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.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ lg(sin(x))lg(cos(x))lg(tan(x))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( lg(sin(x))lg(cos(x))lg(tan(x))\right)}{dx}\\=&\frac{cos(x)lg(cos(x))lg(tan(x))}{ln{10}(sin(x))} + \frac{lg(sin(x))*-sin(x)lg(tan(x))}{ln{10}(cos(x))} + \frac{lg(sin(x))lg(cos(x))sec^{2}(x)(1)}{ln{10}(tan(x))}\\=&\frac{lg(cos(x))lg(tan(x))cos(x)}{ln{10}sin(x)} - \frac{lg(sin(x))lg(tan(x))sin(x)}{ln{10}cos(x)} + \frac{lg(sin(x))lg(cos(x))sec^{2}(x)}{ln{10}tan(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{lg(cos(x))lg(tan(x))cos(x)}{ln{10}sin(x)} - \frac{lg(sin(x))lg(tan(x))sin(x)}{ln{10}cos(x)} + \frac{lg(sin(x))lg(cos(x))sec^{2}(x)}{ln{10}tan(x)}\right)}{dx}\\=&\frac{-0lg(cos(x))lg(tan(x))cos(x)}{ln^{2}{10}sin(x)} + \frac{-sin(x)lg(tan(x))cos(x)}{ln{10}ln{10}(cos(x))sin(x)} + \frac{lg(cos(x))sec^{2}(x)(1)cos(x)}{ln{10}ln{10}(tan(x))sin(x)} + \frac{lg(cos(x))lg(tan(x))*-cos(x)cos(x)}{ln{10}sin^{2}(x)} + \frac{lg(cos(x))lg(tan(x))*-sin(x)}{ln{10}sin(x)} - \frac{-0lg(sin(x))lg(tan(x))sin(x)}{ln^{2}{10}cos(x)} - \frac{cos(x)lg(tan(x))sin(x)}{ln{10}ln{10}(sin(x))cos(x)} - \frac{lg(sin(x))sec^{2}(x)(1)sin(x)}{ln{10}ln{10}(tan(x))cos(x)} - \frac{lg(sin(x))lg(tan(x))cos(x)}{ln{10}cos(x)} - \frac{lg(sin(x))lg(tan(x))sin(x)sin(x)}{ln{10}cos^{2}(x)} + \frac{-0lg(sin(x))lg(cos(x))sec^{2}(x)}{ln^{2}{10}tan(x)} + \frac{cos(x)lg(cos(x))sec^{2}(x)}{ln{10}ln{10}(sin(x))tan(x)} + \frac{lg(sin(x))*-sin(x)sec^{2}(x)}{ln{10}ln{10}(cos(x))tan(x)} + \frac{lg(sin(x))lg(cos(x))*-sec^{2}(x)(1)sec^{2}(x)}{ln{10}tan^{2}(x)} + \frac{lg(sin(x))lg(cos(x))*2sec^{2}(x)tan(x)}{ln{10}tan(x)}\\=&\frac{2lg(cos(x))cos(x)sec^{2}(x)}{ln^{2}{10}sin(x)tan(x)} - \frac{lg(cos(x))lg(tan(x))cos^{2}(x)}{ln{10}sin^{2}(x)} - \frac{lg(sin(x))lg(tan(x))sin^{2}(x)}{ln{10}cos^{2}(x)} - \frac{2lg(sin(x))sin(x)sec^{2}(x)}{ln^{2}{10}cos(x)tan(x)} - \frac{lg(sin(x))lg(cos(x))sec^{4}(x)}{ln{10}tan^{2}(x)} + \frac{2lg(sin(x))lg(cos(x))sec^{2}(x)}{ln{10}} - \frac{lg(tan(x))lg(sin(x))}{ln{10}} - \frac{lg(cos(x))lg(tan(x))}{ln{10}} - \frac{2lg(tan(x))}{ln^{2}{10}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{2lg(cos(x))cos(x)sec^{2}(x)}{ln^{2}{10}sin(x)tan(x)} - \frac{lg(cos(x))lg(tan(x))cos^{2}(x)}{ln{10}sin^{2}(x)} - \frac{lg(sin(x))lg(tan(x))sin^{2}(x)}{ln{10}cos^{2}(x)} - \frac{2lg(sin(x))sin(x)sec^{2}(x)}{ln^{2}{10}cos(x)tan(x)} - \frac{lg(sin(x))lg(cos(x))sec^{4}(x)}{ln{10}tan^{2}(x)} + \frac{2lg(sin(x))lg(cos(x))sec^{2}(x)}{ln{10}} - \frac{lg(tan(x))lg(sin(x))}{ln{10}} - \frac{lg(cos(x))lg(tan(x))}{ln{10}} - \frac{2lg(tan(x))}{ln^{2}{10}}\right)}{dx}\\=&\frac{2*-2*0lg(cos(x))cos(x)sec^{2}(x)}{ln^{3}{10}sin(x)tan(x)} + \frac{2*-sin(x)cos(x)sec^{2}(x)}{ln^{2}{10}ln{10}(cos(x))sin(x)tan(x)} + \frac{2lg(cos(x))*-cos(x)cos(x)sec^{2}(x)}{ln^{2}{10}sin^{2}(x)tan(x)} + \frac{2lg(cos(x))*-sin(x)sec^{2}(x)}{ln^{2}{10}sin(x)tan(x)} + \frac{2lg(cos(x))cos(x)*-sec^{2}(x)(1)sec^{2}(x)}{ln^{2}{10}sin(x)tan^{2}(x)} + \frac{2lg(cos(x))cos(x)*2sec^{2}(x)tan(x)}{ln^{2}{10}sin(x)tan(x)} - \frac{-0lg(cos(x))lg(tan(x))cos^{2}(x)}{ln^{2}{10}sin^{2}(x)} - \frac{-sin(x)lg(tan(x))cos^{2}(x)}{ln{10}ln{10}(cos(x))sin^{2}(x)} - \frac{lg(cos(x))sec^{2}(x)(1)cos^{2}(x)}{ln{10}ln{10}(tan(x))sin^{2}(x)} - \frac{lg(cos(x))lg(tan(x))*-2cos(x)cos^{2}(x)}{ln{10}sin^{3}(x)} - \frac{lg(cos(x))lg(tan(x))*-2cos(x)sin(x)}{ln{10}sin^{2}(x)} - \frac{-0lg(sin(x))lg(tan(x))sin^{2}(x)}{ln^{2}{10}cos^{2}(x)} - \frac{cos(x)lg(tan(x))sin^{2}(x)}{ln{10}ln{10}(sin(x))cos^{2}(x)} - \frac{lg(sin(x))sec^{