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.
\[ \begin{equation}\begin{split}[1/3]Find\ the\ 4th\ derivative\ of\ function\ arcsin(sin(x))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( arcsin(sin(x))\right)}{dx}\\=&(\frac{(cos(x))}{((1 - (sin(x))^{2})^{\frac{1}{2}})})\\=&\frac{cos(x)}{(-sin^{2}(x) + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{cos(x)}{(-sin^{2}(x) + 1)^{\frac{1}{2}}}\right)}{dx}\\=&(\frac{\frac{-1}{2}(-2sin(x)cos(x) + 0)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}})cos(x) + \frac{-sin(x)}{(-sin^{2}(x) + 1)^{\frac{1}{2}}}\\=&\frac{sin(x)cos^{2}(x)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}} - \frac{sin(x)}{(-sin^{2}(x) + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{sin(x)cos^{2}(x)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}} - \frac{sin(x)}{(-sin^{2}(x) + 1)^{\frac{1}{2}}}\right)}{dx}\\=&(\frac{\frac{-3}{2}(-2sin(x)cos(x) + 0)}{(-sin^{2}(x) + 1)^{\frac{5}{2}}})sin(x)cos^{2}(x) + \frac{cos(x)cos^{2}(x)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}} + \frac{sin(x)*-2cos(x)sin(x)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}} - (\frac{\frac{-1}{2}(-2sin(x)cos(x) + 0)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}})sin(x) - \frac{cos(x)}{(-sin^{2}(x) + 1)^{\frac{1}{2}}}\\=&\frac{3sin^{2}(x)cos^{3}(x)}{(-sin^{2}(x) + 1)^{\frac{5}{2}}} + \frac{cos^{3}(x)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}} - \frac{3sin^{2}(x)cos(x)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}} - \frac{cos(x)}{(-sin^{2}(x) + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{3sin^{2}(x)cos^{3}(x)}{(-sin^{2}(x) + 1)^{\frac{5}{2}}} + \frac{cos^{3}(x)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}} - \frac{3sin^{2}(x)cos(x)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}} - \frac{cos(x)}{(-sin^{2}(x) + 1)^{\frac{1}{2}}}\right)}{dx}\\=&3(\frac{\frac{-5}{2}(-2sin(x)cos(x) + 0)}{(-sin^{2}(x) + 1)^{\frac{7}{2}}})sin^{2}(x)cos^{3}(x) + \frac{3*2sin(x)cos(x)cos^{3}(x)}{(-sin^{2}(x) + 1)^{\frac{5}{2}}} + \frac{3sin^{2}(x)*-3cos^{2}(x)sin(x)}{(-sin^{2}(x) + 1)^{\frac{5}{2}}} + (\frac{\frac{-3}{2}(-2sin(x)cos(x) + 0)}{(-sin^{2}(x) + 1)^{\frac{5}{2}}})cos^{3}(x) + \frac{-3cos^{2}(x)sin(x)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}} - 3(\frac{\frac{-3}{2}(-2sin(x)cos(x) + 0)}{(-sin^{2}(x) + 1)^{\frac{5}{2}}})sin^{2}(x)cos(x) - \frac{3*2sin(x)cos(x)cos(x)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}} - \frac{3sin^{2}(x)*-sin(x)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}} - (\frac{\frac{-1}{2}(-2sin(x)cos(x) + 