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.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ log_{tan(x)}^{sin(cos(x))}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( log_{tan(x)}^{sin(cos(x))}\right)}{dx}\\=&(\frac{(\frac{(cos(cos(x))*-sin(x))}{(sin(cos(x)))} - \frac{(sec^{2}(x)(1))log_{tan(x)}^{sin(cos(x))}}{(tan(x))})}{(ln(tan(x)))})\\=&\frac{-sin(x)cos(cos(x))}{ln(tan(x))sin(cos(x))} - \frac{log_{tan(x)}^{sin(cos(x))}sec^{2}(x)}{ln(tan(x))tan(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-sin(x)cos(cos(x))}{ln(tan(x))sin(cos(x))} - \frac{log_{tan(x)}^{sin(cos(x))}sec^{2}(x)}{ln(tan(x))tan(x)}\right)}{dx}\\=&\frac{--sec^{2}(x)(1)sin(x)cos(cos(x))}{ln^{2}(tan(x))(tan(x))sin(cos(x))} - \frac{-cos(cos(x))*-sin(x)sin(x)cos(cos(x))}{ln(tan(x))sin^{2}(cos(x))} - \frac{cos(x)cos(cos(x))}{ln(tan(x))sin(cos(x))} - \frac{sin(x)*-sin(cos(x))*-sin(x)}{ln(tan(x))sin(cos(x))} - \frac{(\frac{(\frac{(cos(cos(x))*-sin(x))}{(sin(cos(x)))} - \frac{(sec^{2}(x)(1))log_{tan(x)}^{sin(cos(x))}}{(tan(x))})}{(ln(tan(x)))})sec^{2}(x)}{ln(tan(x))tan(x)} - \frac{log_{tan(x)}^{sin(cos(x))}*-sec^{2}(x)(1)sec^{2}(x)}{ln^{2}(tan(x))(tan(x))tan(x)} - \frac{log_{tan(x)}^{sin(cos(x))}*-sec^{2}(x)(1)sec^{2}(x)}{ln(tan(x))tan^{2}(x)} - \frac{log_{tan(x)}^{sin(cos(x))}*2sec^{2}(x)tan(x)}{ln(tan(x))tan(x)}\\=&\frac{2sin(x)cos(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin(cos(x))tan(x)} - \frac{sin^{2}(x)cos^{2}(cos(x))}{ln(tan(x))sin^{2}(cos(x))} - \frac{cos(x)cos(cos(x))}{ln(tan(x))sin(cos(x))} - \frac{sin^{2}(x)}{ln(tan(x))} + \frac{2log_{tan(x)}^{sin(cos(x))}sec^{4}(x)}{ln^{2}(tan(x))tan^{2}(x)} + \frac{log_{tan(x)}^{sin(cos(x))}sec^{4}(x)}{ln(tan(x))tan^{2}(x)} - \frac{2log_{tan(x)}^{sin(cos(x))}sec^{2}(x)}{ln(tan(x))}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{2sin(x)cos(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin(cos(x))tan(x)} - \frac{sin^{2}(x)cos^{2}(cos(x))}{ln(tan(x))sin^{2}(cos(x))} - \frac{cos(x)cos(cos(x))}{ln(tan(x))sin(cos(x))} - \frac{sin^{2}(x)}{ln(tan(x))} + \frac{2log_{tan(x)}^{sin(cos(x))}sec^{4}(x)}{ln^{2}(tan(x))tan^{2}(x)} + \frac{log_{tan(x)}^{sin(cos(x))}sec^{4}(x)}{ln(tan(x))tan^{2}(x)} - \frac{2log_{tan(x)}^{sin(cos(x))}sec^{2}(x)}{ln(tan(x))}\right)}{dx}\\=&\frac{2*-2sec^{2}(x)(1)sin(x)cos(cos(x))sec^{2}(x)}{ln^{3}(tan(x))(tan(x))sin(cos(x))tan(x)} + \frac{2*-cos(cos(x))*-sin(x)sin(x)cos(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin^{2}(cos(x))tan(x)} + \frac{2cos(x)cos(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin(cos(x))tan(x)} + \frac{2sin(x)*-sin(cos(x))*-sin(x)sec^{2}(x)}{ln^{2}(tan(x))sin(cos(x))tan(x)} + \frac{2sin(x)cos(cos(x))*-sec^{2}(x)(1)sec^{2}(x)}{ln^{2}(tan(x))sin(cos(x))tan^{2}(x)} + \frac{2sin(x)cos(cos(x))*2sec^{2}(x)tan(x)}{ln^{2}(tan(x))sin(cos(x))tan(x)} - \frac{-sec^{2}(x)(1)sin^{2}(x)cos^{2}(cos(x))}{ln^{2}(tan(x))(tan(x))sin^{2}(cos(x))} - \frac{2sin(x)cos(x)cos^{2}(cos(x))}{ln(tan(x))sin^{2}(cos(x))} - \frac{sin^{2}(x)*-2cos(cos(x))*-sin(x)cos^{2}(cos(x))}{ln(tan(x))sin^{3}(cos(x))} - \frac{sin^{2}(x)*-2cos(cos(x))sin(cos(x))*-sin(x)}{ln(tan(x))sin^{2}(cos(x))} - \frac{-sec^{2}(x)(1)cos(x)cos(cos(x))}{ln^{2}(tan(x))(tan(x))sin(cos(x))} - \frac{-cos(cos(x))*-sin(x)cos(x)cos(cos(x))}{ln(tan(x))sin^{2}(cos(x))} - \frac{-sin(x)cos(cos(x))}{ln(tan(x))sin(cos(x))} - \frac{cos(x)*-sin(cos(x))*-sin(x)}{ln(tan(x))sin(cos(x))} - \frac{-sec^{2}(x)(1)sin^{2}(x)}{ln^{2}(tan(x))(tan(x))} - \frac{2sin(x)cos(x)}{ln(tan(x))} + \frac{2(\frac{(\frac{(cos(cos(x))*-sin(x))}{(sin(cos(x)))} - \frac{(sec^{2}(x)(1))log_{tan(x)}^{sin(cos(x))}}{(tan(x))})}{(ln(tan(x)))})sec^{4}(x)}{ln^{2}(tan(x))tan^{2}(x)} + \frac{2log_{tan(x)}^{sin(cos(x))}*-2sec^{2}(x)(1)sec^{4}(x)}{ln^{3}(tan(x))(tan(x))tan^{2}(x)} + \frac{2log_{tan(x)}^{sin(cos(x))}*-2sec^{2}(x)(1)sec^{4}(x)}{ln^{2}(tan(x))tan^{3}(x)} + \frac{2log_{tan(x)}^{sin(cos(x))}*4sec^{4}(x)tan(x)}{ln^{2}(tan(x))tan^{2}(x)} + \frac{(\frac{(\frac{(cos(cos(x))*-sin(x))}{(sin(cos(x)))} - \frac{(sec^{2}(x)(1))log_{tan(x)}^{sin(cos(x))}}{(tan(x))})}{(ln(tan(x)))})sec^{4}(x)}{ln(tan(x))tan^{2}(x)} + \frac{log_{tan(x)}^{sin(cos(x))}*-sec^{2}(x)(1)sec^{4}(x)}{ln^{2}(tan(x))(tan(x))tan^{2}(x)} + \frac{log_{tan(x)}^{sin(cos(x))}*-2sec^{2}(x)(1)sec^{4}(x)}{ln(tan(x))tan^{3}(x)} + \frac{log_{tan(x)}^{sin(cos(x))}*4sec^{4}(x)tan(x)}{ln(tan(x))tan^{2}(x)} - \frac{2(\frac{(\frac{(cos(cos(x))*-sin(x))}