2}(x)(1)sin^{2}(x)}{ln{10}ln{10}(tan(x))cos^{2}(x)} - \frac{lg(sin(x))lg(tan(x))*2sin(x)cos(x)}{ln{10}cos^{2}(x)} - \frac{lg(sin(x))lg(tan(x))sin^{2}(x)*2sin(x)}{ln{10}cos^{3}(x)} - \frac{2*-2*0lg(sin(x))sin(x)sec^{2}(x)}{ln^{3}{10}cos(x)tan(x)} - \frac{2cos(x)sin(x)sec^{2}(x)}{ln^{2}{10}ln{10}(sin(x))cos(x)tan(x)} - \frac{2lg(sin(x))cos(x)sec^{2}(x)}{ln^{2}{10}cos(x)tan(x)} - \frac{2lg(sin(x))sin(x)sin(x)sec^{2}(x)}{ln^{2}{10}cos^{2}(x)tan(x)} - \frac{2lg(sin(x))sin(x)*-sec^{2}(x)(1)sec^{2}(x)}{ln^{2}{10}cos(x)tan^{2}(x)} - \frac{2lg(sin(x))sin(x)*2sec^{2}(x)tan(x)}{ln^{2}{10}cos(x)tan(x)} - \frac{-0lg(sin(x))lg(cos(x))sec^{4}(x)}{ln^{2}{10}tan^{2}(x)} - \frac{cos(x)lg(cos(x))sec^{4}(x)}{ln{10}ln{10}(sin(x))tan^{2}(x)} - \frac{lg(sin(x))*-sin(x)sec^{4}(x)}{ln{10}ln{10}(cos(x))tan^{2}(x)} - \frac{lg(sin(x))lg(cos(x))*-2sec^{2}(x)(1)sec^{4}(x)}{ln{10}tan^{3}(x)} - \frac{lg(sin(x))lg(cos(x))*4sec^{4}(x)tan(x)}{ln{10}tan^{2}(x)} + \frac{2*-0lg(sin(x))lg(cos(x))sec^{2}(x)}{ln^{2}{10}} + \frac{2cos(x)lg(cos(x))sec^{2}(x)}{ln{10}ln{10}(sin(x))} + \frac{2lg(sin(x))*-sin(x)sec^{2}(x)}{ln{10}ln{10}(cos(x))} + \frac{2lg(sin(x))lg(cos(x))*2sec^{2}(x)tan(x)}{ln{10}} - \frac{-0lg(tan(x))lg(sin(x))}{ln^{2}{10}} - \frac{sec^{2}(x)(1)lg(sin(x))}{ln{10}ln{10}(tan(x))} - \frac{lg(tan(x))cos(x)}{ln{10}ln{10}(sin(x))} - \frac{-0lg(cos(x))lg(tan(x))}{ln^{2}{10}} - \frac{-sin(x)lg(tan(x))}{ln{10}ln{10}(cos(x))} - \frac{lg(cos(x))sec^{2}(x)(1)}{ln{10}ln{10}(tan(x))} - \frac{2*-2*0lg(tan(x))}{ln^{3}{10}} - \frac{2sec^{2}(x)(1)}{ln^{2}{10}ln{10}(tan(x))}\\=&\frac{-6sec^{2}(x)}{ln^{3}{10}tan(x)} - \frac{3lg(cos(x))cos^{2}(x)sec^{2}(x)}{ln^{2}{10}sin^{2}(x)tan(x)} - \frac{3lg(cos(x))sec^{2}(x)}{ln^{2}{10}tan(x)} - \frac{3lg(cos(x))cos(x)sec^{4}(x)}{ln^{2}{10}sin(x)tan^{2}(x)} + \frac{6lg(cos(x))cos(x)sec^{2}(x)}{ln^{2}{10}sin(x)} + \frac{2lg(cos(x))lg(tan(x))cos^{3}(x)}{ln{10}sin^{3}(x)} - \frac{2lg(sin(x))lg(tan(x))sin(x)}{ln{10}cos(x)} + \frac{2lg(cos(x))lg(tan(x))cos(x)}{ln{10}sin(x)} - \frac{3lg(sin(x))sin^{2}(x)sec^{2}(x)}{ln^{2}{10}cos^{2}(x)tan(x)} - \frac{2lg(sin(x))lg(tan(x))sin^{3}(x)}{ln{10}cos^{3}(x)} + \frac{3lg(sin(x))sin(x)sec^{4}(x)}{ln^{2}{10}cos(x)tan^{2}(x)} - \frac{3lg(sin(x))sec^{2}(x)}{ln^{2}{10}tan(x)} - \frac{6lg(sin(x))sin(x)sec^{2}(x)}{ln^{2}{10}cos(x)} + \frac{2lg(sin(x))lg(cos(x))sec^{6}(x)}{ln{10}tan^{3}(x)} - \frac{4lg(sin(x))lg(cos(x))sec^{4}(x)}{ln{10}tan(x)} + \frac{4lg(sin(x))lg(cos(x))tan(x)sec^{2}(x)}{ln{10}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-6sec^{2}(x)}{ln^{3}{10}tan(x)} - \frac{3lg(cos(x))cos^{2}(x)sec^{2}(x)}{ln^{2}{10}sin^{2}(x)tan(x)} - \frac{3lg(cos(x))sec^{2}(x)}{ln^{2}{10}tan(x)} - \frac{3lg(cos(x))cos(x)sec^{4}(x)}{ln^{2}{10}sin(x)tan^{2}(x)} + \frac{6lg(cos(x))cos(x)sec^{2}(x)}{ln^{2}{10}sin(x)} + \frac{2lg(cos(x))lg(tan(x))cos^{3}(x)}{ln{10}sin^{3}(x)} - \frac{2lg(sin(x))lg(tan(x))sin(x)}{ln{10}cos(x)} + \frac{2lg(cos(x))lg(tan(x))cos(x)}{ln{10}sin(x)} - \frac{3lg(sin(x))sin^{2}(x)sec^{2}(x)}{ln^{2}{10}cos^{2}(x)tan(x)} - \frac{2lg(sin(x))lg(tan(x))sin^{3}(x)}{ln{10}cos^{3}(x)} + \frac{3lg(sin(x))sin(x)sec^{4}(x)}{ln^{2}{10}cos(x)tan^{2}(x)} - \frac{3lg(sin(x))sec^{2}(x)}{ln^{2}{10}tan(x)} - \frac{6lg(sin(x))sin(x)sec^{2}(x)}{ln^{2}{10}cos(x)} + \frac{2lg(sin(x))lg(cos(x))sec^{6}(x)}{ln{10}tan^{3}(x)} - \frac{4lg(sin(x))lg(cos(x))sec^{4}(x)}{ln{10}tan(x)} + \frac{4lg(sin(x))lg(cos(x))tan(x)sec^{2}(x)}{ln{10}}\right)}{dx}\\=&\frac{-6*-3*0sec^{2}(x)}{ln^{4}{10}tan(x)} - \frac{6*-sec^{2}(x)(1)sec^{2}(x)}{ln^{3}{10}tan^{2}(x)} - \frac{6*2sec^{2}(x)tan(x)}{ln^{3}{10}