0)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}})cos(x) - \frac{-sin(x)}{(-sin^{2}(x) + 1)^{\frac{1}{2}}}\\=&\frac{15sin^{3}(x)cos^{4}(x)}{(-sin^{2}(x) + 1)^{\frac{7}{2}}} + \frac{9sin(x)cos^{4}(x)}{(-sin^{2}(x) + 1)^{\frac{5}{2}}} - \frac{18sin^{3}(x)cos^{2}(x)}{(-sin^{2}(x) + 1)^{\frac{5}{2}}} - \frac{10sin(x)cos^{2}(x)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}} + \frac{3sin^{3}(x)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}} + \frac{sin(x)}{(-sin^{2}(x) + 1)^{\frac{1}{2}}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/3]Find\ the\ 4th\ derivative\ of\ function\ arccos(cos(x))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( arccos(cos(x))\right)}{dx}\\=&(\frac{-(-sin(x))}{((1 - (cos(x))^{2})^{\frac{1}{2}})})\\=&\frac{sin(x)}{(-cos^{2}(x) + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{sin(x)}{(-cos^{2}(x) + 1)^{\frac{1}{2}}}\right)}{dx}\\=&(\frac{\frac{-1}{2}(--2cos(x)sin(x) + 0)}{(-cos^{2}(x) + 1)^{\frac{3}{2}}})sin(x) + \frac{cos(x)}{(-cos^{2}(x) + 1)^{\frac{1}{2}}}\\=&\frac{-sin^{2}(x)cos(x)}{(-cos^{2}(x) + 1)^{\frac{3}{2}}} + \frac{cos(x)}{(-cos^{2}(x) + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-sin^{2}(x)cos(x)}{(-cos^{2}(x) + 1)^{\frac{3}{2}}} + \frac{cos(x)}{(-cos^{2}(x) + 1)^{\frac{1}{2}}}\right)}{dx}\\=&-(\frac{\frac{-3}{2}(--2cos(x)sin(x) + 0)}{(-cos^{2}(x) + 1)^{\frac{5}{2}}})sin^{2}(x)cos(x) - \frac{2sin(x)cos(x)cos(x)}{(-cos^{2}(x) + 1)^{\frac{3}{2}}} - \frac{sin^{2}(x)*-sin(x)}{(-cos^{2}(x) + 1)^{\frac{3}{2}}} + (\frac{\frac{-1}{2}(--2cos(x)sin(x) + 0)}{(-cos^{2}(x) + 1)^{\frac{3}{2}}})cos(x) + \frac{-sin(x)}{(-cos^{2}(x) + 1)^{\frac{1}{2}}}\\=&\frac{3sin^{3}(x)cos^{2}(x)}{(-cos^{2}(x) + 1)^{\frac{5}{2}}} - \frac{3sin(x)cos^{2}(x)}{(-cos^{2}(x) + 1)^{\frac{3}{2}}} + \frac{sin^{3}(x)}{(-cos^{2}(x) + 1)^{\frac{3}{2}}} - \frac{sin(x)}{(-cos^{2}(x) + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{3sin^{3}(x)cos^{2}(x)}{(-cos^{2}(x) + 1)^{\frac{5}{2}}} - \frac{3sin(x)cos^{2}(x)}{(-cos^{2}(x) + 1)^{\frac{3}{2}}} + \frac{sin^{3}(x)}{(-cos^{2}(x) + 1)^{\frac{3}{2}}} - \frac{sin(x)}{(-cos^{2}(x) + 1)^{\frac{1}{2}}}\right)}{dx}\\=&3(\frac{\frac{-5}{2}(--2cos(x)sin(x) + 0)}{(-cos^{2}(x) + 1)^{\frac{7}{2}}})sin^{3}(x)cos^{2}(x) + \frac{3*3sin^{2}(x)cos(x)cos^{2}(x)}{(-cos^{2}(x) + 1)^{\frac{5}{2}}} + \frac{3sin^{3}(x)*-2cos(x)sin(x)}{(-cos^{2}(x) + 1)^{\frac{5}{2}}} - 