{(sin(cos(x)))} - \frac{(sec^{2}(x)(1))log_{tan(x)}^{sin(cos(x))}}{(tan(x))})}{(ln(tan(x)))})sec^{2}(x)}{ln(tan(x))} - \frac{2log_{tan(x)}^{sin(cos(x))}*-sec^{2}(x)(1)sec^{2}(x)}{ln^{2}(tan(x))(tan(x))} - \frac{2log_{tan(x)}^{sin(cos(x))}*2sec^{2}(x)tan(x)}{ln(tan(x))}\\=&\frac{-6sin(x)cos(cos(x))sec^{4}(x)}{ln^{3}(tan(x))sin(cos(x))tan^{2}(x)} + \frac{3sin^{2}(x)cos^{2}(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin^{2}(cos(x))tan(x)} + \frac{3cos(x)cos(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin(cos(x))tan(x)} + \frac{3sin^{2}(x)sec^{2}(x)}{ln^{2}(tan(x))tan(x)} - \frac{3sin(x)cos(cos(x))sec^{4}(x)}{ln^{2}(tan(x))sin(cos(x))tan^{2}(x)} + \frac{6sin(x)cos(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin(cos(x))} - \frac{2sin(x)cos(x)cos^{2}(cos(x))}{ln(tan(x))sin^{2}(cos(x))} - \frac{sin(x)cos^{2}(cos(x))cos(x)}{ln(tan(x))sin^{2}(cos(x))} - \frac{2sin^{3}(x)cos(cos(x))}{ln(tan(x))sin(cos(x))} - \frac{2sin^{3}(x)cos^{3}(cos(x))}{ln(tan(x))sin^{3}(cos(x))} + \frac{sin(x)cos(cos(x))}{ln(tan(x))sin(cos(x))} - \frac{3sin(x)cos(x)}{ln(tan(x))} - \frac{6log_{tan(x)}^{sin(cos(x))}sec^{6}(x)}{ln^{3}(tan(x))tan^{3}(x)} - \frac{6log_{tan(x)}^{sin(cos(x))}sec^{6}(x)}{ln^{2}(tan(x))tan^{3}(x)} + \frac{12log_{tan(x)}^{sin(cos(x))}sec^{4}(x)}{ln^{2}(tan(x))tan(x)} - \frac{2log_{tan(x)}^{sin(cos(x))}sec^{6}(x)}{ln(tan(x))tan^{3}(x)} + \frac{4log_{tan(x)}^{sin(cos(x))}sec^{4}(x)}{ln(tan(x))tan(x)} - \frac{4log_{tan(x)}^{sin(cos(x))}tan(x)sec^{2}(x)}{ln(tan(x))}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-6sin(x)cos(cos(x))sec^{4}(x)}{ln^{3}(tan(x))sin(cos(x))tan^{2}(x)} + \frac{3sin^{2}(x)cos^{2}(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin^{2}(cos(x))tan(x)} + \frac{3cos(x)cos(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin(cos(x))tan(x)} + \frac{3sin^{2}(x)sec^{2}(x)}{ln^{2}(tan(x))tan(x)} - \frac{3sin(x)cos(cos(x))sec^{4}(x)}{ln^{2}(tan(x))sin(cos(x))tan^{2}(x)} + \frac{6sin(x)cos(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin(cos(x))} - \frac{2sin(x)cos(x)cos^{2}(cos(x))}{ln(tan(x))sin^{2}(cos(x))} - \frac{sin(x)cos^{2}(cos(x))cos(x)}{ln(tan(x))sin^{2}(cos(x))} - \frac{2sin^{3}(x)cos(cos(x))}{ln(tan(x))sin(cos(x))} - \frac{2sin^{3}(x)cos^{3}(cos(x))}{ln(tan(x))sin^{3}(cos(x))} + \frac{sin(x)cos(cos(x))}{ln(tan(x))sin(cos(x))} - \frac{3sin(x)cos(x)}{ln(tan(x))} - \frac{6log_{tan(x)}^{sin(cos(x))}sec^{6}(x)}{ln^{3}(tan(x))tan^{3}(x)} - \frac{6log_{tan(x)}^{sin(cos(x))}sec^{6}(x)}{ln^{2}(tan(x))tan^{3}(x)} + \frac{12log_{tan(x)}^{sin(cos(x))}sec^{4}(x)}{ln^{2}(tan(x))tan(x)} - \frac{2log_{tan(x)}^{sin(cos(x))}sec^{6}(x)}{ln(tan(x))tan^{3}(x)} + \frac{4log_{tan(x)}^{sin(cos(x))}sec^{4}(x)}{ln(tan(x))tan(x)} - \frac{4log_{tan(x)}^{sin(cos(x))}tan(x)sec^{2}(x)}{ln(tan(x))}\right)}{dx}\\=&\frac{-6*-3sec^{2}(x)(1)sin(x)cos(cos(x))sec^{4}(x)}{ln^{4}(tan(x))(tan(x))sin(cos(x))tan^{2}(x)} - \frac{6*-cos(cos(x))*-sin(x)sin(x)cos(cos(x))sec^{4}(x)}{ln^{3}(tan(x))sin^{2}(cos(x))tan^{2}(x)} - \frac{6cos(x)cos(cos(x))sec^{4}(x)}{ln^{3}(tan(x))sin(cos(x))tan^{2}(x)} - \frac{6sin(x)*-sin(cos(x))*-sin(x)sec^{4}(x)}{ln^{3}(tan(x))sin(cos(x))tan^{2}(x)} - \frac{6sin(x)cos(cos(x))*-2sec^{2}(x)(1)sec^{4}(x)}{ln^{3}(tan(x))sin(cos(x))tan^{3}(x)} - \frac{6sin(x)cos(cos(x))*4sec^{4}(x)tan(x)}{ln^{3}(tan(x))sin(cos(x))tan^{2}(x)} + \frac{3*-2sec^{2}(x)(1)sin^{2}(x)cos^{2}(cos(x))sec^{2}(x)}{ln^{3}(tan(x))(tan(x))sin^{2}(cos(x))tan(x)} + \frac{3*2sin(x)cos(x)cos^{2}(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin^{2}(cos(x))tan(x)} + \frac{3sin^{2}(x)*-2cos(cos(x))*-sin(x)cos^{2}(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin^{3}(cos(x))tan(x)} + \frac{3sin^{2}(x)*-2cos(cos(x))sin(cos(x))*-sin(x)sec^{2}(x)}{ln^{2}(tan(x))sin^{2}(cos(x))tan(x)} + \frac{3sin^{2}(x)cos^{2}(cos(x))*-sec^{2}(x)(1)sec^{2}(x)}{ln^{2}(tan(x))sin^{2}(cos(x))tan^{2}(x)} + \frac{3sin^{2}(x)cos^{2}(cos(x))*2sec^{2}(x)tan(x)}{ln^{2}(tan(x))sin^{2}(cos(x))tan(x)} + \frac{3*-2sec^{2}(x)(1)cos(x)cos(cos(x))sec^{2}(x)}{ln^{3}(tan(x))(tan(x))sin(cos(x))tan(x)} + \frac{3*-cos(cos(x))*-sin(x)cos(x)cos(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin^{2}(cos(x))tan(x)} + \frac{3*-sin(x)cos(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin(cos(x))tan(x)} + \frac{3cos(x)*-sin(cos(x))*-sin(x)sec^{2}(x)}{ln^{2}(tan(x))sin(cos(x))tan(x)} + \frac{3cos(x)cos(cos(x))*-sec^{2}(x)(1)sec^{2}(x)}{