tan(x)} - \frac{3*-2*0lg(cos(x))cos^{2}(x)sec^{2}(x)}{ln^{3}{10}sin^{2}(x)tan(x)} - \frac{3*-sin(x)cos^{2}(x)sec^{2}(x)}{ln^{2}{10}ln{10}(cos(x))sin^{2}(x)tan(x)} - \frac{3lg(cos(x))*-2cos(x)cos^{2}(x)sec^{2}(x)}{ln^{2}{10}sin^{3}(x)tan(x)} - \frac{3lg(cos(x))*-2cos(x)sin(x)sec^{2}(x)}{ln^{2}{10}sin^{2}(x)tan(x)} - \frac{3lg(cos(x))cos^{2}(x)*-sec^{2}(x)(1)sec^{2}(x)}{ln^{2}{10}sin^{2}(x)tan^{2}(x)} - \frac{3lg(cos(x))cos^{2}(x)*2sec^{2}(x)tan(x)}{ln^{2}{10}sin^{2}(x)tan(x)} - \frac{3*-2*0lg(cos(x))sec^{2}(x)}{ln^{3}{10}tan(x)} - \frac{3*-sin(x)sec^{2}(x)}{ln^{2}{10}ln{10}(cos(x))tan(x)} - \frac{3lg(cos(x))*-sec^{2}(x)(1)sec^{2}(x)}{ln^{2}{10}tan^{2}(x)} - \frac{3lg(cos(x))*2sec^{2}(x)tan(x)}{ln^{2}{10}tan(x)} - \frac{3*-2*0lg(cos(x))cos(x)sec^{4}(x)}{ln^{3}{10}sin(x)tan^{2}(x)} - \frac{3*-sin(x)cos(x)sec^{4}(x)}{ln^{2}{10}ln{10}(cos(x))sin(x)tan^{2}(x)} - \frac{3lg(cos(x))*-cos(x)cos(x)sec^{4}(x)}{ln^{2}{10}sin^{2}(x)tan^{2}(x)} - \frac{3lg(cos(x))*-sin(x)sec^{4}(x)}{ln^{2}{10}sin(x)tan^{2}(x)} - \frac{3lg(cos(x))cos(x)*-2sec^{2}(x)(1)sec^{4}(x)}{ln^{2}{10}sin(x)tan^{3}(x)} - \frac{3lg(cos(x))cos(x)*4sec^{4}(x)tan(x)}{ln^{2}{10}sin(x)tan^{2}(x)} + \frac{6*-2*0lg(cos(x))cos(x)sec^{2}(x)}{ln^{3}{10}sin(x)} + \frac{6*-sin(x)cos(x)sec^{2}(x)}{ln^{2}{10}ln{10}(cos(x))sin(x)} + \frac{6lg(cos(x))*-cos(x)cos(x)sec^{2}(x)}{ln^{2}{10}sin^{2}(x)} + \frac{6lg(cos(x))*-sin(x)sec^{2}(x)}{ln^{2}{10}sin(x)} + \frac{6lg(cos(x))cos(x)*2sec^{2}(x)tan(x)}{ln^{2}{10}sin(x)} + \frac{2*-0lg(cos(x))lg(tan(x))cos^{3}(x)}{ln^{2}{10}sin^{3}(x)} + \frac{2*-sin(x)lg(tan(x))cos^{3}(x)}{ln{10}ln{10}(cos(x))sin^{3}(x)} + \frac{2lg(cos(x))sec^{2}(x)(1)cos^{3}(x)}{ln{10}ln{10}(tan(x))sin^{3}(x)} + \frac{2lg(cos(x))lg(tan(x))*-3cos(x)cos^{3}(x)}{ln{10}sin^{4}(x)} + \frac{2lg(cos(x))lg(tan(x))*-3cos^{2}(x)sin(x)}{ln{10}sin^{3}(x)} - \frac{2*-0lg(sin(x))lg(tan(x))sin(x)}{ln^{2}{10}cos(x)} - \frac{2cos(x)lg(tan(x))sin(x)}{ln{10}ln{10}(sin(x))cos(x)} - \frac{2lg(sin(x))sec^{2}(x)(1)sin(x)}{ln{10}ln{10}(tan(x))cos(x)} - \frac{2lg(sin(x))lg(tan(x))cos(x)}{ln{10}cos(x)} - \frac{2lg(sin(x))lg(tan(x))sin(x)sin(x)}{ln{10}cos^{2}(x)} + \frac{2*-0lg(cos(x))lg(tan(x))cos(x)}{ln^{2}{10}sin(x)} + \frac{2*-sin(x)lg(tan(x))cos(x)}{ln{10}ln{10}(cos(x))sin(x)} + \frac{2lg(cos(x))sec^{2}(x)(1)cos(x)}{ln{10}ln{10}(tan(x))sin(x)} + \frac{2lg(cos(x))lg(tan(x))*-cos(x)cos(x)}{ln{10}sin^{2}(x)} + \frac{2lg(cos(x))lg(tan(x))*-sin(x)}{ln{10}sin(x)} - \frac{3*-2*0lg(sin(x))sin^{2}(x)sec^{2}(x)}{ln^{3}{10}cos^{2}(x)tan(x)} - \frac{3cos(x)sin^{2}(x)sec^{2}(x)}{ln^{2}{10}ln{10}(sin(x))cos^{2}(x)tan(x)} - \frac{3lg(sin(x))*2sin(x)cos(x)sec^{2}(x)}{ln^{2}{10}cos^{2}(x)tan(x)} - \frac{3lg(sin(x))sin^{2}(x)*2sin(x)sec^{2}(x)}{ln^{2}{10}cos^{3}(x)tan(x)} - \frac{3lg(sin(x))sin^{2}(x)*-sec^{2}(x)(1)sec^{2}(x)}{ln^{2}{10}cos^{2}(x)tan^{2}(x)} - \frac{3lg(sin(x))sin^{2}(x)*2sec^{2}(x)tan(x)}{ln^{2}{10}cos^{2}(x)tan(x)} - \frac{2*-0lg(sin(x))lg(tan(x))sin^{3}(x)}{ln^{2}{10}cos^{3}(x)} - \frac{2cos(x)lg(tan(x))sin^{3}(x)}{ln{10}ln{10}(sin(x))cos^{3}(x)} - \frac{2lg(sin(x))sec^{2}(x)(1)sin^{3}(x)}{ln{10}ln{10}(tan(x))cos^{3}(x)} - \frac{2lg(sin(x))lg(tan(x))*3sin^{2}(x)cos(x)}{ln{10}cos^{3}(x)} - \frac{2lg(sin(x))lg(tan(x))sin^{3}(x)*3sin(x)}{ln{10}cos^{4}(x)} + \frac{3*-2*0lg(sin(x))sin(x)sec^{4}(x)}{ln^{3}{10}cos(x)tan^{2}(x)} + \frac{3cos(x)sin(x)sec^{4}(x)}{ln^{2}{10}ln{10}(sin(x))cos(x)tan^{2}(x)} + \frac{3lg(sin(x))cos(x)sec^{4}(x)}{ln^{2}{10}cos(x)tan^{2}(x)} + \frac{3lg(sin(x))sin(x)sin(x)sec^{4}(x)}{ln^{2}{10}cos^{2}(x)tan^{2}(x)} + \frac{3lg(sin(x))sin(x)*-2sec^{2}