3(\frac{\frac{-3}{2}(--2cos(x)sin(x) + 0)}{(-cos^{2}(x) + 1)^{\frac{5}{2}}})sin(x)cos^{2}(x) - \frac{3cos(x)cos^{2}(x)}{(-cos^{2}(x) + 1)^{\frac{3}{2}}} - \frac{3sin(x)*-2cos(x)sin(x)}{(-cos^{2}(x) + 1)^{\frac{3}{2}}} + (\frac{\frac{-3}{2}(--2cos(x)sin(x) + 0)}{(-cos^{2}(x) + 1)^{\frac{5}{2}}})sin^{3}(x) + \frac{3sin^{2}(x)cos(x)}{(-cos^{2}(x) + 1)^{\frac{3}{2}}} - (\frac{\frac{-1}{2}(--2cos(x)sin(x) + 0)}{(-cos^{2}(x) + 1)^{\frac{3}{2}}})sin(x) - \frac{cos(x)}{(-cos^{2}(x) + 1)^{\frac{1}{2}}}\\=&\frac{-15sin^{4}(x)cos^{3}(x)}{(-cos^{2}(x) + 1)^{\frac{7}{2}}} + \frac{18sin^{2}(x)cos^{3}(x)}{(-cos^{2}(x) + 1)^{\frac{5}{2}}} - \frac{9sin^{4}(x)cos(x)}{(-cos^{2}(x) + 1)^{\frac{5}{2}}} - \frac{3cos^{3}(x)}{(-cos^{2}(x) + 1)^{\frac{3}{2}}} + \frac{10sin^{2}(x)cos(x)}{(-cos^{2}(x) + 1)^{\frac{3}{2}}} - \frac{cos(x)}{(-cos^{2}(x) + 1)^{\frac{1}{2}}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[3/3]Find\ the\ 4th\ derivative\ of\ function\ arctan(tan(x))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( arctan(tan(x))\right)}{dx}\\=&(\frac{(sec^{2}(x)(1))}{(1 + (tan(x))^{2})})\\=&\frac{sec^{2}(x)}{(tan^{2}(x) + 1)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{sec^{2}(x)}{(tan^{2}(x) + 1)}\right)}{dx}\\=&(\frac{-(2tan(x)sec^{2}(x)(1) + 0)}{(tan^{2}(x) + 1)^{2}})sec^{2}(x) + \frac{2sec^{2}(x)tan(x)}{(tan^{2}(x) + 1)}\\=&\frac{-2tan(x)sec^{4}(x)}{(tan^{2}(x) + 1)^{2}} + \frac{2tan(x)sec^{2}(x)}{(tan^{2}(x) + 1)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2tan(x)sec^{4}(x)}{(tan^{2}(x) + 1)^{2}} + \frac{2tan(x)sec^{2}(x)}{(tan^{2}(x) + 1)}\right)}{dx}\\=&-2(\frac{-2(2tan(x)sec^{2}(x)(1) + 0)}{(tan^{2}(x) + 1)^{3}})tan(x)sec^{4}(x) - \frac{2sec^{2}(x)(1)sec^{4}(x)}{(tan^{2}(x) + 1)^{2}} - \frac{2tan(x)*4sec^{4}(x)tan(x)}{(tan^{2}(x) + 1)^{2}} + 2(\frac{-(2tan(x)sec^{2}(x)(1) + 0)}{(tan^{2}(x) + 1)^{2}})tan(x)sec^{2}(x) + \frac{2sec^{2}(x)(1)sec^{2}(x)}{(tan^{2}(x) + 1)} + \frac{2tan(x)*2sec^{2}(x)tan(x)}{(tan^{2}(x) + 1)}\\=&\frac{8tan^{2}(x)sec^{6}(x)}{(tan^{2}(x) + 1)^{3}} - \frac{2sec^{6}(x)}{(tan^{2}(x) + 1)^{2}} - \frac{12tan^{2}(x)sec^{4}(x)}{(tan^{2}(x) + 1)^{2}} + \frac{2sec^{4}(x)}{(tan^{2}(x) + 1)} + \frac{4tan^{2}(x)sec^{2}(x)}{(tan^{2}(x) + 1)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{8tan^{2}(x)sec^{6}(x)}{(tan^{2}(x) + 1)^{3}} - \frac{2sec^{6}(x)}{(tan^{2}(x) + 1)^{2}} - \frac{12tan^{2}(x)sec^{4}(x)}{(tan^{2}(x) + 1)^{2}} + \frac{2sec^{4}(x)}{(tan^{2}(x) + 1)} + \frac{4tan^{2}(x)sec^{2}(x)}{(tan^{2}(x) + 