^{cos(x)}}{x^{3}ln^{3}(x)}\right)}{dx}\\=&\frac{-3*-2sin(x)}{x^{3}ln^{2}(x)cos(x)} - \frac{3*-2sin(x)}{x^{2}ln^{3}(x)(x)cos(x)} - \frac{3cos(x)}{x^{2}ln^{2}(x)cos(x)} - \frac{3sin(x)sin(x)}{x^{2}ln^{2}(x)cos^{2}(x)} - \frac{6*-2sin(x)}{x^{3}ln^{3}(x)cos(x)} - \frac{6*-3sin(x)}{x^{2}ln^{4}(x)(x)cos(x)} - \frac{6cos(x)}{x^{2}ln^{3}(x)cos(x)} - \frac{6sin(x)sin(x)}{x^{2}ln^{3}(x)cos^{2}(x)} + \frac{3*-sin^{2}(x)}{x^{2}ln^{2}(x)cos^{2}(x)} + \frac{3*-2sin^{2}(x)}{xln^{3}(x)(x)cos^{2}(x)} + \frac{3*2sin(x)cos(x)}{xln^{2}(x)cos^{2}(x)} + \frac{3sin^{2}(x)*2sin(x)}{xln^{2}(x)cos^{3}(x)} + \frac{3*-1}{x^{2}ln^{2}(x)} + \frac{3*-2}{xln^{3}(x)(x)} - \frac{2*-sin(x)}{ln^{2}(x)(x)cos(x)} - \frac{2cos(x)}{ln(x)cos(x)} - \frac{2sin(x)sin(x)}{ln(x)cos^{2}(x)} - \frac{2*-sin^{3}(x)}{ln^{2}(x)(x)cos^{3}(x)} - \frac{2*3sin^{2}(x)cos(x)}{ln(x)cos^{3}(x)} - \frac{2sin^{3}(x)*3sin(x)}{ln(x)cos^{4}(x)} - \frac{2*-3log_{x}^{cos(x)}}{x^{4}ln(x)} - \frac{2(\frac{(\frac{(-sin(x))}{(cos(x))} - \frac{(1)log_{x}^{cos(x)}}{(x)})}{(ln(x))})}{x^{3}ln(x)} - \frac{2log_{x}^{cos(x)}*-1}{x^{3}ln^{2}(x)(x)} - \frac{6*-3log_{x}^{cos(x)}}{x^{4}ln^{2}(x)} - \frac{6(\frac{(\frac{(-sin(x))}{(cos(x))} - \frac{(1)log_{x}^{cos(x)}}{(x)})}{(ln(x))})}{x^{3}ln^{2}(x)} - \frac{6log_{x}^{cos(x)}*-2}{x^{3}ln^{3}(x)(x)} - \frac{6*-3log_{x}^{cos(x)}}{x^{4}ln^{3}(x)} - \frac{6(\frac{(\frac{(-sin(x))}{(cos(x))} - \frac{(1)log_{x}^{cos(x)}}{(x)})}{(ln(x))})}{x^{3}ln^{3}(x)} - \frac{6log_{x}^{cos(x)}*-3}{x^{3}ln^{4}(x)(x)}\\=&\frac{8sin(x)}{x^{3}ln^{2}(x)cos(x)} + \frac{24sin(x)}{x^{3}ln^{3}(x)cos(x)} - \frac{6sin^{2}(x)}{x^{2}ln^{2}(x)cos^{2}(x)} + \frac{24sin(x)}{x^{3}ln^{4}(x)cos(x)} - \frac{12sin^{2}(x)}{x^{2}ln^{3}(x)cos^{2}(x)} + \frac{8sin(x)}{xln^{2}(x)cos(x)} + \frac{8sin^{3}(x)}{xln^{2}(x)cos^{3}(x)} - \frac{12}{x^{2}ln^{3}(x)} - \frac{6}{x^{2}ln^{2}(x)} - \frac{8sin^{2}(x)}{ln(x)cos^{2}(x)} - \frac{6sin^{4}(x)}{ln(x)cos^{4}(x)} - \frac{2}{ln(x)} + \frac{6log_{x}^{cos(x)}}{x^{4}ln(x)} + \frac{22log_{x}^{cos(x)}}{x^{4}ln^{2}(x)} + \frac{36log_{x}^{cos(x)}}{x^{4}ln^{3}(x)} + \frac{24log_{x}^{cos(x)}}{x^{4}ln^{4}(x)}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[3/3]Find\ the\ 4th\ derivative\ of\ function\ log_{x}^{tan(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( log_{x}^{tan(x)}\right)}{dx}\\=&(\frac{(\frac{(sec^{2}(x)(1))}{(tan(x))} - \frac{(1)log_{x}^{tan(x)}}{(x)})}{(ln(x))})\\=&\frac{sec^{2}(x)}{ln(x)tan(x)} - \frac{log_{x}^{tan(x)}}{xln(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{sec^{2}(x)}{ln(x)tan(x)} - \frac{log_{x}^{tan(x)}}{xln(x)}\right)}{dx}\\=&\frac{-sec^{2}(x)}{ln^{2}(x)(x)tan(x)} + \frac{-sec^{2}(x)(1)sec^{2}(x)}{ln(x)tan^{2}(x)} + \frac{2sec^{2}(x)tan(x)}{ln(x)tan(x)} - \frac{-log_{x}^{tan(x)}}{x^{2}ln(x)} - \frac{(\frac{(\frac{(sec^{2}(x)(1))}{(tan(x))} - \frac{(1)log_{x}^{tan(x)}}{(x)})}{(ln(x))})}{xln(x)} - \frac{log_{x}^{tan(x)}*-1}{xln^{2}(x)(x)}\\=&\frac{-2sec^{2}(x)}{xln^{2}(x)tan(x)} - \frac{sec^{4}(x)}{ln(x)tan^{2}(x)} + \frac{2sec^{2}(x)}{ln(x)} + \frac{log_{x}^{tan(x)}}{x^{2}ln(x)} + \frac{2log_{x}^{tan(x)}}{x^{2}ln^{2}(x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2sec^{2}(x)}{xln^{2}(x)tan(x)} - \frac{sec^{4}(x)}{ln(x)tan^{2}(x)} + \frac{2sec^{2}(x)}{ln(x)} + \frac{log_{x}^{tan(x)}}{x^{2}ln(x)} + \frac{2log_{x}^{tan(x)}}{x^{2}ln^{2}(x)}\right)}{dx}\\=&\frac{-2*-sec^{2}(x)}{x^{2}ln^{2}(x)tan(x)} - \frac{2*-2sec^{2}(x)}{xln^{3}(x)(x)tan(x)} - \frac{2*-sec^{2}(x)(1)sec^{2}(x)}{xln^{2}(x)tan^{2}(x)} - \frac{2*2sec^{2}(x)tan(x)}{xln^{2}(x)tan(x)} - \frac{-sec^{4}(x)}{ln^{2}(x)(x)tan^{2}(x)} - \frac{-2sec^{2}(x)(1)sec^{4}(x)}{ln(x)tan^{3}(x)} - \frac{4sec^{4}(x)tan(x)}{ln(x)tan^{2}(x)} + \frac{2*-sec^{2}(x)}{ln^{2}(x)(x)} + \frac{2*2sec^{2}(x)tan(x)}{ln(x)} + \frac{-2log_{x}^{tan(x)}}{x^{3}ln(x)} + \frac{(\frac{(\frac{(sec^{2}(x)(1))}{(tan(x))} - \frac{(1)log_{x}^{tan(x)}}{(x)})}{(ln(x))})}{x^{2}ln(x)} + \frac{log_{x}^{tan(x)}*-1}{x^{2}ln^{2}(x)(x)} + \frac{2*-2log_{x}^{tan(x)}}{x^{3}ln^{2}(x)} + \frac{2(\frac{(\frac{(sec^{2}(x)(1))}{(tan(x))} - \frac{(1)log_{x}^{tan(x)}}{(x)})}{(ln(x))})}{x^{2}ln^{2}(x)} + \frac{2log_{x}^{tan(x)}*-2}{x^{2}ln^{3}(x)(x)}\\=&\frac{3sec^{2}(x)}{x^{2}ln^{2}(x)tan(x)} + \frac{6sec^{2}(x)}{x^{2}ln^{3}(x)tan(x)} + \frac{3sec^{4}(x)}{xln^{2}(x)tan^{2}(x)} - \frac{6sec^{2}(x)}{xln^{2}(x)} + \frac{2sec^{6}(x)}{ln(x)tan^{3}(x)} - \frac{4sec^{4}(x)}{ln(x)tan(x)} + \frac{4tan(x)sec^{2}(x)}{ln(x)} - \frac{2log_{x}^{tan(x)}}{x^{3}ln(x)} - \frac{6log_{x}^{tan(x)}}{x^{3}ln^{2}(x)} - \frac{6log_{x}^{tan(x)}}{x^{3}ln^{3}(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{3sec^{2}(x)}{x^{2}ln^{2}(x)tan(x)} + \frac{6sec^{2}(x)}{x^{2}ln^{3}(x)tan(x)} + \frac{3sec^{4}(x)}{xln^{2}(x)tan^{2}(x)} - \frac{6sec^{2}(x)}{xln^{2}(x)} + \frac{2sec^{6}(x)}{ln(x)tan^{3}(x)} - \frac{4sec^{4}(x)}{ln(x)tan(x)} + \frac{4tan(x)sec^{2}(x)}{ln(x)} - \frac{2log_{x}^{tan(x)}}{x^{3}ln(x)} - \frac{6log_{x}^{tan(x)}}{x^{3}ln^{2}(x)} - \frac{6log_{x}^{tan(x)}}{x^{3}ln^{3}(x)}\right)}{dx}\\=&\frac{3*-2sec^{2}(x)}{x^{3}ln^{2