(x)(1)sec^{4}(x)}{ln^{2}{10}cos(x)tan^{3}(x)} + \frac{3lg(sin(x))sin(x)*4sec^{4}(x)tan(x)}{ln^{2}{10}cos(x)tan^{2}(x)} - \frac{3*-2*0lg(sin(x))sec^{2}(x)}{ln^{3}{10}tan(x)} - \frac{3cos(x)sec^{2}(x)}{ln^{2}{10}ln{10}(sin(x))tan(x)} - \frac{3lg(sin(x))*-sec^{2}(x)(1)sec^{2}(x)}{ln^{2}{10}tan^{2}(x)} - \frac{3lg(sin(x))*2sec^{2}(x)tan(x)}{ln^{2}{10}tan(x)} - \frac{6*-2*0lg(sin(x))sin(x)sec^{2}(x)}{ln^{3}{10}cos(x)} - \frac{6cos(x)sin(x)sec^{2}(x)}{ln^{2}{10}ln{10}(sin(x))cos(x)} - \frac{6lg(sin(x))cos(x)sec^{2}(x)}{ln^{2}{10}cos(x)} - \frac{6lg(sin(x))sin(x)sin(x)sec^{2}(x)}{ln^{2}{10}cos^{2}(x)} - \frac{6lg(sin(x))sin(x)*2sec^{2}(x)tan(x)}{ln^{2}{10}cos(x)} + \frac{2*-0lg(sin(x))lg(cos(x))sec^{6}(x)}{ln^{2}{10}tan^{3}(x)} + \frac{2cos(x)lg(cos(x))sec^{6}(x)}{ln{10}ln{10}(sin(x))tan^{3}(x)} + \frac{2lg(sin(x))*-sin(x)sec^{6}(x)}{ln{10}ln{10}(cos(x))tan^{3}(x)} + \frac{2lg(sin(x))lg(cos(x))*-3sec^{2}(x)(1)sec^{6}(x)}{ln{10}tan^{4}(x)} + \frac{2lg(sin(x))lg(cos(x))*6sec^{6}(x)tan(x)}{ln{10}tan^{3}(x)} - \frac{4*-0lg(sin(x))lg(cos(x))sec^{4}(x)}{ln^{2}{10}tan(x)} - \frac{4cos(x)lg(cos(x))sec^{4}(x)}{ln{10}ln{10}(sin(x))tan(x)} - \frac{4lg(sin(x))*-sin(x)sec^{4}(x)}{ln{10}ln{10}(cos(x))tan(x)} - \frac{4lg(sin(x))lg(cos(x))*-sec^{2}(x)(1)sec^{4}(x)}{ln{10}tan^{2}(x)} - \frac{4lg(sin(x))lg(cos(x))*4sec^{4}(x)tan(x)}{ln{10}tan(x)} + \frac{4*-0lg(sin(x))lg(cos(x))tan(x)sec^{2}(x)}{ln^{2}{10}} + \frac{4cos(x)lg(cos(x))tan(x)sec^{2}(x)}{ln{10}ln{10}(sin(x))} + \frac{4lg(sin(x))*-sin(x)tan(x)sec^{2}(x)}{ln{10}ln{10}(cos(x))} + \frac{4lg(sin(x))lg(cos(x))sec^{2}(x)(1)sec^{2}(x)}{ln{10}} + \frac{4lg(sin(x))lg(cos(x))tan(x)*2sec^{2}(x)tan(x)}{ln{10}}\\=&\frac{12sec^{4}(x)}{ln^{3}{10}tan^{2}(x)} - \frac{24sec^{2}(x)}{ln^{3}{10}} + \frac{6lg(cos(x))cos^{2}(x)sec^{4}(x)}{ln^{2}{10}sin^{2}(x)tan^{2}(x)} + \frac{8lg(cos(x))cos^{3}(x)sec^{2}(x)}{ln^{2}{10}sin^{3}(x)tan(x)} + \frac{8lg(cos(x))cos(x)sec^{2}(x)}{ln^{2}{10}sin(x)tan(x)} + \frac{8lg(cos(x))cos(x)sec^{6}(x)}{ln^{2}{10}sin(x)tan^{3}(x)} - \frac{12lg(cos(x))cos^{2}(x)sec^{2}(x)}{ln^{2}{10}sin^{2}(x)} + \frac{6lg(cos(x))sec^{4}(x)}{ln^{2}{10}tan^{2}(x)} + \frac{6lg(sin(x))sec^{4}(x)}{ln^{2}{10}tan^{2}(x)} - \frac{12lg(cos(x))sec^{2}(x)}{ln^{2}{10}} + \frac{6lg(sin(x))sin^{2}(x)sec^{4}(x)}{ln^{2}{10}cos^{2}(x)tan^{2}(x)} - \frac{8lg(sin(x))sin(x)sec^{6}(x)}{ln^{2}{10}cos(x)tan^{3}(x)} - \frac{16lg(cos(x))cos(x)sec^{4}(x)}{ln^{2}{10}sin(x)tan(x)} + \frac{16lg(cos(x))cos(x)tan(x)sec^{2}(x)}{ln^{2}{10}sin(x)} - \frac{6lg(cos(x))lg(tan(x))cos^{4}(x)}{ln{10}sin^{4}(x)} - \frac{8lg(sin(x))sin(x)sec^{2}(x)}{ln^{2}{10}cos(x)tan(x)} - \frac{8lg(cos(x))lg(tan(x))cos^{2}(x)}{ln{10}sin^{2}(x)} - \frac{8lg(sin(x))lg(tan(x))sin^{2}(x)}{ln{10}cos^{2}(x)} - \frac{8lg(sin(x))sin^{3}(x)sec^{2}(x)}{ln^{2}{10}cos^{3}(x)tan(x)} - \frac{6lg(sin(x))lg(tan(x))sin^{4}(x)}{ln{10}cos^{4}(x)} - \frac{6lg(sin(x))lg(cos(x))sec^{8}(x)}{ln{10}tan^{4}(x)} - \frac{12lg(sin(x))sin^{2}(x)sec^{2}(x)}{ln^{2}{10}cos^{2}(x)} + \frac{16lg(sin(x))sin(x)sec^{4}(x)}{ln^{2}{10}cos(x)tan(x)} - \frac{16lg(sin(x))sin(x)tan(x)sec^{2}(x)}{ln^{2}{10}cos(x)} + \frac{16lg(sin(x))lg(cos(x))sec^{6}(x)}{ln{10}tan^{2}(x)} - \frac{2lg(tan(x))cos^{2}(x)}{ln^{2}{10}sin^{2}(x)} - \frac{12lg(sin(x))sec^{2}(x)}{ln^{2}{10}} - \frac{2lg(tan(x))sin^{2}(x)}{ln^{2}{10}cos^{2}(x)} - \frac{16lg(sin(x))lg(cos(x))sec^{4}(x)}{ln{10}} + \frac{4lg(cos(x))lg(sin(x))sec^{4}(x)}{ln{10}} + \frac{8lg(sin(x))lg(cos(x))tan^{2}(x)sec^{2}(x)}{ln{10}} - \frac{2lg(cos(x))lg(tan(x))}{ln{10}} - \frac{2lg(tan(x))lg(sin(x))}{ln{10}} - \frac{4lg(tan(x))}{ln^{2}{10}}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ lg(sin(x))lg(cos(x))lg(tan(x))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( lg(sin(x))lg(cos(x))lg(tan(x))\right)}{dx}\\=&\frac{cos(x)lg(cos(x))lg(tan(x))}{ln{10}(sin(x))} + \frac{lg(sin(x))*-sin(x)lg(tan(x))}{ln{10}(cos(x))} + \frac{lg(sin(x))lg(cos(x))sec^{2}(x)(1)}{ln{10}(tan(x))}\\=&\frac{lg(cos(x))lg(tan(x))cos(x)}{ln{10}sin(x)} - \frac{lg(sin(x))lg(tan(x))sin(x)}{ln{10}cos(x)} + \frac{lg(sin(x))lg(cos(x))sec^{2}(x)}{ln{10}tan(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{lg(cos(x))lg(tan(x))cos(x)}{ln{10}sin(x)} - \frac{lg(sin(x))lg(tan(x))sin(x)}{ln{10}cos(x)} + \frac{lg(sin(x))lg(cos(x))sec^{2}(x)}{ln{10}tan(x)}\right)}{dx}\\=&\frac{-0lg(cos(x))lg(tan(x))cos(x)}{ln^{2}{10}sin(x)} + \frac{-sin(x)lg(tan(x))cos(x)}{ln{10}ln{10}(cos(x))sin(x)} + \frac{lg(cos(x))sec^{2}(x)(1)cos(x)}{ln{10}ln{10}(tan(x))sin(x)} + \frac{lg(cos(x))lg(tan(x))*-cos(x)cos(x)}{ln{10}sin^{2}(x)} + \frac{lg(cos(x))lg(tan(x))*-sin(x)}{ln{10}sin(x)} - \frac{-0lg(sin(x))lg(tan(x))sin(x)}{ln^{2}{10}cos(x)} - \frac{cos(x)lg(tan(x))sin(x)}{ln{10}ln{10}(sin(x))cos(x)} - \frac{lg(sin(x))sec^{2}(x)(1)sin(x)}{ln{10}ln{10}(tan(x))cos(x)} - \frac{lg(sin(x))lg(tan(x))cos(x)}{ln{10}cos(x)} - \frac{lg(sin(x))lg(tan(x))sin(x)sin(x)}{ln{10}cos^{2}(x)} + \frac{-0lg(sin(x))lg(cos(x))sec^{2}(x)}{ln^{2}{10}tan(x)} + \frac{cos(x)lg(cos(x))sec^{2}(x)}{ln{10}ln{10}(sin(x))tan(x)} + \frac{lg(sin(x))*-sin(x)sec^{2}(x)}{ln{10}ln{10}(cos(x))tan(x)} + \frac{lg(sin(x))lg(cos(x))*-sec^{2}(x)(1)sec^{2}(x)}{ln{10}tan^{2}(x)} + \frac{lg(sin(x))lg(cos(x))*2sec^{2}(x)tan(x)}{ln{10}tan(x)}\\=&\frac{2lg(cos(x))cos(x)sec^{2}(x)}{ln^{2}{10}sin(x)tan(x)} - \frac{lg(cos(x))lg(tan(x))cos^{2}(x)}{ln{10}sin^{2}(x)} - \frac{lg(sin(x))lg(tan(x))sin^{2}(x)}{ln{10}cos^{2}(x)} - \frac{2lg(sin(x))sin(x)sec^{2}(x)}{ln^{2}{10}cos(x)tan(x)} - \frac{lg(sin(x))lg(cos(x))sec^{4}(x)}{ln{10}tan^{2}(x)} + \frac{2lg(sin(x))lg(cos(x))sec^{2}(x)}{ln{10}} - \frac{lg(tan(x))lg(sin(x))}{ln{10}} - \frac{lg(cos(x))lg(tan(x))}{ln{10}} - \frac{2lg(tan(x))}{ln^{2}{10}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{2lg(cos(x))cos(x)sec^{2}(x)}{ln^{2}{10}sin(x)tan(x)} - \frac{lg(cos(x))lg(tan(x))cos^{2}(x)}{ln{10}sin^{2}(x)} - \frac{lg(sin(x))lg(tan(x))sin^{2}(x)}{ln{10}cos^{2}(x)} - \frac{2lg(sin(x))sin(x)sec^{2}(x)}{ln^{2}{10}cos(x)tan(x)} - \frac{lg(sin(x))lg(cos(x))sec^{4}(x)}{ln{10}tan^{2}(x)} + \frac{2lg(sin(x))lg(cos(x))sec^{2}(x)}{ln{10}} - \frac{lg(tan(x))lg(sin(x))}{ln{10}} - \frac{lg(cos(x))lg(tan(x))}{ln{10}} - \frac{2lg(tan(x))}{ln^{2}{10}}\right)}{dx}\\=&\frac{2*-2*0lg(cos(x))cos(x)sec^{2}(x)}{ln^{3}{10}sin(x)tan(x)} + \frac{2*-sin(x)cos(x)sec^{2}(x)}{ln^{2}{10}ln{10}(cos(x))sin(x)tan(x)} + \frac{2lg(cos(x))*-cos(x)cos(x)sec^{2}(x)}{ln^{2}{10}sin^{2}(x)tan(x)} + \frac{2lg(cos(x))*-sin(x)sec^{2}(x)}{ln^{2}{10}sin(x)tan(x)} + \frac{2lg(cos(x))cos(x)*-sec^{2}(x)(1)sec^{2}(x)}{ln^{2}{10}sin(x)tan^{2}(x)} + \frac{2lg(cos(x))cos(x)*2sec^{2}(x)tan(x)}{ln^{2}{10}sin(x)tan(x)} - \frac{-0lg(cos(x))lg(tan(x))cos^{2}(x)}{ln^{2}{10}sin^{2}(x)} - \frac{-sin(x)lg(tan(x))cos^{2}(x)}{ln{10}ln{10}(cos(x))sin^{2}(x)} - \frac{lg(cos(x))sec^{2}(x)(1)cos^{2}(x)}{ln{10}ln{10}(tan(x))sin^{2}(x)} - \frac{lg(cos(x))lg(tan(x))*-2cos(x)cos^{2}(x)}{ln{10}sin^{3}(x)} - \frac{lg(cos(x))lg(tan(x))*-2cos(x)sin(x)}{ln{10}sin^{2}(x)} - \frac{-0lg(sin(x))lg(tan(x))sin^{2}(x)}{ln^{2}{10}cos^{2}(x)} - \frac{cos(x)lg(tan(x))sin^{2}(x)}{ln{10}ln{10}(sin(x))cos^{2}(x)} - \frac{lg(sin(x))sec^{2}(x)(1)sin^{2}(x)}{ln{10}ln{10}(tan(x))cos