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.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ lg({(sin(x))}^{6} + {(cos(x))}^{6})\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = lg(sin^{6}(x) + cos^{6}(x))\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( lg(sin^{6}(x) + cos^{6}(x))\right)}{dx}\\=&\frac{(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{ln{10}(sin^{6}(x) + cos^{6}(x))}\\=&\frac{6sin^{5}(x)cos(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} - \frac{6sin(x)cos^{5}(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{6sin^{5}(x)cos(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} - \frac{6sin(x)cos^{5}(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}}\right)}{dx}\\=&\frac{6(\frac{-(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{(sin^{6}(x) + cos^{6}(x))^{2}})sin^{5}(x)cos(x)}{ln{10}} + \frac{6*-0sin^{5}(x)cos(x)}{(sin^{6}(x) + cos^{6}(x))ln^{2}{10}} + \frac{6*5sin^{4}(x)cos(x)cos(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} + \frac{6sin^{5}(x)*-sin(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} - \frac{6(\frac{-(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{(sin^{6}(x) + cos^{6}(x))^{2}})sin(x)cos^{5}(x)}{ln{10}} - \frac{6*-0sin(x)cos^{5}(x)}{(sin^{6}(x) + cos^{6}(x))ln^{2}{10}} - \frac{6cos(x)cos^{5}(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} - \frac{6sin(x)*-5cos^{4}(x)sin(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}}\\=&\frac{-36sin^{10}(x)cos^{2}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{72sin^{6}(x)cos^{6}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{30sin^{4}(x)cos^{2}(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} + \frac{30sin^{2}(x)cos^{4}(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} - \frac{36sin^{2}(x)cos^{10}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} - \frac{6cos^{6}(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} - \frac{6sin^{6}(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-36sin^{10}(x)cos^{2}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{72sin^{6}(x)cos^{6}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{30sin^{4}(x)cos^{2}(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} + \frac{30sin^{2}(x)cos^{4}(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} - \frac{36sin^{2}(x)cos^{10}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} - \frac{6cos^{6}(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} - \frac{6sin^{6}(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}}\right)}{dx}\\=&\frac{-36(\frac{-2(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{(sin^{6}(x) + cos^{6}(x))^{3}})sin^{10}(x)cos^{2}(x)}{ln{10}} - \frac{36*-0sin^{10}(x)cos^{2}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln^{2}{10}} - \frac{36*10sin^{9}(x)cos(x)cos^{2}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} - \frac{36sin^{10}(x)*-2cos(x)sin(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{72(\frac{-2(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{(sin^{6}(x) + cos^{6}(x))^{3}})sin^{6}(x)cos^{6}(x)}{ln{10}} + \frac{72*-0sin^{6}(x)cos^{6}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln^{2}{10}} + \frac{72*6sin^{5}(x)cos(x)cos^{6}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{72sin^{6}(x)*-6cos^{5}(x)sin(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{30(\frac{-(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{(sin^{6}(x) + cos^{6}(x))^{2}})sin^{4}(x)cos^{2}(x)}{ln{10}} + \frac{30*-0sin^{4}(x)cos^{2}(x)}{(sin^{6}(x) + cos^{6}(x))ln^{2}{10}} + \frac{30*4sin^{3}(x)cos(x)cos^{2}(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} + \frac{30sin^{4}(x)*-2cos(x)sin(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} + \frac{30(\frac{-(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{(sin^{6}(x) + cos^{6}(x))^{2}})sin^{2}(x)cos^{4}(x)}{ln{10}} + \frac{30*-0sin^{2}(x)cos^{4}(x)}{(sin^{6}(x) + cos^{6}(x))ln^{2}{10}} + \frac{30*2sin(x)cos(x)cos^{4}(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} + \frac{30sin^{2}(x)*-4cos^{3}(x)sin(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} - \frac{36(\frac{-2(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{(sin^{6}(x) + cos^{6}(x))^{3}})sin^{2}(x)cos^{10}(x)}{ln{10}} - \frac{36*-0sin^{2}(x)cos^{10}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln^{2}{10}} - \frac{36*2sin(x)cos(x)cos^{10}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} - \frac{36sin^{2}(x)*-10cos^{9}(x)sin(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} - \frac{6(\frac{-(