1)}\right)}{dx}\\=&8(\frac{-3(2tan(x)sec^{2}(x)(1) + 0)}{(tan^{2}(x) + 1)^{4}})tan^{2}(x)sec^{6}(x) + \frac{8*2tan(x)sec^{2}(x)(1)sec^{6}(x)}{(tan^{2}(x) + 1)^{3}} + \frac{8tan^{2}(x)*6sec^{6}(x)tan(x)}{(tan^{2}(x) + 1)^{3}} - 2(\frac{-2(2tan(x)sec^{2}(x)(1) + 0)}{(tan^{2}(x) + 1)^{3}})sec^{6}(x) - \frac{2*6sec^{6}(x)tan(x)}{(tan^{2}(x) + 1)^{2}} - 12(\frac{-2(2tan(x)sec^{2}(x)(1) + 0)}{(tan^{2}(x) + 1)^{3}})tan^{2}(x)sec^{4}(x) - \frac{12*2tan(x)sec^{2}(x)(1)sec^{4}(x)}{(tan^{2}(x) + 1)^{2}} - \frac{12tan^{2}(x)*4sec^{4}(x)tan(x)}{(tan^{2}(x) + 1)^{2}} + 2(\frac{-(2tan(x)sec^{2}(x)(1) + 0)}{(tan^{2}(x) + 1)^{2}})sec^{4}(x) + \frac{2*4sec^{4}(x)tan(x)}{(tan^{2}(x) + 1)} + 4(\frac{-(2tan(x)sec^{2}(x)(1) + 0)}{(tan^{2}(x) + 1)^{2}})tan^{2}(x)sec^{2}(x) + \frac{4*2tan(x)sec^{2}(x)(1)sec^{2}(x)}{(tan^{2}(x) + 1)} + \frac{4tan^{2}(x)*2sec^{2}(x)tan(x)}{(tan^{2}(x) + 1)}\\=&\frac{-48tan^{3}(x)sec^{8}(x)}{(tan^{2}(x) + 1)^{4}} + \frac{24tan(x)sec^{8}(x)}{(tan^{2}(x) + 1)^{3}} + \frac{96tan^{3}(x)sec^{6}(x)}{(tan^{2}(x) + 1)^{3}} - \frac{40tan(x)sec^{6}(x)}{(tan^{2}(x) + 1)^{2}} + \frac{16tan(x)sec^{4}(x)}{(tan^{2}(x) + 1)} - \frac{56tan^{3}(x)sec^{4}(x)}{(tan^{2}(x) + 1)^{2}} + \frac{8tan^{3}(x)sec^{2}(x)}{(tan^{2}(x) + 1)}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/3]Find\ the\ 4th\ derivative\ of\ function\ arcsin(sin(x))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( arcsin(sin(x))\right)}{dx}\\=&(\frac{(cos(x))}{((1 - (sin(x))^{2})^{\frac{1}{2}})})\\=&\frac{cos(x)}{(-sin^{2}(x) + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{cos(x)}{(-sin^{2}(x) + 1)^{\frac{1}{2}}}\right)}{dx}\\=&(\frac{\frac{-1}{2}(-2sin(x)cos(x) + 0)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}})cos(x) + \frac{-sin(x)}{(-sin^{2}(x) + 1)^{\frac{1}{2}}}\\=&\frac{sin(x)cos^{2}(x)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}} - \frac{sin(x)}{(-sin^{2}(x) + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{sin(x)cos^{2}(x)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}} - \frac{sin(x)}{(-sin^{2}(x) + 1)^{\frac{1}{2}}}\right)}{dx}\\=&(\frac{\frac{-3}{2}(-2sin(x)cos(x) + 0)}{(-sin^{2}(x) + 1)^{\frac{5}{2}}})sin(x)cos^{2}(x) + \frac{cos(x)cos^{2}(x)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}} + \frac{sin(x)*-2cos(x)sin(x)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}} - (\frac{\frac{-1}{2}(-2sin(x)cos(x) + 0)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}})sin(x) - \frac{cos(x)}{(-sin^{2}(x) + 