ln^{2}(tan(x))sin(cos(x))tan^{2}(x)} + \frac{3cos(x)cos(cos(x))*2sec^{2}(x)tan(x)}{ln^{2}(tan(x))sin(cos(x))tan(x)} + \frac{3*-2sec^{2}(x)(1)sin^{2}(x)sec^{2}(x)}{ln^{3}(tan(x))(tan(x))tan(x)} + \frac{3*2sin(x)cos(x)sec^{2}(x)}{ln^{2}(tan(x))tan(x)} + \frac{3sin^{2}(x)*-sec^{2}(x)(1)sec^{2}(x)}{ln^{2}(tan(x))tan^{2}(x)} + \frac{3sin^{2}(x)*2sec^{2}(x)tan(x)}{ln^{2}(tan(x))tan(x)} - \frac{3*-2sec^{2}(x)(1)sin(x)cos(cos(x))sec^{4}(x)}{ln^{3}(tan(x))(tan(x))sin(cos(x))tan^{2}(x)} - \frac{3*-cos(cos(x))*-sin(x)sin(x)cos(cos(x))sec^{4}(x)}{ln^{2}(tan(x))sin^{2}(cos(x))tan^{2}(x)} - \frac{3cos(x)cos(cos(x))sec^{4}(x)}{ln^{2}(tan(x))sin(cos(x))tan^{2}(x)} - \frac{3sin(x)*-sin(cos(x))*-sin(x)sec^{4}(x)}{ln^{2}(tan(x))sin(cos(x))tan^{2}(x)} - \frac{3sin(x)cos(cos(x))*-2sec^{2}(x)(1)sec^{4}(x)}{ln^{2}(tan(x))sin(cos(x))tan^{3}(x)} - \frac{3sin(x)cos(cos(x))*4sec^{4}(x)tan(x)}{ln^{2}(tan(x))sin(cos(x))tan^{2}(x)} + \frac{6*-2sec^{2}(x)(1)sin(x)cos(cos(x))sec^{2}(x)}{ln^{3}(tan(x))(tan(x))sin(cos(x))} + \frac{6*-cos(cos(x))*-sin(x)sin(x)cos(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin^{2}(cos(x))} + \frac{6cos(x)cos(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin(cos(x))} + \frac{6sin(x)*-sin(cos(x))*-sin(x)sec^{2}(x)}{ln^{2}(tan(x))sin(cos(x))} + \frac{6sin(x)cos(cos(x))*2sec^{2}(x)tan(x)}{ln^{2}(tan(x))sin(cos(x))} - \frac{2*-sec^{2}(x)(1)sin(x)cos(x)cos^{2}(cos(x))}{ln^{2}(tan(x))(tan(x))sin^{2}(cos(x))} - \frac{2cos(x)cos(x)cos^{2}(cos(x))}{ln(tan(x))sin^{2}(cos(x))} - \frac{2sin(x)*-2cos(cos(x))*-sin(x)cos(x)cos^{2}(cos(x))}{ln(tan(x))sin^{3}(cos(x))} - \frac{2sin(x)*-sin(x)cos^{2}(cos(x))}{ln(tan(x))sin^{2}(cos(x))} - \frac{2sin(x)cos(x)*-2cos(cos(x))sin(cos(x))*-sin(x)}{ln(tan(x))sin^{2}(cos(x))} - \frac{-sec^{2}(x)(1)sin(x)cos^{2}(cos(x))cos(x)}{ln^{2}(tan(x))(tan(x))sin^{2}(cos(x))} - \frac{cos(x)cos^{2}(cos(x))cos(x)}{ln(tan(x))sin^{2}(cos(x))} - \frac{sin(x)*-2cos(cos(x))*-sin(x)cos^{2}(cos(x))cos(x)}{ln(tan(x))sin^{3}(cos(x))} - \frac{sin(x)*-2cos(cos(x))sin(cos(x))*-sin(x)cos(x)}{ln(tan(x))sin^{2}(cos(x))} - \frac{sin(x)cos^{2}(cos(x))*-sin(x)}{ln(tan(x))sin^{2}(cos(x))} - \frac{2*-sec^{2}(x)(1)sin^{3}(x)cos(cos(x))}{ln^{2}(tan(x))(tan(x))sin(cos(x))} - \frac{2*-cos(cos(x))*-sin(x)sin^{3}(x)cos(cos(x))}{ln(tan(x))sin^{2}(cos(x))} - \frac{2*3sin^{2}(x)cos(x)cos(cos(x))}{ln(tan(x))sin(cos(x))} - \frac{2sin^{3}(x)*-sin(cos(x))*-sin(x)}{ln(tan(x))sin(cos(x))} - \frac{2*-sec^{2}(x)(1)sin^{3}(x)cos^{3}(cos(x))}{ln^{2}(tan(x))(tan(x))sin^{3}(cos(x))} - \frac{2*-3cos(cos(x))*-sin(x)sin^{3}(x)cos^{3}(cos(x))}{ln(tan(x))sin^{4}(cos(x))} - \frac{2*3sin^{2}(x)cos(x)cos^{3}(cos(x))}{ln(tan(x))sin^{3}(cos(x))} - \frac{2sin^{3}(x)*-3cos^{2}(cos(x))sin(cos(x))*-sin(x)}{ln(tan(x))sin^{3}(cos(x))} + \frac{-sec^{2}(x)(1)sin(x)cos(cos(x))}{ln^{2}(tan(x))(tan(x))sin(cos(x))} + \frac{cos(x)cos(cos(x))}{ln(tan(x))sin(cos(x))} + \frac{sin(x)*-cos(cos(x))*-sin(x)cos(cos(x))}{ln(tan(x))sin^{2}(cos(x))} + \frac{sin(x)*-sin(cos(x))*-sin(x)}{ln(tan(x))sin(cos(x))} - \frac{3*-sec^{2}(x)(1)sin(x)cos(x)}{ln^{2}(tan(x))(tan(x))} - \frac{3cos(x)cos(x)}{ln(tan(x))} - \frac{3sin(x)*-sin(x)}{ln(tan(x))} - \frac{6(\frac{(\frac{(cos(cos(x))*-sin(x))}{(sin(cos(x)))} - \frac{(sec^{2}(x)(1))log_{tan(x)}^{sin(cos(x))}}{(tan(x))})}{(ln(tan(x)))})sec^{6}(x)}{ln^{3}(tan(x))tan^{3}(x)} - \frac{6log_{tan(x)}^{sin(cos(x))}*-3sec^{2}(x)(1)sec^{6}(x)}{ln^{4}(tan(x))(tan(x))tan^{3}(x)} - \frac{6log_{tan(x)}^{sin(cos(x))}*-3sec^{2}(x)(1)sec^{6}(x)}{ln^{3}(tan(x))tan^{4}(x)} - \frac{6log_{tan(x)}^{sin(cos(x))}*6sec^{6}(x)tan(x)}{ln^{3}(tan(x))tan^{3}(x)} - \frac{6(\frac{(\frac{(cos(cos(x))*-sin(x))}{(sin(cos(x)))} - \frac{(sec^{2}(x)(1))log_{tan(x)}^{sin(cos(x))}}{(tan(x))})}{(ln(tan(x)))})sec^{6}(x)}{ln^{2}(tan(x))tan^{3}(x)} - \frac{6log_{tan(x)}^{sin(cos(x))}*-2sec^{2}(x)(1)sec^{6}(x)}{ln^{3}(tan(x))(tan(x))tan^{3}(x)} - \frac{6log_{tan(x)}^{sin(cos(x))}*-3sec^{2}(x)(1)sec^{6}(x)}{ln^{2}(tan(x))tan^{4}(x)} - \frac{6log_{tan(x)}^{sin(cos(x))}*6sec^{6}(x)tan(x)}{ln^{2}(tan(x))tan^{3}(x)} + \frac{12(\frac{(\frac{(cos(cos(x))*-sin(x))}{(sin(cos(x)))} - \frac{(sec^{2}(x)(1))log_{tan(x)}^{sin(cos(x))}}{(tan(x))})}{(ln(tan(x)))})sec^{4}(x)}{ln^{2}(tan(x))tan(x)} + \frac{12log_{tan(x)}^{sin(cos(x))}*-2sec^{2}(x)(1)sec^{4}(x)}{ln^{3}(tan(x))(tan(x))tan(x)} + \frac{12log_{tan(x)}^{sin(cos(x))}*-sec^{2}(x)(1)sec^{4}(x)}{ln^{2}(tan(x))tan^{2}(x