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{(sin^{6}(x) + cos^{6}(x))^{2}})cos^{6}(x)}{ln{10}} - \frac{6*-0cos^{6}(x)}{(sin^{6}(x) + cos^{6}(x))ln^{2}{10}} - \frac{6*-6cos^{5}(x)sin(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} - \frac{6(\frac{-(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{(sin^{6}(x) + cos^{6}(x))^{2}})sin^{6}(x)}{ln{10}} - \frac{6*-0sin^{6}(x)}{(sin^{6}(x) + cos^{6}(x))ln^{2}{10}} - \frac{6*6sin^{5}(x)cos(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}}\\=&\frac{432sin^{15}(x)cos^{3}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}} - \frac{1296sin^{11}(x)cos^{7}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}} - \frac{540sin^{9}(x)cos^{3}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{108sin^{11}(x)cos(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{1296sin^{7}(x)cos^{11}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}} + \frac{648sin^{5}(x)cos^{7}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} - \frac{648sin^{7}(x)cos^{5}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{96sin(x)cos^{5}(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} - \frac{108sin(x)cos^{11}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} - \frac{96sin^{5}(x)cos(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} + \frac{540sin^{3}(x)cos^{9}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} - \frac{432sin^{3}(x)cos^{15}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{432sin^{15}(x)cos^{3}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}} - \frac{1296sin^{11}(x)cos^{7}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}} - \frac{540sin^{9}(x)cos^{3}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{108sin^{11}(x)cos(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{1296sin^{7}(x)cos^{11}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}} + \frac{648sin^{5}(x)cos^{7}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} - \frac{648sin^{7}(x)cos^{5}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{96sin(x)cos^{5}(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} - \frac{108sin(x)cos^{11}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} - \frac{96sin^{5}(x)cos(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} + \frac{540sin^{3}(x)cos^{9}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} - \frac{432sin^{3}(x)cos^{15}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}}\right)}{dx}\\=&\frac{432(\frac{-3(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{(sin^{6}(x) + cos^{6}(x))^{4}})sin^{15}(x)cos^{3}(x)}{ln{10}} + \frac{432*-0sin^{15}(x)cos^{3}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln^{2}{10}} + \frac{432*15sin^{14}(x)cos(x)cos^{3}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}} + \frac{432sin^{15}(x)*-3cos^{2}(x)sin(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}} - \frac{1296(\frac{-3(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{(sin^{6}(x) + cos^{6}(x))^{4}})sin^{11}(x)cos^{7}(x)}{ln{10}} - \frac{1296*-0sin^{11}(x)cos^{7}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln^{2}{10}} - \frac{1296*11sin^{10}(x)cos(x)cos^{7}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}} - \frac{1296sin^{11}(x)*-7cos^{6}(x)sin(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}} - \frac{540(\frac{-2(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{(sin^{6}(x) + cos^{6}(x))^{3}})sin^{9}(x)cos^{3}(x)}{ln{10}} - \frac{540*-0sin^{9}(x)cos^{3}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln^{2}{10}} - \frac{540*9sin^{8}(x)cos(x)cos^{3}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} - \frac{540sin^{9}(x)*-3cos^{2}(x)sin(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{108(\frac{-2(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{(sin^{6}(x) + cos^{6}(x))^{3}})sin^{11}(x)cos(x)}{ln{10}} + \frac{108*-0sin^{11}(x)cos(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln^{2}{10}} + \frac{108*11sin^{10}(x)cos(x)cos(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{108sin^{11}(x)*-sin(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{1296(\frac{-3(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{(sin^{6}(x) + cos^{6}(x))^{4}})sin^{7}(x)cos^{11}(x)}{ln{10}} + \frac{1296*-0sin^{7}(x)cos^{11}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln^{2}{10}} + \frac{1296*7sin^{6}(x)cos(x)cos^{11}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}} + \frac{1296sin^{7}(x)*-11cos^{10}(x)sin(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}} + \frac{648(\frac{-2(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