1)^{\frac{1}{2}}}\\=&\frac{3sin^{2}(x)cos^{3}(x)}{(-sin^{2}(x) + 1)^{\frac{5}{2}}} + \frac{cos^{3}(x)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}} - \frac{3sin^{2}(x)cos(x)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}} - \frac{cos(x)}{(-sin^{2}(x) + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{3sin^{2}(x)cos^{3}(x)}{(-sin^{2}(x) + 1)^{\frac{5}{2}}} + \frac{cos^{3}(x)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}} - \frac{3sin^{2}(x)cos(x)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}} - \frac{cos(x)}{(-sin^{2}(x) + 1)^{\frac{1}{2}}}\right)}{dx}\\=&3(\frac{\frac{-5}{2}(-2sin(x)cos(x) + 0)}{(-sin^{2}(x) + 1)^{\frac{7}{2}}})sin^{2}(x)cos^{3}(x) + \frac{3*2sin(x)cos(x)cos^{3}(x)}{(-sin^{2}(x) + 1)^{\frac{5}{2}}} + \frac{3sin^{2}(x)*-3cos^{2}(x)sin(x)}{(-sin^{2}(x) + 1)^{\frac{5}{2}}} + (\frac{\frac{-3}{2}(-2sin(x)cos(x) + 0)}{(-sin^{2}(x) + 1)^{\frac{5}{2}}})cos^{3}(x) + \frac{-3cos^{2}(x)sin(x)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}} - 3(\frac{\frac{-3}{2}(-2sin(x)cos(x) + 0)}{(-sin^{2}(x) + 1)^{\frac{5}{2}}})sin^{2}(x)cos(x) - \frac{3*2sin(x)cos(x)cos(x)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}} - \frac{3sin^{2}(x)*-sin(x)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}} - (\frac{\frac{-1}{2}(-2sin(x)cos(x) + 0)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}})cos(x) - \frac{-sin(x)}{(-sin^{2}(x) + 1)^{\frac{1}{2}}}\\=&\frac{15sin^{3}(x)cos^{4}(x)}{(-sin^{2}(x) + 1)^{\frac{7}{2}}} + \frac{9sin(x)cos^{4}(x)}{(-sin^{2}(x) + 1)^{\frac{5}{2}}} - \frac{18sin^{3}(x)cos^{2}(x)}{(-sin^{2}(x) + 1)^{\frac{5}{2}}} - \frac{10sin(x)cos^{2}(x)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}} + \frac{3sin^{3}(x)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}} + \frac{sin(x)}{(-sin^{2}(x) + 1)^{\frac{1}{2}}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/3]Find\ the\ 4th\ derivative\ of\ function\ arccos(cos(x))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( arccos(cos(x))\right)}{dx}\\=&(\frac{-(-sin(x))}{((1 - (cos(x))^{2})^{\frac{1}{2}})})\\=&\frac{sin(x)}{(-cos^{2}(x) + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{sin(x)}{(-cos^{2}(x) + 1)^{\frac{1}{2}}}\right)}{dx}\\=&(\frac{\frac{-1}{2}(--2cos(x)sin(x) + 0)}{(-cos^{2}(x) + 1)^{\frac{3}{2}}})sin(x) + \frac{cos(x)}{(-cos^{2}(x) + 1)^{\frac{1}{2}}}\\=&\frac{-sin^{2}(x)cos(x)}{(-cos^{2}(x) + 1)^{\frac{3}{2}}} + \frac{cos(x)}{(-cos^{2}(x) + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-sin^{2}(x)cos(x)}{(-cos^{2}(x) + 1)^{\frac{3}{2}}} + \frac{cos(x)}{(-cos^{2}(x) + 1)^{\frac{1}{2}}}\right)}{dx}\\=&-(\frac{\frac{-3}{2}(--2cos(x)sin(x) + 0)}{(-cos^{2}(x) + 1)^{\frac{5}{2}}})sin^{2}(x)cos(x) - \frac{2sin(x)cos(x)cos(x)}{(-cos^{2}(x) + 1)^{\frac{3}{2}}} - \frac{sin^{2}(x)*-sin(x)}{(-cos^{2}(x) + 1)^{\frac{3}{2}}} + (\frac{\frac{-1}{2}(--2cos(x)sin(x) + 0)}{(-cos^{2}(x) + 1)^{\frac{3}{2}}})cos(x) + \frac{-sin(x)}{(-cos^{2}(x) + 1)^{\frac{1}{2}}}\\=&\frac{3sin^{3}(x)cos^{2}(x)}{(-cos^{2}(x) + 1)^{\frac{5}{2}}} - \frac{3sin(x)cos^{2}(x)}{(-cos^{2}(x) + 1)^{\frac{3}{2}}} + \frac{sin^{3}(x)}{(-cos^{2}(x) + 1)^{\frac{3}{2}}} - \frac{sin(x)}{(-cos^{2}(x) + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{3sin^{3}(x)cos^{2}(x)}{(-cos^{2}(x) + 1)^{\frac{5}{2}}} - \frac{3sin(x)cos^{2}(x)}{(-cos^{2}(x) + 1)^{\frac{3}{2}}} + \frac{sin^{3}(x)}{(-cos^{2}(x) + 1)^{\frac{3}{2}}} - \frac{sin(x)}{(-cos^{2}(x) + 1)^{\frac{1}{2}}}\right)}{dx}\\=&3(\frac{\frac{-5}{2}(--2cos(x)sin(x) + 0)}{(-cos^{2}(x) + 1)^{\frac{7}{2}}})sin^{3}(x)cos^{2}(x) + \frac{3*3sin^{2}(x)cos(x)cos^{2}(x)}{(-cos^{2}(x) + 1)^{\frac{5}{2}}} + \frac{3sin^{3}(x)*-2cos(x)sin(x)}{(-cos^{2}(x) + 1)^{\frac{5}{2}}} - 3(\frac{\frac{-3}{2}(--2cos(x)sin(x) + 0)}{(-cos^{2}(x) + 1)^{\frac{5}{2}}})sin(x)cos^{2}(x) - \frac{3cos(x)cos^{2}(x)}{(-cos^{2}(x) + 1)^{\frac{3}{2}}} - \frac{3sin(x)*-2cos(x)sin(x)}{(-cos^{2}(x) + 1)^{\frac{3}{2}}} + (\frac{\frac{-3}{2}(--2cos(x)sin(x) + 0)}{(-cos^{2}(x) + 1)^{\frac{5}{2}}})sin^{3}(x) + \frac{3sin^{2}(x)cos(x)}{(-cos^{2}(x) + 1)^{\frac{3}{2}}} - (\frac{\frac{-1}{2}(--2cos(x)sin(x) + 0)}{(-cos^{2}(x) + 1)^{\frac{3}{2}}})sin(x) - \frac{cos(x)}{(-cos^{2}(x) + 1)^{\frac{1}{2}}}\\=&\frac{-15sin^{4}(x)cos^{3}(x)}{(-cos^{2}(x) + 1)^{\frac{7}{2}}} + \frac{18sin^{2}(x)cos^{3}(x)}{(-cos^{2}(x) + 1)^{\frac{5}{2}}} - \frac{9sin^{4}(x)cos(x)}{(-cos^{2}(x) + 1)^{\frac{5}{2}}} - \frac{3cos^{3}(x)}{(-cos^{2}(x) + 1)^{\frac{3}{2}}} + \frac{10sin^{2}(x)cos(x)}{(-cos^{2}(x) + 1)^{\frac{3}{2}}} - \frac{cos(x)}{(-cos^{2}(x) + 1)^{\frac{1}{2}}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[3/3]Find\ the\ 4th\ derivative\ of\ function\ arctan(tan(x))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( arctan(tan(x))\right)}{dx}\\=&(\frac{(sec^{2}(x)(1))}{(1 + (tan(x))^{2})})\\=&\frac{sec^{2}(x)}{(tan^{2}(x) + 1)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{sec^{2}(x)}{(tan^{2}(x) + 1)}\right)}{dx}\\=&(\frac{-(2tan(x)sec^{2}(x)(1) + 0)}{(tan^{2}(x) + 