)} + \frac{12log_{tan(x)}^{sin(cos(x))}*4sec^{4}(x)tan(x)}{ln^{2}(tan(x))tan(x)} - \frac{2(\frac{(\frac{(cos(cos(x))*-sin(x))}{(sin(cos(x)))} - \frac{(sec^{2}(x)(1))log_{tan(x)}^{sin(cos(x))}}{(tan(x))})}{(ln(tan(x)))})sec^{6}(x)}{ln(tan(x))tan^{3}(x)} - \frac{2log_{tan(x)}^{sin(cos(x))}*-sec^{2}(x)(1)sec^{6}(x)}{ln^{2}(tan(x))(tan(x))tan^{3}(x)} - \frac{2log_{tan(x)}^{sin(cos(x))}*-3sec^{2}(x)(1)sec^{6}(x)}{ln(tan(x))tan^{4}(x)} - \frac{2log_{tan(x)}^{sin(cos(x))}*6sec^{6}(x)tan(x)}{ln(tan(x))tan^{3}(x)} + \frac{4(\frac{(\frac{(cos(cos(x))*-sin(x))}{(sin(cos(x)))} - \frac{(sec^{2}(x)(1))log_{tan(x)}^{sin(cos(x))}}{(tan(x))})}{(ln(tan(x)))})sec^{4}(x)}{ln(tan(x))tan(x)} + \frac{4log_{tan(x)}^{sin(cos(x))}*-sec^{2}(x)(1)sec^{4}(x)}{ln^{2}(tan(x))(tan(x))tan(x)} + \frac{4log_{tan(x)}^{sin(cos(x))}*-sec^{2}(x)(1)sec^{4}(x)}{ln(tan(x))tan^{2}(x)} + \frac{4log_{tan(x)}^{sin(cos(x))}*4sec^{4}(x)tan(x)}{ln(tan(x))tan(x)} - \frac{4(\frac{(\frac{(cos(cos(x))*-sin(x))}{(sin(cos(x)))} - \frac{(sec^{2}(x)(1))log_{tan(x)}^{sin(cos(x))}}{(tan(x))})}{(ln(tan(x)))})tan(x)sec^{2}(x)}{ln(tan(x))} - \frac{4log_{tan(x)}^{sin(cos(x))}*-sec^{2}(x)(1)tan(x)sec^{2}(x)}{ln^{2}(tan(x))(tan(x))} - \frac{4log_{tan(x)}^{sin(cos(x))}sec^{2}(x)(1)sec^{2}(x)}{ln(tan(x))} - \frac{4log_{tan(x)}^{sin(cos(x))}tan(x)*2sec^{2}(x)tan(x)}{ln(tan(x))}\\=&\frac{24sin(x)cos(cos(x))sec^{6}(x)}{ln^{4}(tan(x))sin(cos(x))tan^{3}(x)} - \frac{12sin^{2}(x)cos^{2}(cos(x))sec^{4}(x)}{ln^{3}(tan(x))sin^{2}(cos(x))tan^{2}(x)} - \frac{12cos(x)cos(cos(x))sec^{4}(x)}{ln^{3}(tan(x))sin(cos(x))tan^{2}(x)} - \frac{12sin^{2}(x)sec^{4}(x)}{ln^{3}(tan(x))tan^{2}(x)} + \frac{24sin(x)cos(cos(x))sec^{6}(x)}{ln^{3}(tan(x))sin(cos(x))tan^{3}(x)} - \frac{48sin(x)cos(cos(x))sec^{4}(x)}{ln^{3}(tan(x))sin(cos(x))tan(x)} + \frac{8sin(x)cos(x)cos^{2}(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin^{2}(cos(x))tan(x)} + \frac{8sin^{3}(x)cos^{3}(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin^{3}(cos(x))tan(x)} + \frac{8sin^{3}(x)cos(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin(cos(x))tan(x)} - \frac{6sin^{2}(x)cos^{2}(cos(x))sec^{4}(x)}{ln^{2}(tan(x))sin^{2}(cos(x))tan^{2}(x)} + \frac{12sin^{2}(x)cos^{2}(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin^{2}(cos(x))} + \frac{4sin(x)cos^{2}(cos(x))cos(x)sec^{2}(x)}{ln^{2}(tan(x))sin^{2}(cos(x))tan(x)} + \frac{8sin(x)cos(cos(x))sec^{6}(x)}{ln^{2}(tan(x))sin(cos(x))tan^{3}(x)} + \frac{12sin(x)cos(x)sec^{2}(x)}{ln^{2}(tan(x))tan(x)} - \frac{6cos(x)cos(cos(x))sec^{4}(x)}{ln^{2}(tan(x))sin(cos(x))tan^{2}(x)} + \frac{12cos(x)cos(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin(cos(x))} - \frac{6sin^{2}(x)sec^{4}(x)}{ln^{2}(tan(x))tan^{2}(x)} + \frac{12sin^{2}(x)sec^{2}(x)}{ln^{2}(tan(x))} - \frac{4sin(x)cos(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin(cos(x))tan(x)} - \frac{16sin(x)cos(cos(x))sec^{4}(x)}{ln^{2}(tan(x))sin(cos(x))tan(x)} + \frac{16sin(x)cos(cos(x))tan(x)sec^{2}(x)}{ln^{2}(tan(x))sin(cos(x))} - \frac{3cos^{2}(x)cos^{2}(cos(x))}{ln(tan(x))sin^{2}(cos(x))} - \frac{6sin^{2}(x)cos^{3}(cos(x))cos(x)}{ln(tan(x))sin^{3}(cos(x))} - \frac{6sin^{2}(x)cos(cos(x))cos(x)}{ln(tan(x))sin(cos(x))} - \frac{6sin^{2}(x)cos(x)cos(cos(x))}{ln(tan(x))sin(cos(x))} - \frac{6sin^{2}(x)cos(x)cos^{3}(cos(x))}{ln(tan(x))sin^{3}(cos(x))} + \frac{3sin^{2}(x)cos^{2}(cos(x))}{ln(tan(x))sin^{2}(cos(x))} - \frac{6sin^{4}(x)cos^{4}(cos(x))}{ln(tan(x))sin^{4}(cos(x))} - \frac{8sin^{4}(x)cos^{2}(cos(x))}{ln(tan(x))sin^{2}(cos(x))} + \frac{cos(x)cos(cos(x))}{ln(tan(x))sin(cos(x))} + \frac{sin^{2}(x)cos^{2}(cos(x))}{ln(tan(x))sin^{2}(cos(x))} - \frac{2sin^{4}(x)}{ln(tan(x))} + \frac{4sin^{2}(x)}{ln(tan(x))} - \frac{3cos^{2}(x)}{ln(tan(x))} + \frac{24log_{tan(x)}^{sin(cos(x))}sec^{8}(x)}{ln^{4}(tan(x))tan^{4}(x)} + \frac{36log_{tan(x)}^{sin(cos(x))}sec^{8}(x)}{ln^{3}(tan(x))tan^{4}(x)} - \frac{72log_{tan(x)}^{sin(cos(x))}sec^{6}(x)}{ln^{3}(tan(x))tan^{2}(x)} + \frac{22log_{tan(x)}^{sin(cos(x))}sec^{8}(x)}{ln^{2}(tan(x))tan^{4}(x)} - \frac{56log_{tan(x)}^{sin(cos(x))}sec^{6}(x)}{ln^{2}(tan(x))tan^{2}(x)} + \frac{56log_{tan(x)}^{sin(cos(x))}sec^{4}(x)}{ln^{2}(tan(x))} + \frac{6log_{tan(x)}^{sin(cos(x))}sec^{8}(x)}{ln(tan(x))tan^{4}(x)} - \frac{16log_{tan(x)}^{sin(cos(x))}sec^{6}(x)}{ln(tan(x))tan^{2}(x)} + \frac{12log_{tan(x)}^{sin(cos(x))}sec^{4}(x)}{ln(tan(x))} - \frac{8log_{tan(x)}^{sin(cos(x))}tan^{2}(x)sec^{2}(x)}{ln(tan(x))}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ log_{tan(x)}^{sin(cos(x))}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( log_{tan(x)}^{sin(cos(x))}\right)}{dx}\\=&(\frac{(\frac{(cos(cos(x))*-sin(x))}{(sin(cos(x)))} - \frac{(sec^{2}(x)(1))log_{tan(x)}^{sin(cos(x))}}{(tan(x))})}{(ln(tan(x)))})\\=&\frac{-sin(x)cos(cos(x))}{ln(tan(x))sin(cos(x))} - \frac{log_{tan(x)}^{sin(cos(x))}sec^{2}(x)}{ln(tan(x))tan(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-sin(x)cos(cos(x))}{ln(tan(x))sin(cos(x))} - \frac{log_{tan(x)}^{sin(cos(x))}sec^{2}(x)}{ln(tan(x))tan(x)}\right)}{dx}\\=&\frac{--sec^{2}(x)(1)sin(x)cos(cos(x))}{ln^{2}(tan(x))(tan(x))sin(cos(x))} - \frac{-cos(cos(x))*-sin(x)sin(x)cos(cos(x))}{ln(tan(x))sin^{2}(cos(x))} - \frac{cos(x)cos(cos(x))}{ln(tan(x))sin(cos(x))} - \frac{sin(x)*-sin(cos(x))*-sin(x)}{ln(tan(x))sin(cos(x))} - \frac{(\frac{(\frac{(cos(cos(x))*-sin(x))}{(sin(cos(x)))} - \frac{(sec^{2}(x)(1))log_{tan(x)}^{sin(cos(x))}}{(tan(x))})}{(ln(tan(x)))})sec^{2}(x)}{ln(tan(x))tan(x)} - \frac{log_{tan(x)}^{sin(cos(x))}*-sec^{2}(x)(1)sec^{2}(x)}{ln^{2}(tan(x))(tan(x))tan(x)} - \frac{log_{tan(x)}^{sin(cos(x))}*-sec^{2}(x)(1)sec^{2}(x)}{ln(tan(x))tan^{2}(x)} - \frac{log_{tan(x)}^{sin(cos(x))}*2sec^{2}(x)tan(x)}{ln(tan(x))tan(x)}\\=&\frac{2sin(x)cos(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin(cos(x))tan(x)} - \frac{sin^{2}(x)cos^{2}(cos(x))}{ln(tan(x))sin^{2}(cos(x))} - \frac{cos(x)cos(cos(x))}{ln(tan(x))sin(cos(x))} - \frac{sin^{2}(x)}{ln(tan(x))} + \frac{2log_{tan(x)}^{sin(cos(x))}sec^{4}(x)}{ln^{2}(tan(x))tan^{2}(x)} + \frac{log_{tan(x)}^{sin(cos(x))}sec^{4}(x)}{ln(tan(x))tan^{2}(x)} - \frac{2log_{tan(x)}^{sin(cos(x))}sec^{2}(x)}{ln(tan(x))}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{2sin(x)cos(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin(cos(x))tan(x)} - \frac{sin^{2}(x)cos^{2}(cos(x))}{ln(tan(x))sin^{2}(cos(x))} - \frac{cos(x)cos(cos(x))}{ln(tan(x))sin(cos(x))} - \frac{sin^{2}(x)}{ln(tan(x))} + \frac{2log_{tan(x)}^{sin(cos(x))}sec^{4}(x)}{ln^{2}(tan(x))tan^{2}(x)} + \frac{log_{tan(x)}^{sin(cos(x))}sec^{4}(x)}{ln(tan(x))tan^{2}(x)} - \frac{2log_{tan(x)}^{sin(cos(x))}sec^{2}(x)}{ln(tan(x))}\right)}{dx}\\=&\frac{2*-2sec^{2}(x)(1)sin(x)cos(cos(x))sec^{2}(x)}{ln^{3}(tan(x))(tan(x))sin(cos(x))tan(x)} + \frac{2*-cos(cos(x))*-sin(x)sin(x)cos(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin^{2}(cos(x))tan(x)} + \frac{2cos(x)cos(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin(cos(x))tan(x)} + \frac{2sin(x)*-sin(cos(x))*-sin(x)sec^{2}(x)}{ln^{2}(tan(x))sin(cos(x))tan(x)} + \frac{2sin(x)cos(cos(x))*-sec^{2}(x)(1)sec^{2}(x)}{ln^{2}(tan(x))sin(cos(x))tan^{2}(x)} + \frac{2sin(x)cos(cos(x))*2sec^{2}(x)tan(x)}{ln^{2}(tan(x))sin(cos(x))tan(x)} - \frac{-sec^{2}(x)(1)sin^{2}(x)cos^{2}(cos(x))}{ln^{2}(tan(x))(tan(x))sin^{2}(cos(x))} - \frac{2sin(x)cos(x)cos^{2}(cos(x))}{ln(tan(x))sin^{2}(cos(x))} - \frac{sin^{2}(x)*-2cos(cos(x))*-sin(x)cos^{2}(cos(x))}{ln(tan(x))sin^{3}(cos(x))} - \frac{sin^{2}(x)*-2cos(cos(x))sin(cos(x))*-sin(x)}{ln(tan(x))sin^{2}(cos(x))} - \frac{-sec^{2}(x)(1)cos(x)cos(cos(x))}{ln^{2}(tan(x))(tan(x))sin(cos(x))} - \frac{-cos(cos(x))*-sin(x)cos(x)cos(cos(x))}{ln(tan(x))sin^{2}(cos(x))} - \frac{-sin(x)cos(cos(x))}{ln(tan(x))sin(cos(x))} - \frac{cos(x)*-sin(cos(x))*-sin(x)}{ln(tan(x))sin(cos(x))} - \frac{-sec^{2}(x)(1)sin^{2}(x)}{ln^{2}(tan(x))(tan(x))} - \frac{2sin(x)cos(x)}{ln(tan(x))} + \frac{2(\frac{(\frac{(cos(cos(x))*-sin(x))}{(sin(cos(x)))} - \frac{(sec^{2}(x)(1))log_{tan(x)}^{sin(cos(x))}}{(tan(x))})}{(ln(tan(x)))})sec^{4}(x)}{ln^{2}(tan(x))tan^{2}(x)} + \frac{2log_{tan(x)}^{sin(cos(x))}*-2sec^{2}(x)(1)sec^{4}(x)}{ln^{3}(tan(x))(tan(x))tan^{2}(x)} + \frac{2log_{tan(x)}^{sin(cos(x))}*-2sec^{2}(x)(1)sec^{4}(x)}{ln^{2}(tan(x))tan^{3}(x)} + \frac{2log_{tan(x)}^{sin(cos(x))}*4sec^{4}(x)tan(x)}{ln^{2}(tan(x))tan^{2}(x)} + \frac{(\frac{(\frac{(cos(cos(x))*-sin(x))}{(sin(cos(x)))} - \frac{(sec^{2}(x)(1))log_{tan(x)}^{sin(cos(x))}}{(tan(x))})}{(ln(tan(x)))})sec^{4}(x)}{ln(tan(x))tan^{2}(x)} + \frac{log_{tan(x)}^{sin(cos(x))}*-sec^{2}(x)(1)sec^{4}(x)}{ln^{2}(tan(x))(tan(x))tan^{2}(x)} + \frac{log_{tan(x)}^{sin(cos(x))}*-2sec^{2}(x)(1)sec^{4}(x)}{ln(tan(x))tan^{3}(x)} + \frac{log_{tan(x)}^{sin(cos(x))}*4sec^{4}(x)tan(x)}{ln(tan(x))tan^{2}(x)} - \frac{2(\frac{(\frac{(cos(cos(x))*-sin(x))}{(sin(cos(x)))} - \frac{(sec^{2}(