{(sin^{6}(x) + cos^{6}(x))^{3}})sin^{5}(x)cos^{7}(x)}{ln{10}} + \frac{648*-0sin^{5}(x)cos^{7}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln^{2}{10}} + \frac{648*5sin^{4}(x)cos(x)cos^{7}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{648sin^{5}(x)*-7cos^{6}(x)sin(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} - \frac{648(\frac{-2(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{(sin^{6}(x) + cos^{6}(x))^{3}})sin^{7}(x)cos^{5}(x)}{ln{10}} - \frac{648*-0sin^{7}(x)cos^{5}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln^{2}{10}} - \frac{648*7sin^{6}(x)cos(x)cos^{5}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} - \frac{648sin^{7}(x)*-5cos^{4}(x)sin(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{96(\frac{-(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{(sin^{6}(x) + cos^{6}(x))^{2}})sin(x)cos^{5}(x)}{ln{10}} + \frac{96*-0sin(x)cos^{5}(x)}{(sin^{6}(x) + cos^{6}(x))ln^{2}{10}} + \frac{96cos(x)cos^{5}(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} + \frac{96sin(x)*-5cos^{4}(x)sin(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} - \frac{108(\frac{-2(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{(sin^{6}(x) + cos^{6}(x))^{3}})sin(x)cos^{11}(x)}{ln{10}} - \frac{108*-0sin(x)cos^{11}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln^{2}{10}} - \frac{108cos(x)cos^{11}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} - \frac{108sin(x)*-11cos^{10}(x)sin(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} - \frac{96(\frac{-(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{(sin^{6}(x) + cos^{6}(x))^{2}})sin^{5}(x)cos(x)}{ln{10}} - \frac{96*-0sin^{5}(x)cos(x)}{(sin^{6}(x) + cos^{6}(x))ln^{2}{10}} - \frac{96*5sin^{4}(x)cos(x)cos(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} - \frac{96sin^{5}(x)*-sin(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} + \frac{540(\frac{-2(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{(sin^{6}(x) + cos^{6}(x))^{3}})sin^{3}(x)cos^{9}(x)}{ln{10}} + \frac{540*-0sin^{3}(x)cos^{9}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln^{2}{10}} + \frac{540*3sin^{2}(x)cos(x)cos^{9}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{540sin^{3}(x)*-9cos^{8}(x)sin(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} - \frac{432(\frac{-3(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{(sin^{6}(x) + cos^{6}(x))^{4}})sin^{3}(x)cos^{15}(x)}{ln{10}} - \frac{432*-0sin^{3}(x)cos^{15}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln^{2}{10}} - \frac{432*3sin^{2}(x)cos(x)cos^{15}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}} - \frac{432sin^{3}(x)*-15cos^{14}(x)sin(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}}\\=&\frac{-7776sin^{20}(x)cos^{4}(x)}{(sin^{6}(x) + cos^{6}(x))^{4}ln{10}} + \frac{31104sin^{16}(x)cos^{8}(x)}{(sin^{6}(x) + cos^{6}(x))^{4}ln{10}} + \frac{12960sin^{14}(x)cos^{4}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}} - \frac{2592sin^{16}(x)cos^{2}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}} - \frac{46656sin^{12}(x)cos^{12}(x)}{(sin^{6}(x) + cos^{6}(x))^{4}ln{10}} - \frac{28512sin^{10}(x)cos^{8}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}} - \frac{1620sin^{8}(x)cos^{4}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{18144sin^{12}(x)cos^{6}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}} + \frac{3384sin^{10}(x)cos^{2}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{18144sin^{6}(x)cos^{12}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}} - \frac{1620sin^{4}(x)cos^{8}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} - \frac{10224sin^{6}(x)cos^{6}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} - \frac{28512sin^{8}(x)cos^{10}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}} + \frac{31104sin^{8}(x)cos^{16}(x)}{(sin^{6}(x) + cos^{6}(x))^{4}ln{10}} - \frac{480sin^{4}(x)cos^{2}(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} + \frac{3384sin^{2}(x)cos^{10}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} - \frac{2592sin^{2}(x)cos^{16}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}} - \frac{480sin^{2}(x)cos^{4}(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} + \frac{96cos^{6}(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} - \frac{108cos^{12}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{12960sin^{4}(x)cos^{14}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}} - \frac{7776sin^{4}(x)cos^{20}(x)}{(sin^{6}(x) + cos^{6}(x))^{4}ln{10}} - \frac{108sin^{12}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{96sin^{6}(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ lg({(sin(x))}^{6} + {(cos(x))}^{6})\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = lg(sin^{6}(x) + cos^{6}(x))\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( lg(sin^{6}(x) + cos^{6}(x))\right)}{dx}\\=&\frac{(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{ln{10}(sin^{6}(x) + cos^{6}(x))}\\=&\frac{6sin^{5}(x)cos(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} - \frac{6sin(x)cos^{5}(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{6sin^{5}(x)cos(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} - \frac{6sin(x)cos^{5}(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}}\right)}{dx}\\=&\frac{6(\frac{-(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{(sin^{6}(x) + cos^{6}(x))^{2}})sin^{5}(x)cos(x)}{ln{10}} + \frac{6*-0sin^{5}(x)cos(x)}{(sin^{6}(x) + cos^{6}(x))ln^{2}{10}} + \frac{6*5sin^{4}(x)cos(x)cos(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} + \frac{6sin^{5}(x)*-sin(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} - \frac{6(\frac{-(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{(sin^{6}(x) + cos^{6}(x))^{2}})sin(x)cos^{5}(x)}{ln{10}} - \frac{6*-0sin(x)cos^{5}(x)}{(sin^{6}(x) + cos^{6}(x))ln^{2}{10}} - \frac{6cos(x)cos^{5}(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} - \frac{6sin(x)*-5cos^{4}(x)sin(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}}\\=&\frac{-36sin^{10}(x)cos^{2}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{72sin^{6}(x)cos^{6}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{30sin^{4}(x)cos^{2}(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} + \frac{30sin^{2}(x)cos^{4}(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} - \frac{36sin^{2}(x)cos^{10}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} - \frac{6cos^{6}(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} - \frac{6sin^{6}(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-36sin^{10}(x)cos^{2}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{72sin^{6}(x)cos^{6}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{30sin^{4}(x)cos^{2}(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} + \frac{30sin^{2}(x)cos^{4}(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} - \frac{36sin^{2}(x)cos^{10}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} - \frac{6cos^{6}(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} - \frac{6sin^{6}(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}}\right)}{dx}\\=&\frac{-36(\frac{-2(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{(sin^{6}(x) + cos^{6}(x))^{3}})sin^{10}(x)cos^{2}(x)}{ln{10}} - \frac{36*-0sin^{10}(x)cos^{2}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln^{2}{10}} - \frac{36*10sin^{9}(x)cos(x)cos^{2}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} - \frac{36sin^{10}(x)*-2cos(x)sin(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{72(\frac{-2(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{(sin^{6}(x) + cos^{6}(x))^{3}})sin^{6}(x)cos^{6}(x)}{ln{10}} + \frac{72*-0sin^{6}(x)cos^{6}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln^{2}{10}} + \frac{72*6sin^{5}(x)cos(x)cos^{6}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{72sin^{6}(x)*-6cos^{5}(x)sin(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{30(\frac{-(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{(sin^{6}(x) + cos^{6}(x))^{2}})sin^{4}(x)cos^{2}(x)}{ln{10}} + \frac{30*-0sin^{4}(x)cos^{2}(x)}{(sin^{6}(x) + cos^{6}(x))ln^{2}{10}} + \frac{30*4sin^{3}(x)cos(x)cos^{2}(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} + \frac{30sin^{4}(x)*-2cos(x)sin(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} + \frac{30(\frac{-(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{(sin^{6}(x) + cos^{6}(x))^{2}})sin^{2}(x)cos^{4}(x)}{ln{10}} + \frac{30*-0sin^{2}(x)cos^{4}(x)}{(sin^{6}(x) + cos^{6}(x))ln^{2}{10}} + \frac{30*2sin(x)cos(x)cos^{4}(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} + \frac{30sin^{2}(x)*-4cos^{3}(x)sin(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} - \frac{36(\frac{-2(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{(sin^{6}(x) + cos^{6}(x))^{3}})sin^{2}(x)cos^{10}(x)}{ln{10}} - \frac{36*-0sin^{2}(x)cos^{10}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln^{2}{10}} - \frac{36*2sin(x)cos(x)cos^{10}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} - \frac{36sin^{2}(x)*-10cos^{9}(x)sin(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} - \frac{6(\frac{-(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{(sin^