}(x)tan(x)} + \frac{3*-2sec^{2}(x)}{x^{2}ln^{3}(x)(x)tan(x)} + \frac{3*-sec^{2}(x)(1)sec^{2}(x)}{x^{2}ln^{2}(x)tan^{2}(x)} + \frac{3*2sec^{2}(x)tan(x)}{x^{2}ln^{2}(x)tan(x)} + \frac{6*-2sec^{2}(x)}{x^{3}ln^{3}(x)tan(x)} + \frac{6*-3sec^{2}(x)}{x^{2}ln^{4}(x)(x)tan(x)} + \frac{6*-sec^{2}(x)(1)sec^{2}(x)}{x^{2}ln^{3}(x)tan^{2}(x)} + \frac{6*2sec^{2}(x)tan(x)}{x^{2}ln^{3}(x)tan(x)} + \frac{3*-sec^{4}(x)}{x^{2}ln^{2}(x)tan^{2}(x)} + \frac{3*-2sec^{4}(x)}{xln^{3}(x)(x)tan^{2}(x)} + \frac{3*-2sec^{2}(x)(1)sec^{4}(x)}{xln^{2}(x)tan^{3}(x)} + \frac{3*4sec^{4}(x)tan(x)}{xln^{2}(x)tan^{2}(x)} - \frac{6*-sec^{2}(x)}{x^{2}ln^{2}(x)} - \frac{6*-2sec^{2}(x)}{xln^{3}(x)(x)} - \frac{6*2sec^{2}(x)tan(x)}{xln^{2}(x)} + \frac{2*-sec^{6}(x)}{ln^{2}(x)(x)tan^{3}(x)} + \frac{2*-3sec^{2}(x)(1)sec^{6}(x)}{ln(x)tan^{4}(x)} + \frac{2*6sec^{6}(x)tan(x)}{ln(x)tan^{3}(x)} - \frac{4*-sec^{4}(x)}{ln^{2}(x)(x)tan(x)} - \frac{4*-sec^{2}(x)(1)sec^{4}(x)}{ln(x)tan^{2}(x)} - \frac{4*4sec^{4}(x)tan(x)}{ln(x)tan(x)} + \frac{4*-tan(x)sec^{2}(x)}{ln^{2}(x)(x)} + \frac{4sec^{2}(x)(1)sec^{2}(x)}{ln(x)} + \frac{4tan(x)*2sec^{2}(x)tan(x)}{ln(x)} - \frac{2*-3log_{x}^{tan(x)}}{x^{4}ln(x)} - \frac{2(\frac{(\frac{(sec^{2}(x)(1))}{(tan(x))} - \frac{(1)log_{x}^{tan(x)}}{(x)})}{(ln(x))})}{x^{3}ln(x)} - \frac{2log_{x}^{tan(x)}*-1}{x^{3}ln^{2}(x)(x)} - \frac{6*-3log_{x}^{tan(x)}}{x^{4}ln^{2}(x)} - \frac{6(\frac{(\frac{(sec^{2}(x)(1))}{(tan(x))} - \frac{(1)log_{x}^{tan(x)}}{(x)})}{(ln(x))})}{x^{3}ln^{2}(x)} - \frac{6log_{x}^{tan(x)}*-2}{x^{3}ln^{3}(x)(x)} - \frac{6*-3log_{x}^{tan(x)}}{x^{4}ln^{3}(x)} - \frac{6(\frac{(\frac{(sec^{2}(x)(1))}{(tan(x))} - \frac{(1)log_{x}^{tan(x)}}{(x)})}{(ln(x))})}{x^{3}ln^{3}(x)} - \frac{6log_{x}^{tan(x)}*-3}{x^{3}ln^{4}(x)(x)}\\=&\frac{-8sec^{2}(x)}{x^{3}ln^{2}(x)tan(x)} - \frac{24sec^{2}(x)}{x^{3}ln^{3}(x)tan(x)} - \frac{6sec^{4}(x)}{x^{2}ln^{2}(x)tan^{2}(x)} + \frac{12sec^{2}(x)}{x^{2}ln^{2}(x)} - \frac{24sec^{2}(x)}{x^{3}ln^{4}(x)tan(x)} - \frac{12sec^{4}(x)}{x^{2}ln^{3}(x)tan^{2}(x)} + \frac{24sec^{2}(x)}{x^{2}ln^{3}(x)} - \frac{8sec^{6}(x)}{xln^{2}(x)tan^{3}(x)} + \frac{16sec^{4}(x)}{xln^{2}(x)tan(x)} - \frac{16tan(x)sec^{2}(x)}{xln^{2}(x)} - \frac{6sec^{8}(x)}{ln(x)tan^{4}(x)} + \frac{16sec^{6}(x)}{ln(x)tan^{2}(x)} - \frac{12sec^{4}(x)}{ln(x)} + \frac{8tan^{2}(x)sec^{2}(x)}{ln(x)} + \frac{6log_{x}^{tan(x)}}{x^{4}ln(x)} + \frac{22log_{x}^{tan(x)}}{x^{4}ln^{2}(x)} + \frac{36log_{x}^{tan(x)}}{x^{4}ln^{3}(x)} + \frac{24log_{x}^{tan(x)}}{x^{4}ln^{4}(x)}\\ \end{split}\end{equation} \]