x)(1))log_{tan(x)}^{sin(cos(x))}}{(tan(x))})}{(ln(tan(x)))})sec^{2}(x)}{ln(tan(x))} - \frac{2log_{tan(x)}^{sin(cos(x))}*-sec^{2}(x)(1)sec^{2}(x)}{ln^{2}(tan(x))(tan(x))} - \frac{2log_{tan(x)}^{sin(cos(x))}*2sec^{2}(x)tan(x)}{ln(tan(x))}\\=&\frac{-6sin(x)cos(cos(x))sec^{4}(x)}{ln^{3}(tan(x))sin(cos(x))tan^{2}(x)} + \frac{3sin^{2}(x)cos^{2}(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin^{2}(cos(x))tan(x)} + \frac{3cos(x)cos(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin(cos(x))tan(x)} + \frac{3sin^{2}(x)sec^{2}(x)}{ln^{2}(tan(x))tan(x)} - \frac{3sin(x)cos(cos(x))sec^{4}(x)}{ln^{2}(tan(x))sin(cos(x))tan^{2}(x)} + \frac{6sin(x)cos(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin(cos(x))} - \frac{2sin(x)cos(x)cos^{2}(cos(x))}{ln(tan(x))sin^{2}(cos(x))} - \frac{sin(x)cos^{2}(cos(x))cos(x)}{ln(tan(x))sin^{2}(cos(x))} - \frac{2sin^{3}(x)cos(cos(x))}{ln(tan(x))sin(cos(x))} - \frac{2sin^{3}(x)cos^{3}(cos(x))}{ln(tan(x))sin^{3}(cos(x))} + \frac{sin(x)cos(cos(x))}{ln(tan(x))sin(cos(x))} - \frac{3sin(x)cos(x)}{ln(tan(x))} - \frac{6log_{tan(x)}^{sin(cos(x))}sec^{6}(x)}{ln^{3}(tan(x))tan^{3}(x)} - \frac{6log_{tan(x)}^{sin(cos(x))}sec^{6}(x)}{ln^{2}(tan(x))tan^{3}(x)} + \frac{12log_{tan(x)}^{sin(cos(x))}sec^{4}(x)}{ln^{2}(tan(x))tan(x)} - \frac{2log_{tan(x)}^{sin(cos(x))}sec^{6}(x)}{ln(tan(x))tan^{3}(x)} + \frac{4log_{tan(x)}^{sin(cos(x))}sec^{4}(x)}{ln(tan(x))tan(x)} - \frac{4log_{tan(x)}^{sin(cos(x))}tan(x)sec^{2}(x)}{ln(tan(x))}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-6sin(x)cos(cos(x))sec^{4}(x)}{ln^{3}(tan(x))sin(cos(x))tan^{2}(x)} + \frac{3sin^{2}(x)cos^{2}(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin^{2}(cos(x))tan(x)} + \frac{3cos(x)cos(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin(cos(x))tan(x)} + \frac{3sin^{2}(x)sec^{2}(x)}{ln^{2}(tan(x))tan(x)} - \frac{3sin(x)cos(cos(x))sec^{4}(x)}{ln^{2}(tan(x))sin(cos(x))tan^{2}(x)} + \frac{6sin(x)cos(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin(cos(x))} - \frac{2sin(x)cos(x)cos^{2}(cos(x))}{ln(tan(x))sin^{2}(cos(x))} - \frac{sin(x)cos^{2}(cos(x))cos(x)}{ln(tan(x))sin^{2}(cos(x))} - \frac{2sin^{3}(x)cos(cos(x))}{ln(tan(x))sin(cos(x))} - \frac{2sin^{3}(x)cos^{3}(cos(x))}{ln(tan(x))sin^{3}(cos(x))} + \frac{sin(x)cos(cos(x))}{ln(tan(x))sin(cos(x))} - \frac{3sin(x)cos(x)}{ln(tan(x))} - \frac{6log_{tan(x)}^{sin(cos(x))}sec^{6}(x)}{ln^{3}(tan(x))tan^{3}(x)} - \frac{6log_{tan(x)}^{sin(cos(x))}sec^{6}(x)}{ln^{2}(tan(x))tan^{3}(x)} + \frac{12log_{tan(x)}^{sin(cos(x))}sec^{4}(x)}{ln^{2}(tan(x))tan(x)} - \frac{2log_{tan(x)}^{sin(cos(x))}sec^{6}(x)}{ln(tan(x))tan^{3}(x)} + \frac{4log_{tan(x)}^{sin(cos(x))}sec^{4}(x)}{ln(tan(x))tan(x)} - \frac{4log_{tan(x)}^{sin(cos(x))}tan(x)sec^{2}(x)}{ln(tan(x))}\right)}{dx}\\=&\frac{-6*-3sec^{2}(x)(1)sin(x)cos(cos(x))sec^{4}(x)}{ln^{4}(tan(x))(tan(x))sin(cos(x))tan^{2}(x)} - \frac{6*-cos(cos(x))*-sin(x)sin(x)cos(cos(x))sec^{4}(x)}{ln^{3}(tan(x))sin^{2}(cos(x))tan^{2}(x)} - \frac{6cos(x)cos(cos(x))sec^{4}(x)}{ln^{3}(tan(x))sin(cos(x))tan^{2}(x)} - \frac{6sin(x)*-sin(cos(x))*-sin(x)sec^{4}(x)}{ln^{3}(tan(x))sin(cos(x))tan^{2}(x)} - \frac{6sin(x)cos(cos(x))*-2sec^{2}(x)(1)sec^{4}(x)}{ln^{3}(tan(x))sin(cos(x))tan^{3}(x)} - \frac{6sin(x)cos(cos(x))*4sec^{4}(x)tan(x)}{ln^{3}(tan(x))sin(cos(x))tan^{2}(x)} + \frac{3*-2sec^{2}(x)(1)sin^{2}(x)cos^{2}(cos(x))sec^{2}(x)}{ln^{3}(tan(x))(tan(x))sin^{2}(cos(x))tan(x)} + \frac{3*2sin(x)cos(x)cos^{2}(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin^{2}(cos(x))tan(x)} + \frac{3sin^{2}(x)*-2cos(cos(x))*-sin(x)cos^{2}(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin^{3}(cos(x))tan(x)} + \frac{3sin^{2}(x)*-2cos(cos(x))sin(cos(x))*-sin(x)sec^{2}(x)}{ln^{2}(tan(x))sin^{2}(cos(x))tan(x)} + \frac{3sin^{2}(x)cos^{2}(cos(x))*-sec^{2}(x)(1)sec^{2}(x)}{ln^{2}(tan(x))sin^{2}(cos(x))tan^{2}(x)} + \frac{3sin^{2}(x)cos^{2}(cos(x))*2sec^{2}(x)tan(x)}{ln^{2}(tan(x))sin^{2}(cos(x))tan(x)} + \frac{3*-2sec^{2}(x)(1)cos(x)cos(cos(x))sec^{2}(x)}{ln^{3}(tan(x))(tan(x))sin(cos(x))tan(x)} + \frac{3*-cos(cos(x))*-sin(x)cos(x)cos(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin^{2}(cos(x))tan(x)} + \frac{3*-sin(x)cos(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin(cos(x))tan(x)} + \frac{3cos(x)*-sin(cos(x))*-sin(x)sec^{2}(x)}{ln^{2}(tan(x))sin(cos(x))tan(x)} + \frac{3cos(x)cos(cos(x))*-sec^{2}(x)(1)sec^{2}(x)}{ln^{2}(tan(x))sin(cos(x))tan^{2}(