{6}(x) + cos^{6}(x))^{2}})cos^{6}(x)}{ln{10}} - \frac{6*-0cos^{6}(x)}{(sin^{6}(x) + cos^{6}(x))ln^{2}{10}} - \frac{6*-6cos^{5}(x)sin(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} - \frac{6(\frac{-(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{(sin^{6}(x) + cos^{6}(x))^{2}})sin^{6}(x)}{ln{10}} - \frac{6*-0sin^{6}(x)}{(sin^{6}(x) + cos^{6}(x))ln^{2}{10}} - \frac{6*6sin^{5}(x)cos(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}}\\=&\frac{432sin^{15}(x)cos^{3}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}} - \frac{1296sin^{11}(x)cos^{7}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}} - \frac{540sin^{9}(x)cos^{3}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{108sin^{11}(x)cos(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{1296sin^{7}(x)cos^{11}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}} + \frac{648sin^{5}(x)cos^{7}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} - \frac{648sin^{7}(x)cos^{5}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{96sin(x)cos^{5}(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} - \frac{108sin(x)cos^{11}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} - \frac{96sin^{5}(x)cos(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} + \frac{540sin^{3}(x)cos^{9}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} - \frac{432sin^{3}(x)cos^{15}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{432sin^{15}(x)cos^{3}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}} - \frac{1296sin^{11}(x)cos^{7}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}} - \frac{540sin^{9}(x)cos^{3}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{108sin^{11}(x)cos(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{1296sin^{7}(x)cos^{11}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}} + \frac{648sin^{5}(x)cos^{7}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} - \frac{648sin^{7}(x)cos^{5}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{96sin(x)cos^{5}(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} - \frac{108sin(x)cos^{11}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} - \frac{96sin^{5}(x)cos(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} + \frac{540sin^{3}(x)cos^{9}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} - \frac{432sin^{3}(x)cos^{15}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}}\right)}{dx}\\=&\frac{432(\frac{-3(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{(sin^{6}(x) + cos^{6}(x))^{4}})sin^{15}(x)cos^{3}(x)}{ln{10}} + \frac{432*-0sin^{15}(x)cos^{3}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln^{2}{10}} + \frac{432*15sin^{14}(x)cos(x)cos^{3}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}} + \frac{432sin^{15}(x)*-3cos^{2}(x)sin(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}} - \frac{1296(\frac{-3(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{(sin^{6}(x) + cos^{6}(x))^{4}})sin^{11}(x)cos^{7}(x)}{ln{10}} - \frac{1296*-0sin^{11}(x)cos^{7}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln^{2}{10}} - \frac{1296*11sin^{10}(x)cos(x)cos^{7}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}} - \frac{1296sin^{11}(x)*-7cos^{6}(x)sin(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}} - \frac{540(\frac{-2(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{(sin^{6}(x) + cos^{6}(x))^{3}})sin^{9}(x)cos^{3}(x)}{ln{10}} - \frac{540*-0sin^{9}(x)cos^{3}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln^{2}{10}} - \frac{540*9sin^{8}(x)cos(x)cos^{3}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} - \frac{540sin^{9}(x)*-3cos^{2}(x)sin(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{108(\frac{-2(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{(sin^{6}(x) + cos^{6}(x))^{3}})sin^{11}(x)cos(x)}{ln{10}} + \frac{108*-0sin^{11}(x)cos(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln^{2}{10}} + \frac{108*11sin^{10}(x)cos(x)cos(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{108sin^{11}(x)*-sin(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{1296(\frac{-3(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{(sin^{6}(x) + cos^{6}(x))^{4}})sin^{7}(x)cos^{11}(x)}{ln{10}} + \frac{1296*-0sin^{7}(x)cos^{11}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln^{2}{10}} + \frac{1296*7sin^{6}(x)cos(x)cos^{11}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}} + \frac{1296sin^{7}(x)*-11cos^{10}(x)sin(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}} + \frac{648(\frac{-2(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{(sin^{6}(x) + cos^{6}(x))^{3}