1)^{2}})sec^{2}(x) + \frac{2sec^{2}(x)tan(x)}{(tan^{2}(x) + 1)}\\=&\frac{-2tan(x)sec^{4}(x)}{(tan^{2}(x) + 1)^{2}} + \frac{2tan(x)sec^{2}(x)}{(tan^{2}(x) + 1)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2tan(x)sec^{4}(x)}{(tan^{2}(x) + 1)^{2}} + \frac{2tan(x)sec^{2}(x)}{(tan^{2}(x) + 1)}\right)}{dx}\\=&-2(\frac{-2(2tan(x)sec^{2}(x)(1) + 0)}{(tan^{2}(x) + 1)^{3}})tan(x)sec^{4}(x) - \frac{2sec^{2}(x)(1)sec^{4}(x)}{(tan^{2}(x) + 1)^{2}} - \frac{2tan(x)*4sec^{4}(x)tan(x)}{(tan^{2}(x) + 1)^{2}} + 2(\frac{-(2tan(x)sec^{2}(x)(1) + 0)}{(tan^{2}(x) + 1)^{2}})tan(x)sec^{2}(x) + \frac{2sec^{2}(x)(1)sec^{2}(x)}{(tan^{2}(x) + 1)} + \frac{2tan(x)*2sec^{2}(x)tan(x)}{(tan^{2}(x) + 1)}\\=&\frac{8tan^{2}(x)sec^{6}(x)}{(tan^{2}(x) + 1)^{3}} - \frac{2sec^{6}(x)}{(tan^{2}(x) + 1)^{2}} - \frac{12tan^{2}(x)sec^{4}(x)}{(tan^{2}(x) + 1)^{2}} + \frac{2sec^{4}(x)}{(tan^{2}(x) + 1)} + \frac{4tan^{2}(x)sec^{2}(x)}{(tan^{2}(x) + 1)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{8tan^{2}(x)sec^{6}(x)}{(tan^{2}(x) + 1)^{3}} - \frac{2sec^{6}(x)}{(tan^{2}(x) + 1)^{2}} - \frac{12tan^{2}(x)sec^{4}(x)}{(tan^{2}(x) + 1)^{2}} + \frac{2sec^{4}(x)}{(tan^{2}(x) + 1)} + \frac{4tan^{2}(x)sec^{2}(x)}{(tan^{2}(x) + 1)}\right)}{dx}\\=&8(\frac{-3(2tan(x)sec^{2}(x)(1) + 0)}{(tan^{2}(x) + 1)^{4}})tan^{2}(x)sec^{6}(x) + \frac{8*2tan(x)sec^{2}(x)(1)sec^{6}(x)}{(tan^{2}(x) + 1)^{3}} + \frac{8tan^{2}(x)*6sec^{6}(x)tan(x)}{(tan^{2}(x) + 1)^{3}} - 2(\frac{-2(2tan(x)sec^{2}(x)(1) + 0)}{(tan^{2}(x) + 1)^{3}})sec^{6}(x) - \frac{2*6sec^{6}(x)tan(x)}{(tan^{2}(x) + 1)^{2}} - 12(\frac{-2(2tan(x)sec^{2}(x)(1) + 0)}{(tan^{2}(x) + 1)^{3}})tan^{2}(x)sec^{4}(x) - \frac{12*2tan(x)sec^{2}(x)(1)sec^{4}(x)}{(tan^{2}(x) + 1)^{2}} - \frac{12tan^{2}(x)*4sec^{4}(x)tan(x)}{(tan^{2}(x) + 1)^{2}} + 2(\frac{-(2tan(x)sec^{2}(x)(1) + 0)}{(tan^{2}(x) + 1)^{2}})sec^{4}(x) + \frac{2*4sec^{4}(x)tan(x)}{(tan^{2}(x) + 1)} + 4(\frac{-(2tan(x)sec^{2}(x)(1) + 0)}{(tan^{2}(x) + 1)^{2}})tan^{2}(x)sec^{2}(x) + \frac{4*2tan(x)sec^{2}(x)(1)sec^{2}(x)}{(tan^{2}(x) + 1)} + \frac{4tan^{2}(x)*2sec^{2}(x)tan(x)}{(tan^{2}(x) + 1)}\\=&\frac{-48tan^{3}(x)sec^{8}(x)}{(tan^{2}(x) + 1)^{4}} + \frac{24tan(x)sec^{8}(x)}{(tan^{2}(x) + 1)^{3}} + \frac{96tan^{3}(x)sec^{6}(x)}{(tan^{2}(x) + 1)^{3}} - \frac{40tan(x)sec^{6}(x)}{(tan^{2}(x) + 1)^{2}} + \frac{16tan(x)sec^{4}(x)}{(tan^{2}(x) + 1)} - \frac{56tan^{3}(x)sec^{4}(x)}{(tan^{2}(x) + 1)^{2}} + \frac{8tan^{3}(x)sec^{2}(x)}{(tan^{2}(x) + 1)}\\ \end{split}\end{equation} \]