x)} + \frac{3cos(x)cos(cos(x))*2sec^{2}(x)tan(x)}{ln^{2}(tan(x))sin(cos(x))tan(x)} + \frac{3*-2sec^{2}(x)(1)sin^{2}(x)sec^{2}(x)}{ln^{3}(tan(x))(tan(x))tan(x)} + \frac{3*2sin(x)cos(x)sec^{2}(x)}{ln^{2}(tan(x))tan(x)} + \frac{3sin^{2}(x)*-sec^{2}(x)(1)sec^{2}(x)}{ln^{2}(tan(x))tan^{2}(x)} + \frac{3sin^{2}(x)*2sec^{2}(x)tan(x)}{ln^{2}(tan(x))tan(x)} - \frac{3*-2sec^{2}(x)(1)sin(x)cos(cos(x))sec^{4}(x)}{ln^{3}(tan(x))(tan(x))sin(cos(x))tan^{2}(x)} - \frac{3*-cos(cos(x))*-sin(x)sin(x)cos(cos(x))sec^{4}(x)}{ln^{2}(tan(x))sin^{2}(cos(x))tan^{2}(x)} - \frac{3cos(x)cos(cos(x))sec^{4}(x)}{ln^{2}(tan(x))sin(cos(x))tan^{2}(x)} - \frac{3sin(x)*-sin(cos(x))*-sin(x)sec^{4}(x)}{ln^{2}(tan(x))sin(cos(x))tan^{2}(x)} - \frac{3sin(x)cos(cos(x))*-2sec^{2}(x)(1)sec^{4}(x)}{ln^{2}(tan(x))sin(cos(x))tan^{3}(x)} - \frac{3sin(x)cos(cos(x))*4sec^{4}(x)tan(x)}{ln^{2}(tan(x))sin(cos(x))tan^{2}(x)} + \frac{6*-2sec^{2}(x)(1)sin(x)cos(cos(x))sec^{2}(x)}{ln^{3}(tan(x))(tan(x))sin(cos(x))} + \frac{6*-cos(cos(x))*-sin(x)sin(x)cos(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin^{2}(cos(x))} + \frac{6cos(x)cos(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin(cos(x))} + \frac{6sin(x)*-sin(cos(x))*-sin(x)sec^{2}(x)}{ln^{2}(tan(x))sin(cos(x))} + \frac{6sin(x)cos(cos(x))*2sec^{2}(x)tan(x)}{ln^{2}(tan(x))sin(cos(x))} - \frac{2*-sec^{2}(x)(1)sin(x)cos(x)cos^{2}(cos(x))}{ln^{2}(tan(x))(tan(x))sin^{2}(cos(x))} - \frac{2cos(x)cos(x)cos^{2}(cos(x))}{ln(tan(x))sin^{2}(cos(x))} - \frac{2sin(x)*-2cos(cos(x))*-sin(x)cos(x)cos^{2}(cos(x))}{ln(tan(x))sin^{3}(cos(x))} - \frac{2sin(x)*-sin(x)cos^{2}(cos(x))}{ln(tan(x))sin^{2}(cos(x))} - \frac{2sin(x)cos(x)*-2cos(cos(x))sin(cos(x))*-sin(x)}{ln(tan(x))sin^{2}(cos(x))} - \frac{-sec^{2}(x)(1)sin(x)cos^{2}(cos(x))cos(x)}{ln^{2}(tan(x))(tan(x))sin^{2}(cos(x))} - \frac{cos(x)cos^{2}(cos(x))cos(x)}{ln(tan(x))sin^{2}(cos(x))} - \frac{sin(x)*-2cos(cos(x))*-sin(x)cos^{2}(cos(x))cos(x)}{ln(tan(x))sin^{3}(cos(x))} - \frac{sin(x)*-2cos(cos(x))sin(cos(x))*-sin(x)cos(x)}{ln(tan(x))sin^{2}(cos(x))} - \frac{sin(x)cos^{2}(cos(x))*-sin(x)}{ln(tan(x))sin^{2}(cos(x))} - \frac{2*-sec^{2}(x)(1)sin^{3}(x)cos(cos(x))}{ln^{2}(tan(x))(tan(x))sin(cos(x))} - \frac{2*-cos(cos(x))*-sin(x)sin^{3}(x)cos(cos(x))}{ln(tan(x))sin^{2}(cos(x))} - \frac{2*3sin^{2}(x)cos(x)cos(cos(x))}{ln(tan(x))sin(cos(x))} - \frac{2sin^{3}(x)*-sin(cos(x))*-sin(x)}{ln(tan(x))sin(cos(x))} - \frac{2*-sec^{2}(x)(1)sin^{3}(x)cos^{3}(cos(x))}{ln^{2}(tan(x))(tan(x))sin^{3}(cos(x))} - \frac{2*-3cos(cos(x))*-sin(x)sin^{3}(x)cos^{3}(cos(x))}{ln(tan(x))sin^{4}(cos(x))} - \frac{2*3sin^{2}(x)cos(x)cos^{3}(cos(x))}{ln(tan(x))sin^{3}(cos(x))} - \frac{2sin^{3}(x)*-3cos^{2}(cos(x))sin(cos(x))*-sin(x)}{ln(tan(x))sin^{3}(cos(x))} + \frac{-sec^{2}(x)(1)sin(x)cos(cos(x))}{ln^{2}(tan(x))(tan(x))sin(cos(x))} + \frac{cos(x)cos(cos(x))}{ln(tan(x))sin(cos(x))} + \frac{sin(x)*-cos(cos(x))*-sin(x)cos(cos(x))}{ln(tan(x))sin^{2}(cos(x))} + \frac{sin(x)*-sin(cos(x))*-sin(x)}{ln(tan(x))sin(cos(x))} - \frac{3*-sec^{2}(x)(1)sin(x)cos(x)}{ln^{2}(tan(x))(tan(x))} - \frac{3cos(x)cos(x)}{ln(tan(x))} - \frac{3sin(x)*-sin(x)}{ln(tan(x))} - \frac{6(\frac{(\frac{(cos(cos(x))*-sin(x))}{(sin(cos(x)))} - \frac{(sec^{2}(x)(1))log_{tan(x)}^{sin(cos(x))}}{(tan(x))})}{(ln(tan(x)))})sec^{6}(x)}{ln^{3}(tan(x))tan^{3}(x)} - \frac{6log_{tan(x)}^{sin(cos(x))}*-3sec^{2}(x)(1)sec^{6}(x)}{ln^{4}(tan(x))(tan(x))tan^{3}(x)} - \frac{6log_{tan(x)}^{sin(cos(x))}*-3sec^{2}(x)(1)sec^{6}(x)}{ln^{3}(tan(x))tan^{4}(x)} - \frac{6log_{tan(x)}^{sin(cos(x))}*6sec^{6}(x)tan(x)}{ln^{3}(tan(x))tan^{3}(x)} - \frac{6(\frac{(\frac{(cos(cos(x))*-sin(x))}{(sin(cos(x)))} - \frac{(sec^{2}(x)(1))log_{tan(x)}^{sin(cos(x))}}{(tan(x))})}{(ln(tan(x)))})sec^{6}(x)}{ln^{2}(tan(x))tan^{3}(x)} - \frac{6log_{tan(x)}^{sin(cos(x))}*-2sec^{2}(x)(1)sec^{6}(x)}{ln^{3}(tan(x))(tan(x))tan^{3}(x)} - \frac{6log_{tan(x)}^{sin(cos(x))}*-3sec^{2}(x)(1)sec^{6}(x)}{ln^{2}(tan(x))tan^{4}(x)} - \frac{6log_{tan(x)}^{sin(cos(x))}*6sec^{6}(x)tan(x)}{ln^{2}(tan(x))tan^{3}(x)} + \frac{12(\frac{(\frac{(cos(cos(x))*-sin(x))}{(sin(cos(x)))} - \frac{(sec^{2}(x)(1))log_{tan(x)}^{sin(cos(x))}}{(tan(x))})}{(ln(tan(x)))})sec^{4}(x)}{ln^{2}(tan(x))tan(x)} + \frac{12log_{tan(x)}^{sin(cos(x))}*-2sec^{2}(x)(1)sec^{4}(x)}{ln^{3}(tan(x))(tan(x))tan(x)} + \frac{12log_{tan(x)}^{sin(cos(x))}*-sec^{2}(x)(1)sec^{4}(x)}{ln^{2}(tan(x))tan^{2}(x)} + \frac{12log_{tan(x)}^{sin(cos