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\ sin(arctan(cot(x)) + 5)\ with\ respect\ to\ x：\\\end{split}\end{equation} \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( sin(arctan(cot(x)) + 5)\right)}{dx}\\=&cos(arctan(cot(x)) + 5)((\frac{(-csc^{2}(x))}{(1 + (cot(x))^{2})}) + 0)\\=&\frac{-cos(arctan(cot(x)) + 5)csc^{2}(x)}{(cot^{2}(x) + 1)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-cos(arctan(cot(x)) + 5)csc^{2}(x)}{(cot^{2}(x) + 1)}\right)}{dx}\\=&-(\frac{-(-2cot(x)csc^{2}(x) + 0)}{(cot^{2}(x) + 1)^{2}})cos(arctan(cot(x)) + 5)csc^{2}(x) - \frac{-sin(arctan(cot(x)) + 5)((\frac{(-csc^{2}(x))}{(1 + (cot(x))^{2})}) + 0)csc^{2}(x)}{(cot^{2}(x) + 1)} - \frac{cos(arctan(cot(x)) + 5)*-2csc^{2}(x)cot(x)}{(cot^{2}(x) + 1)}\\=&\frac{-2cos(arctan(cot(x)) + 5)cot(x)csc^{4}(x)}{(cot^{2}(x) + 1)^{2}} - \frac{sin(arctan(cot(x)) + 5)csc^{4}(x)}{(cot^{2}(x) + 1)^{2}} + \frac{2cos(arctan(cot(x)) + 5)cot(x)csc^{2}(x)}{(cot^{2}(x) + 1)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2cos(arctan(cot(x)) + 5)cot(x)csc^{4}(x)}{(cot^{2}(x) + 1)^{2}} - \frac{sin(arctan(cot(x)) + 5)csc^{4}(x)}{(cot^{2}(x) + 1)^{2}} + \frac{2cos(arctan(cot(x)) + 5)cot(x)csc^{2}(x)}{(cot^{2}(x) + 1)}\right)}{dx}\\=&-2(\frac{-2(-2cot(x)csc^{2}(x) + 0)}{(cot^{2}(x) + 1)^{3}})cos(arctan(cot(x)) + 5)cot(x)csc^{4}(x) - \frac{2*-sin(arctan(cot(x)) + 5)((\frac{(-csc^{2}(x))}{(1 + (cot(x))^{2})}) + 0)cot(x)csc^{4}(x)}{(cot^{2}(x) + 1)^{2}} - \frac{2cos(arctan(cot(x)) + 5)*-csc^{2}(x)csc^{4}(x)}{(cot^{2}(x) + 1)^{2}} - \frac{2cos(arctan(cot(x)) + 5)cot(x)*-4csc^{4}(x)cot(x)}{(cot^{2}(x) + 1)^{2}} - (\frac{-2(-2cot(x)csc^{2}(x) + 0)}{(cot^{2}(x) + 1)^{3}})sin(arctan(cot(x)) + 5)csc^{4}(x) - \frac{cos(arctan(cot(x)) + 5)((\frac{(-csc^{2}(x))}{(1 + (cot(x))^{2})}) + 0)csc^{4}(x)}{(cot^{2}(x) + 1)^{2}} - \frac{sin(arctan(cot(x)) + 5)*-4csc^{4}(x)cot(x)}{(cot^{2}(x) + 1)^{2}} + 2(\frac{-(-2cot(x)csc^{2}(x) + 0)}{(cot^{2}(x) + 1)^{2}})cos(arctan(cot(x)) + 5)cot(x)csc^{2}(x) + \frac{2*-sin(arctan(cot(x)) + 5)((\frac{(-csc^{2}(x))}{(1 + (cot(x))^{2})}) + 0)cot(x)csc^{2}(x)}{(cot^{2}(x) + 1)} + \frac{2cos(arctan(cot(x)) + 5)*-csc^{2}(x)csc^{2}(x)}{(cot^{2}(x) + 1)} + \frac{2cos(arctan(cot(x)) + 