^{2}(x)} - \frac{lg(sin(x))lg(tan(x))*2sin(x)cos(x)}{ln{10}cos^{2}(x)} - \frac{lg(sin(x))lg(tan(x))sin^{2}(x)*2sin(x)}{ln{10}cos^{3}(x)} - \frac{2*-2*0lg(sin(x))sin(x)sec^{2}(x)}{ln^{3}{10}cos(x)tan(x)} - \frac{2cos(x)sin(x)sec^{2}(x)}{ln^{2}{10}ln{10}(sin(x))cos(x)tan(x)} - \frac{2lg(sin(x))cos(x)sec^{2}(x)}{ln^{2}{10}cos(x)tan(x)} - \frac{2lg(sin(x))sin(x)sin(x)sec^{2}(x)}{ln^{2}{10}cos^{2}(x)tan(x)} - \frac{2lg(sin(x))sin(x)*-sec^{2}(x)(1)sec^{2}(x)}{ln^{2}{10}cos(x)tan^{2}(x)} - \frac{2lg(sin(x))sin(x)*2sec^{2}(x)tan(x)}{ln^{2}{10}cos(x)tan(x)} - \frac{-0lg(sin(x))lg(cos(x))sec^{4}(x)}{ln^{2}{10}tan^{2}(x)} - \frac{cos(x)lg(cos(x))sec^{4}(x)}{ln{10}ln{10}(sin(x))tan^{2}(x)} - \frac{lg(sin(x))*-sin(x)sec^{4}(x)}{ln{10}ln{10}(cos(x))tan^{2}(x)} - \frac{lg(sin(x))lg(cos(x))*-2sec^{2}(x)(1)sec^{4}(x)}{ln{10}tan^{3}(x)} - \frac{lg(sin(x))lg(cos(x))*4sec^{4}(x)tan(x)}{ln{10}tan^{2}(x)} + \frac{2*-0lg(sin(x))lg(cos(x))sec^{2}(x)}{ln^{2}{10}} + \frac{2cos(x)lg(cos(x))sec^{2}(x)}{ln{10}ln{10}(sin(x))} + \frac{2lg(sin(x))*-sin(x)sec^{2}(x)}{ln{10}ln{10}(cos(x))} + \frac{2lg(sin(x))lg(cos(x))*2sec^{2}(x)tan(x)}{ln{10}} - \frac{-0lg(tan(x))lg(sin(x))}{ln^{2}{10}} - \frac{sec^{2}(x)(1)lg(sin(x))}{ln{10}ln{10}(tan(x))} - \frac{lg(tan(x))cos(x)}{ln{10}ln{10}(sin(x))} - \frac{-0lg(cos(x))lg(tan(x))}{ln^{2}{10}} - \frac{-sin(x)lg(tan(x))}{ln{10}ln{10}(cos(x))} - \frac{lg(cos(x))sec^{2}(x)(1)}{ln{10}ln{10}(tan(x))} - \frac{2*-2*0lg(tan(x))}{ln^{3}{10}} - \frac{2sec^{2}(x)(1)}{ln^{2}{10}ln{10}(tan(x))}\\=&\frac{-6sec^{2}(x)}{ln^{3}{10}tan(x)} - \frac{3lg(cos(x))cos^{2}(x)sec^{2}(x)}{ln^{2}{10}sin^{2}(x)tan(x)} - \frac{3lg(cos(x))sec^{2}(x)}{ln^{2}{10}tan(x)} - \frac{3lg(cos(x))cos(x)sec^{4}(x)}{ln^{2}{10}sin(x)tan^{2}(x)} + \frac{6lg(cos(x))cos(x)sec^{2}(x)}{ln^{2}{10}sin(x)} + \frac{2lg(cos(x))lg(tan(x))cos^{3}(x)}{ln{10}sin^{3}(x)} - \frac{2lg(sin(x))lg(tan(x))sin(x)}{ln{10}cos(x)} + \frac{2lg(cos(x))lg(tan(x))cos(x)}{ln{10}sin(x)} - \frac{3lg(sin(x))sin^{2}(x)sec^{2}(x)}{ln^{2}{10}cos^{2}(x)tan(x)} - \frac{2lg(sin(x))lg(tan(x))sin^{3}(x)}{ln{10}cos^{3}(x)} + \frac{3lg(sin(x))sin(x)sec^{4}(x)}{ln^{2}{10}cos(x)tan^{2}(x)} - \frac{3lg(sin(x))sec^{2}(x)}{ln^{2}{10}tan(x)} - \frac{6lg(sin(x))sin(x)sec^{2}(x)}{ln^{2}{10}cos(x)} + \frac{2lg(sin(x))lg(cos(x))sec^{6}(x)}{ln{10}tan^{3}(x)} - \frac{4lg(sin(x))lg(cos(x))sec^{4}(x)}{ln{10}tan(x)} + \frac{4lg(sin(x))lg(cos(x))tan(x)sec^{2}(x)}{ln{10}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-6sec^{2}(x)}{ln^{3}{10}tan(x)} - \frac{3lg(cos(x))cos^{2}(x)sec^{2}(x)}{ln^{2}{10}sin^{2}(x)tan(x)} - \frac{3lg(cos(x))sec^{2}(x)}{ln^{2}{10}tan(x)} - \frac{3lg(cos(x))cos(x)sec^{4}(x)}{ln^{2}{10}sin(x)tan^{2}(x)} + \frac{6lg(cos(x))cos(x)sec^{2}(x)}{ln^{2}{10}sin(x)} + \frac{2lg(cos(x))lg(tan(x))cos^{3}(x)}{ln{10}sin^{3}(x)} - \frac{2lg(sin(x))lg(tan(x))sin(x)}{ln{10}cos(x)} + \frac{2lg(cos(x))lg(tan(x))cos(x)}{ln{10}sin(x)} - \frac{3lg(sin(x))sin^{2}(x)sec^{2}(x)}{ln^{2}{10}cos^{2}(x)tan(x)} - \frac{2lg(sin(x))lg(tan(x))sin^{3}(x)}{ln{10}cos^{3}(x)} + \frac{3lg(sin(x))sin(x)sec^{4}(x)}{ln^{2}{10}cos(x)tan^{2}(x)} - \frac{3lg(sin(x))sec^{2}(x)}{ln^{2}{10}tan(x)} - \frac{6lg(sin(x))sin(x)sec^{2}(x)}{ln^{2}{10}cos(x)} + \frac{2lg(sin(x))lg(cos(x))sec^{6}(x)}{ln{10}tan^{3}(x)} - \frac{4lg(sin(x))lg(cos(x))sec^{4}(x)}{ln{10}tan(x)} + \frac{4lg(sin(x))lg(cos(x))tan(x)sec^{2}(x)}{ln{10}}\right)}{dx}\\=&\frac{-6*-3*0sec^{2}(x)}{ln^{4}{10}tan(x)} - \frac{6*-sec^{2}(x)(1)sec^{2}(x)}{ln^{3}{10}tan^{2}(x)} - \frac{6*2sec^{2}(x)tan(x)}{ln^{3}{10}tan(x)} - \frac{3*-2*0lg(cos(x))cos^{2}(x)sec^{2}(x)}{ln^{3}{10}sin^{2}(x)tan(x)} - \frac{3*-sin(x)cos^{2}(x)sec^{2}(x)}{ln^{2}{10}ln{10}(cos(x))sin^{2}(x)tan(x)} - \frac{3lg(cos(x))*-2cos(x)cos^{2}(x)sec^{2}(x)}{ln^{2}{10}sin^{3}(x)tan(x)} - \frac{3lg(cos(x))*-2cos(x)sin(x)sec^{2}(x)}{ln^{2}{10}sin^{2}(x)tan(x)} - \frac{3lg(cos(x))cos^{2}(x)*-sec^{2}(x)(1)sec^{2}(x)}{ln^{2}{10}sin^{2}(x)tan^{2}(x)} - \frac{3lg(cos(x))cos^{2}(x)*2sec^{2}(x)tan(x)}{ln^{2}{10}sin^{2}(x)tan(x)} - \frac{3*-2*0lg(cos(x))sec^{2}(x)}{ln^{3}{10}tan(x)} - \frac{3*-sin(x)sec^{2}(x)}{ln^{2}{10}ln{10}(cos(x))tan(x)} - \frac{3lg(cos(x))*-sec^{2}(x)(1)sec^{2}(x)}{ln^{2}{10}tan^{2}(x)} - \frac{3lg(cos(x))*2sec^{2}(x)tan(x)}{ln^{2}{10}tan(x)} - \frac{3*-2*0lg(cos(x))cos(x)sec^{4}(x)}{ln^{3}{10}sin(x)tan^{2}(x)} - \frac{3*-sin(x)cos(x)sec^{4}(x)}{ln^{2}{10}ln{10}(cos(x))sin(x)tan^{2}(x)} - \frac{3lg(cos(x))*-cos(x)cos(x)sec^{4}(x)}{ln^{2}{10}sin^{2}(x)tan^{2}(x)} - \frac{3lg(cos(x))*-sin(x)sec^{4}(x)}{ln^{2}{10}sin(x)tan^{2}(x)} - \frac{3lg(cos(x))cos(x)*-2sec^{2}(x)(1)sec^{4}(x)}{ln^{2}{10}sin(x)tan^{3}(x)} - \frac{3lg(cos(x))cos(x)*4sec^{4}(x)tan(x)}{ln^{2}{10}sin(x)tan^{2}(x)} + \frac{6*-2*0lg(cos(x))cos(x)sec^{2}(x)}{ln^{3}{10}sin(x)} + \frac{6*-sin(x)cos(x)sec^{2}(x)}{ln^{2}{10}ln{10}(cos(x))sin(x)} + \frac{6lg(cos(x))*-cos(x)cos(x)sec^{2}(x)}{ln^{2}{10}sin^{2}(x)} + \frac{6lg(cos(x))*-sin(x)sec^{2}(x)}{ln^{2}{10}sin(x)} + \frac{6lg(cos(x))cos(x)*2sec^{2}(x)tan(x)}{ln^{2}{10}sin(x)} + \frac{2*-0lg(cos(x))lg(tan(x))cos^{3}(x)}{ln^{2}{10}sin^{3}(x)} + \frac{2*-sin(x)lg(tan(x))cos^{3}(x)}{ln{10}ln{10}(cos(x))sin^{3}(x)} + \frac{2lg(cos(x))sec^{2}(x)(1)cos^{3}(x)}{ln{10}ln{10}(tan(x))sin^{3}(x)} + \frac{2lg(cos(x))lg(tan(x))*-3cos(x)cos^{3}(x)}{ln{10}sin^{4}(x)} + \frac{2lg(cos(x))lg(tan(x))*-3cos^{2}(x)sin(x)}{ln{10}sin^{3}(x)} - \frac{2*-0lg(sin(x))lg(tan(x))sin(x)}{ln^{2}{10}cos(x)} - \frac{2cos(x)lg(tan(x))sin(x)}{ln{10}ln{10}(sin(x))cos(x)} - \frac{2lg(sin(x))sec^{2}(x)(1)sin(x)}{ln{10}ln{10}(tan(x))cos(x)} - \frac{2lg(sin(x))lg(tan(x))cos(x)}{ln{10}cos(x)} - \frac{2lg(sin(x))lg(tan(x))sin(x)sin(x)}{ln{10}cos^{2}(x)} + \frac{2*-0lg(cos(x))lg(tan(x))cos(x)}{ln^{2}{10}sin(x)} + \frac{2*-sin(x)lg(tan(x))cos(x)}{ln{10}ln{10}(cos(x))sin(x)} + \frac{2lg(cos(x))sec^{2}(x)(1)cos(x)}{ln{10}ln{10}(tan(x))sin(x)} + \frac{2lg(cos(x))lg(tan(x))*-cos(x)cos(x)}{ln{10}sin^{2}(x)} + \frac{2lg(cos(x))lg(tan(x))*-sin(x)}{ln{10}sin(x)} - \frac{3*-2*0lg(sin(x))sin^{2}(x)sec^{2}(x)}{ln^{3}{10}cos^{2}(x)tan(x)} - \frac{3cos(x)sin^{2}(x)sec^{2}(x)}{ln^{2}{10}ln{10}(sin(x))cos^{2}(x)tan(x)} - \frac{3lg(sin(x))*2sin(x)cos(x)sec^{2}(x)}{ln^{2}{10}cos^{2}(x)tan(x)} - \frac{3lg(sin(x))sin^{2}(x)*2sin(x)sec^{2}(x)}{ln^{2}{10}cos^{3}(x)tan(x)} - \frac{3lg(sin(x))sin^{2}(x)*-sec^{2}(x)(1)sec^{2}(x)}{ln^{2}{10}cos^{2}(x)tan^{2}(x)} - \frac{3lg(sin(x))sin^{2}(x)*2sec^{2}(x)tan(x)}{ln^{2}{10}cos^{2}(x)tan(x)} - \frac{2*-0lg(sin(x))lg(tan(x))sin^{3}(x)}{ln^{2}{10}cos^{3}(x)} - \frac{2cos(x)lg(tan(x))sin^{3}(x)}{ln{10}ln{10}(sin(x))cos^{3}(x)} - \frac{2lg(sin(x))sec^{2}(x)(1)sin^{3}(x)}{ln{10}ln{10}(tan(x))cos^{3}(x)} - \frac{2lg(sin(x))lg(tan(x))*3sin^{2}(x)cos(x)}{ln{10}cos^{3}(x)} - \frac{2lg(sin(x))lg(tan(x))sin^{3}(x)*3sin(x)}{ln{10}cos^{4}(x)} + \frac{3*-2*0lg(sin(x))sin(x)sec^{4}(x)}{ln^{3}{10}cos(x)tan^{2}(x)} + \frac{3cos(x)sin(x)sec^{4}(x)}{ln^{2}{10}ln{10}(sin(x))cos(x)tan^{2}(x)} + \frac{3lg(sin(x))cos(x)sec^{4}(x)}{ln^{2}{10}cos(x)tan^{2}(x)} + \frac{3lg(sin(x))sin(x)sin(x)sec^{4}(x)}{ln^{2}{10}cos^{2}(x)tan^{2}(x)} + \frac{3lg(sin(x))sin(x)*-2sec^{2}(x)(1)sec^{4}(x)}{ln^{2}{10}cos(x)tan^{3}(x