})sin^{5}(x)cos^{7}(x)}{ln{10}} + \frac{648*-0sin^{5}(x)cos^{7}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln^{2}{10}} + \frac{648*5sin^{4}(x)cos(x)cos^{7}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{648sin^{5}(x)*-7cos^{6}(x)sin(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} - \frac{648(\frac{-2(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{(sin^{6}(x) + cos^{6}(x))^{3}})sin^{7}(x)cos^{5}(x)}{ln{10}} - \frac{648*-0sin^{7}(x)cos^{5}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln^{2}{10}} - \frac{648*7sin^{6}(x)cos(x)cos^{5}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} - \frac{648sin^{7}(x)*-5cos^{4}(x)sin(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{96(\frac{-(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{(sin^{6}(x) + cos^{6}(x))^{2}})sin(x)cos^{5}(x)}{ln{10}} + \frac{96*-0sin(x)cos^{5}(x)}{(sin^{6}(x) + cos^{6}(x))ln^{2}{10}} + \frac{96cos(x)cos^{5}(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} + \frac{96sin(x)*-5cos^{4}(x)sin(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} - \frac{108(\frac{-2(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{(sin^{6}(x) + cos^{6}(x))^{3}})sin(x)cos^{11}(x)}{ln{10}} - \frac{108*-0sin(x)cos^{11}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln^{2}{10}} - \frac{108cos(x)cos^{11}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} - \frac{108sin(x)*-11cos^{10}(x)sin(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} - \frac{96(\frac{-(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{(sin^{6}(x) + cos^{6}(x))^{2}})sin^{5}(x)cos(x)}{ln{10}} - \frac{96*-0sin^{5}(x)cos(x)}{(sin^{6}(x) + cos^{6}(x))ln^{2}{10}} - \frac{96*5sin^{4}(x)cos(x)cos(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} - \frac{96sin^{5}(x)*-sin(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} + \frac{540(\frac{-2(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{(sin^{6}(x) + cos^{6}(x))^{3}})sin^{3}(x)cos^{9}(x)}{ln{10}} + \frac{540*-0sin^{3}(x)cos^{9}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln^{2}{10}} + \frac{540*3sin^{2}(x)cos(x)cos^{9}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{540sin^{3}(x)*-9cos^{8}(x)sin(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} - \frac{432(\frac{-3(6sin^{5}(x)cos(x) + -6cos^{5}(x)sin(x))}{(sin^{6}(x) + cos^{6}(x))^{4}})sin^{3}(x)cos^{15}(x)}{ln{10}} - \frac{432*-0sin^{3}(x)cos^{15}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln^{2}{10}} - \frac{432*3sin^{2}(x)cos(x)cos^{15}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}} - \frac{432sin^{3}(x)*-15cos^{14}(x)sin(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}}\\=&\frac{-7776sin^{20}(x)cos^{4}(x)}{(sin^{6}(x) + cos^{6}(x))^{4}ln{10}} + \frac{31104sin^{16}(x)cos^{8}(x)}{(sin^{6}(x) + cos^{6}(x))^{4}ln{10}} + \frac{12960sin^{14}(x)cos^{4}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}} - \frac{2592sin^{16}(x)cos^{2}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}} - \frac{46656sin^{12}(x)cos^{12}(x)}{(sin^{6}(x) + cos^{6}(x))^{4}ln{10}} - \frac{28512sin^{10}(x)cos^{8}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}} - \frac{1620sin^{8}(x)cos^{4}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{18144sin^{12}(x)cos^{6}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}} + \frac{3384sin^{10}(x)cos^{2}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{18144sin^{6}(x)cos^{12}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}} - \frac{1620sin^{4}(x)cos^{8}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} - \frac{10224sin^{6}(x)cos^{6}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} - \frac{28512sin^{8}(x)cos^{10}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}} + \frac{31104sin^{8}(x)cos^{16}(x)}{(sin^{6}(x) + cos^{6}(x))^{4}ln{10}} - \frac{480sin^{4}(x)cos^{2}(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} + \frac{3384sin^{2}(x)cos^{10}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} - \frac{2592sin^{2}(x)cos^{16}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}} - \frac{480sin^{2}(x)cos^{4}(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} + \frac{96cos^{6}(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}} - \frac{108cos^{12}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{12960sin^{4}(x)cos^{14}(x)}{(sin^{6}(x) + cos^{6}(x))^{3}ln{10}} - \frac{7776sin^{4}(x)cos^{20}(x)}{(sin^{6}(x) + cos^{6}(x))^{4}ln{10}} - \frac{108sin^{12}(x)}{(sin^{6}(x) + cos^{6}(x))^{2}ln{10}} + \frac{96sin^{6}(x)}{(sin^{6}(x) + cos^{6}(x))ln{10}}\\ \end{split}\end{equation} \]