(x))}*4sec^{4}(x)tan(x)}{ln^{2}(tan(x))tan(x)} - \frac{2(\frac{(\frac{(cos(cos(x))*-sin(x))}{(sin(cos(x)))} - \frac{(sec^{2}(x)(1))log_{tan(x)}^{sin(cos(x))}}{(tan(x))})}{(ln(tan(x)))})sec^{6}(x)}{ln(tan(x))tan^{3}(x)} - \frac{2log_{tan(x)}^{sin(cos(x))}*-sec^{2}(x)(1)sec^{6}(x)}{ln^{2}(tan(x))(tan(x))tan^{3}(x)} - \frac{2log_{tan(x)}^{sin(cos(x))}*-3sec^{2}(x)(1)sec^{6}(x)}{ln(tan(x))tan^{4}(x)} - \frac{2log_{tan(x)}^{sin(cos(x))}*6sec^{6}(x)tan(x)}{ln(tan(x))tan^{3}(x)} + \frac{4(\frac{(\frac{(cos(cos(x))*-sin(x))}{(sin(cos(x)))} - \frac{(sec^{2}(x)(1))log_{tan(x)}^{sin(cos(x))}}{(tan(x))})}{(ln(tan(x)))})sec^{4}(x)}{ln(tan(x))tan(x)} + \frac{4log_{tan(x)}^{sin(cos(x))}*-sec^{2}(x)(1)sec^{4}(x)}{ln^{2}(tan(x))(tan(x))tan(x)} + \frac{4log_{tan(x)}^{sin(cos(x))}*-sec^{2}(x)(1)sec^{4}(x)}{ln(tan(x))tan^{2}(x)} + \frac{4log_{tan(x)}^{sin(cos(x))}*4sec^{4}(x)tan(x)}{ln(tan(x))tan(x)} - \frac{4(\frac{(\frac{(cos(cos(x))*-sin(x))}{(sin(cos(x)))} - \frac{(sec^{2}(x)(1))log_{tan(x)}^{sin(cos(x))}}{(tan(x))})}{(ln(tan(x)))})tan(x)sec^{2}(x)}{ln(tan(x))} - \frac{4log_{tan(x)}^{sin(cos(x))}*-sec^{2}(x)(1)tan(x)sec^{2}(x)}{ln^{2}(tan(x))(tan(x))} - \frac{4log_{tan(x)}^{sin(cos(x))}sec^{2}(x)(1)sec^{2}(x)}{ln(tan(x))} - \frac{4log_{tan(x)}^{sin(cos(x))}tan(x)*2sec^{2}(x)tan(x)}{ln(tan(x))}\\=&\frac{24sin(x)cos(cos(x))sec^{6}(x)}{ln^{4}(tan(x))sin(cos(x))tan^{3}(x)} - \frac{12sin^{2}(x)cos^{2}(cos(x))sec^{4}(x)}{ln^{3}(tan(x))sin^{2}(cos(x))tan^{2}(x)} - \frac{12cos(x)cos(cos(x))sec^{4}(x)}{ln^{3}(tan(x))sin(cos(x))tan^{2}(x)} - \frac{12sin^{2}(x)sec^{4}(x)}{ln^{3}(tan(x))tan^{2}(x)} + \frac{24sin(x)cos(cos(x))sec^{6}(x)}{ln^{3}(tan(x))sin(cos(x))tan^{3}(x)} - \frac{48sin(x)cos(cos(x))sec^{4}(x)}{ln^{3}(tan(x))sin(cos(x))tan(x)} + \frac{8sin(x)cos(x)cos^{2}(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin^{2}(cos(x))tan(x)} + \frac{8sin^{3}(x)cos^{3}(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin^{3}(cos(x))tan(x)} + \frac{8sin^{3}(x)cos(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin(cos(x))tan(x)} - \frac{6sin^{2}(x)cos^{2}(cos(x))sec^{4}(x)}{ln^{2}(tan(x))sin^{2}(cos(x))tan^{2}(x)} + \frac{12sin^{2}(x)cos^{2}(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin^{2}(cos(x))} + \frac{4sin(x)cos^{2}(cos(x))cos(x)sec^{2}(x)}{ln^{2}(tan(x))sin^{2}(cos(x))tan(x)} + \frac{8sin(x)cos(cos(x))sec^{6}(x)}{ln^{2}(tan(x))sin(cos(x))tan^{3}(x)} + \frac{12sin(x)cos(x)sec^{2}(x)}{ln^{2}(tan(x))tan(x)} - \frac{6cos(x)cos(cos(x))sec^{4}(x)}{ln^{2}(tan(x))sin(cos(x))tan^{2}(x)} + \frac{12cos(x)cos(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin(cos(x))} - \frac{6sin^{2}(x)sec^{4}(x)}{ln^{2}(tan(x))tan^{2}(x)} + \frac{12sin^{2}(x)sec^{2}(x)}{ln^{2}(tan(x))} - \frac{4sin(x)cos(cos(x))sec^{2}(x)}{ln^{2}(tan(x))sin(cos(x))tan(x)} - \frac{16sin(x)cos(cos(x))sec^{4}(x)}{ln^{2}(tan(x))sin(cos(x))tan(x)} + \frac{16sin(x)cos(cos(x))tan(x)sec^{2}(x)}{ln^{2}(tan(x))sin(cos(x))} - \frac{3cos^{2}(x)cos^{2}(cos(x))}{ln(tan(x))sin^{2}(cos(x))} - \frac{6sin^{2}(x)cos^{3}(cos(x))cos(x)}{ln(tan(x))sin^{3}(cos(x))} - \frac{6sin^{2}(x)cos(cos(x))cos(x)}{ln(tan(x))sin(cos(x))} - \frac{6sin^{2}(x)cos(x)cos(cos(x))}{ln(tan(x))sin(cos(x))} - \frac{6sin^{2}(x)cos(x)cos^{3}(cos(x))}{ln(tan(x))sin^{3}(cos(x))} + \frac{3sin^{2}(x)cos^{2}(cos(x))}{ln(tan(x))sin^{2}(cos(x))} - \frac{6sin^{4}(x)cos^{4}(cos(x))}{ln(tan(x))sin^{4}(cos(x))} - \frac{8sin^{4}(x)cos^{2}(cos(x))}{ln(tan(x))sin^{2}(cos(x))} + \frac{cos(x)cos(cos(x))}{ln(tan(x))sin(cos(x))} + \frac{sin^{2}(x)cos^{2}(cos(x))}{ln(tan(x))sin^{2}(cos(x))} - \frac{2sin^{4}(x)}{ln(tan(x))} + \frac{4sin^{2}(x)}{ln(tan(x))} - \frac{3cos^{2}(x)}{ln(tan(x))} + \frac{24log_{tan(x)}^{sin(cos(x))}sec^{8}(x)}{ln^{4}(tan(x))tan^{4}(x)} + \frac{36log_{tan(x)}^{sin(cos(x))}sec^{8}(x)}{ln^{3}(tan(x))tan^{4}(x)} - \frac{72log_{tan(x)}^{sin(cos(x))}sec^{6}(x)}{ln^{3}(tan(x))tan^{2}(x)} + \frac{22log_{tan(x)}^{sin(cos(x))}sec^{8}(x)}{ln^{2}(tan(x))tan^{4}(x)} - \frac{56log_{tan(x)}^{sin(cos(x))}sec^{6}(x)}{ln^{2}(tan(x))tan^{2}(x)} + \frac{56log_{tan(x)}^{sin(cos(x))}sec^{4}(x)}{ln^{2}(tan(x))} + \frac{6log_{tan(x)}^{sin(cos(x))}sec^{8}(x)}{ln(tan(x))tan^{4}(x)} - \frac{16log_{tan(x)}^{sin(cos(x))}sec^{6}(x)}{ln(tan(x))tan^{2}(x)} + \frac{12log_{tan(x)}^{sin(cos(x))}sec^{4}(x)}{ln(tan(x))} - \frac{8log_{tan(x)}^{sin(cos(x))}tan^{2}(x)sec^{2}(x)}{ln(tan(x))}\\ \end{split}\end{equation} \]