)} + \frac{3lg(sin(x))sin(x)*4sec^{4}(x)tan(x)}{ln^{2}{10}cos(x)tan^{2}(x)} - \frac{3*-2*0lg(sin(x))sec^{2}(x)}{ln^{3}{10}tan(x)} - \frac{3cos(x)sec^{2}(x)}{ln^{2}{10}ln{10}(sin(x))tan(x)} - \frac{3lg(sin(x))*-sec^{2}(x)(1)sec^{2}(x)}{ln^{2}{10}tan^{2}(x)} - \frac{3lg(sin(x))*2sec^{2}(x)tan(x)}{ln^{2}{10}tan(x)} - \frac{6*-2*0lg(sin(x))sin(x)sec^{2}(x)}{ln^{3}{10}cos(x)} - \frac{6cos(x)sin(x)sec^{2}(x)}{ln^{2}{10}ln{10}(sin(x))cos(x)} - \frac{6lg(sin(x))cos(x)sec^{2}(x)}{ln^{2}{10}cos(x)} - \frac{6lg(sin(x))sin(x)sin(x)sec^{2}(x)}{ln^{2}{10}cos^{2}(x)} - \frac{6lg(sin(x))sin(x)*2sec^{2}(x)tan(x)}{ln^{2}{10}cos(x)} + \frac{2*-0lg(sin(x))lg(cos(x))sec^{6}(x)}{ln^{2}{10}tan^{3}(x)} + \frac{2cos(x)lg(cos(x))sec^{6}(x)}{ln{10}ln{10}(sin(x))tan^{3}(x)} + \frac{2lg(sin(x))*-sin(x)sec^{6}(x)}{ln{10}ln{10}(cos(x))tan^{3}(x)} + \frac{2lg(sin(x))lg(cos(x))*-3sec^{2}(x)(1)sec^{6}(x)}{ln{10}tan^{4}(x)} + \frac{2lg(sin(x))lg(cos(x))*6sec^{6}(x)tan(x)}{ln{10}tan^{3}(x)} - \frac{4*-0lg(sin(x))lg(cos(x))sec^{4}(x)}{ln^{2}{10}tan(x)} - \frac{4cos(x)lg(cos(x))sec^{4}(x)}{ln{10}ln{10}(sin(x))tan(x)} - \frac{4lg(sin(x))*-sin(x)sec^{4}(x)}{ln{10}ln{10}(cos(x))tan(x)} - \frac{4lg(sin(x))lg(cos(x))*-sec^{2}(x)(1)sec^{4}(x)}{ln{10}tan^{2}(x)} - \frac{4lg(sin(x))lg(cos(x))*4sec^{4}(x)tan(x)}{ln{10}tan(x)} + \frac{4*-0lg(sin(x))lg(cos(x))tan(x)sec^{2}(x)}{ln^{2}{10}} + \frac{4cos(x)lg(cos(x))tan(x)sec^{2}(x)}{ln{10}ln{10}(sin(x))} + \frac{4lg(sin(x))*-sin(x)tan(x)sec^{2}(x)}{ln{10}ln{10}(cos(x))} + \frac{4lg(sin(x))lg(cos(x))sec^{2}(x)(1)sec^{2}(x)}{ln{10}} + \frac{4lg(sin(x))lg(cos(x))tan(x)*2sec^{2}(x)tan(x)}{ln{10}}\\=&\frac{12sec^{4}(x)}{ln^{3}{10}tan^{2}(x)} - \frac{24sec^{2}(x)}{ln^{3}{10}} + \frac{6lg(cos(x))cos^{2}(x)sec^{4}(x)}{ln^{2}{10}sin^{2}(x)tan^{2}(x)} + \frac{8lg(cos(x))cos^{3}(x)sec^{2}(x)}{ln^{2}{10}sin^{3}(x)tan(x)} + \frac{8lg(cos(x))cos(x)sec^{2}(x)}{ln^{2}{10}sin(x)tan(x)} + \frac{8lg(cos(x))cos(x)sec^{6}(x)}{ln^{2}{10}sin(x)tan^{3}(x)} - \frac{12lg(cos(x))cos^{2}(x)sec^{2}(x)}{ln^{2}{10}sin^{2}(x)} + \frac{6lg(cos(x))sec^{4}(x)}{ln^{2}{10}tan^{2}(x)} + \frac{6lg(sin(x))sec^{4}(x)}{ln^{2}{10}tan^{2}(x)} - \frac{12lg(cos(x))sec^{2}(x)}{ln^{2}{10}} + \frac{6lg(sin(x))sin^{2}(x)sec^{4}(x)}{ln^{2}{10}cos^{2}(x)tan^{2}(x)} - \frac{8lg(sin(x))sin(x)sec^{6}(x)}{ln^{2}{10}cos(x)tan^{3}(x)} - \frac{16lg(cos(x))cos(x)sec^{4}(x)}{ln^{2}{10}sin(x)tan(x)} + \frac{16lg(cos(x))cos(x)tan(x)sec^{2}(x)}{ln^{2}{10}sin(x)} - \frac{6lg(cos(x))lg(tan(x))cos^{4}(x)}{ln{10}sin^{4}(x)} - \frac{8lg(sin(x))sin(x)sec^{2}(x)}{ln^{2}{10}cos(x)tan(x)} - \frac{8lg(cos(x))lg(tan(x))cos^{2}(x)}{ln{10}sin^{2}(x)} - \frac{8lg(sin(x))lg(tan(x))sin^{2}(x)}{ln{10}cos^{2}(x)} - \frac{8lg(sin(x))sin^{3}(x)sec^{2}(x)}{ln^{2}{10}cos^{3}(x)tan(x)} - \frac{6lg(sin(x))lg(tan(x))sin^{4}(x)}{ln{10}cos^{4}(x)} - \frac{6lg(sin(x))lg(cos(x))sec^{8}(x)}{ln{10}tan^{4}(x)} - \frac{12lg(sin(x))sin^{2}(x)sec^{2}(x)}{ln^{2}{10}cos^{2}(x)} + \frac{16lg(sin(x))sin(x)sec^{4}(x)}{ln^{2}{10}cos(x)tan(x)} - \frac{16lg(sin(x))sin(x)tan(x)sec^{2}(x)}{ln^{2}{10}cos(x)} + \frac{16lg(sin(x))lg(cos(x))sec^{6}(x)}{ln{10}tan^{2}(x)} - \frac{2lg(tan(x))cos^{2}(x)}{ln^{2}{10}sin^{2}(x)} - \frac{12lg(sin(x))sec^{2}(x)}{ln^{2}{10}} - \frac{2lg(tan(x))sin^{2}(x)}{ln^{2}{10}cos^{2}(x)} - \frac{16lg(sin(x))lg(cos(x))sec^{4}(x)}{ln{10}} + \frac{4lg(cos(x))lg(sin(x))sec^{4}(x)}{ln{10}} + \frac{8lg(sin(x))lg(cos(x))tan^{2}(x)sec^{2}(x)}{ln{10}} - \frac{2lg(cos(x))lg(tan(x))}{ln{10}} - \frac{2lg(tan(x))lg(sin(x))}{ln{10}} - \frac{4lg(tan(x))}{ln^{2}{10}}\\ \end{split}\end{equation} \]