5)cot(x)*-2csc^{2}(x)cot(x)}{(cot^{2}(x) + 1)}\\=&\frac{-8cos(arctan(cot(x)) + 5)cot^{2}(x)csc^{6}(x)}{(cot^{2}(x) + 1)^{3}} - \frac{6sin(arctan(cot(x)) + 5)cot(x)csc^{6}(x)}{(cot^{2}(x) + 1)^{3}} + \frac{2cos(arctan(cot(x)) + 5)csc^{6}(x)}{(cot^{2}(x) + 1)^{2}} + \frac{12cos(arctan(cot(x)) + 5)cot^{2}(x)csc^{4}(x)}{(cot^{2}(x) + 1)^{2}} + \frac{cos(arctan(cot(x)) + 5)csc^{6}(x)}{(cot^{2}(x) + 1)^{3}} + \frac{6sin(arctan(cot(x)) + 5)cot(x)csc^{4}(x)}{(cot^{2}(x) + 1)^{2}} - \frac{2cos(arctan(cot(x)) + 5)csc^{4}(x)}{(cot^{2}(x) + 1)} - \frac{4cos(arctan(cot(x)) + 5)cot^{2}(x)csc^{2}(x)}{(cot^{2}(x) + 1)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-8cos(arctan(cot(x)) + 5)cot^{2}(x)csc^{6}(x)}{(cot^{2}(x) + 1)^{3}} - \frac{6sin(arctan(cot(x)) + 5)cot(x)csc^{6}(x)}{(cot^{2}(x) + 1)^{3}} + \frac{2cos(arctan(cot(x)) + 5)csc^{6}(x)}{(cot^{2}(x) + 1)^{2}} + \frac{12cos(arctan(cot(x)) + 5)cot^{2}(x)csc^{4}(x)}{(cot^{2}(x) + 1)^{2}} + \frac{cos(arctan(cot(x)) + 5)csc^{6}(x)}{(cot^{2}(x) + 1)^{3}} + \frac{6sin(arctan(cot(x)) + 5)cot(x)csc^{4}(x)}{(cot^{2}(x) + 1)^{2}} - \frac{2cos(arctan(cot(x)) + 5)csc^{4}(x)}{(cot^{2}(x) + 1)} - \frac{4cos(arctan(cot(x)) + 5)cot^{2}(x)csc^{2}(x)}{(cot^{2}(x) + 1)}\right)}{dx}\\=&-8(\frac{-3(-2cot(x)csc^{2}(x) + 0)}{(cot^{2}(x) + 1)^{4}})cos(arctan(cot(x)) + 5)cot^{2}(x)csc^{6}(x) - \frac{8*-sin(arctan(cot(x)) + 5)((\frac{(-csc^{2}(x))}{(1 + (cot(x))^{2})}) + 0)cot^{2}(x)csc^{6}(x)}{(cot^{2}(x) + 1)^{3}} - \frac{8cos(arctan(cot(x)) + 5)*-2cot(x)csc^{2}(x)csc^{6}(x)}{(cot^{2}(x) + 1)^{3}} - \frac{8cos(arctan(cot(x)) + 5)cot^{2}(x)*-6csc^{6}(x)cot(x)}{(cot^{2}(x) + 1)^{3}} - 6(\frac{-3(-2cot(x)csc^{2}(x) + 0)}{(cot^{2}(x) + 1)^{4}})sin(arctan(cot(x)) + 5)cot(x)csc^{6}(x) - \frac{6cos(arctan(cot(x)) + 5)((\frac{(-csc^{2}(x))}{(1 + (cot(x))^{2})}) + 0)cot(x)csc^{6}(x)}{(cot^{2}(x) + 1)^{3}} - \frac{6sin(arctan(cot(x)) + 5)*-csc^{2}(x)csc^{6}(x)}{(cot^{2}(x) + 1)^{3}} - \frac{6sin(arctan(cot(x)) + 5)cot(x)*-6csc^{6}(x)cot(x)}{(cot^{2}(x) + 1)^{3}} + 2(\frac{-2(-2cot(x)csc^{2}(x) + 0)}{(cot^{2}(x) + 1)^{3}})cos(arctan(cot(x)) + 5)csc^{6}(x) + \frac{2*-sin(arctan(cot(x)) + 5)((\frac{(-csc^{2}(x))}{(1 + (cot(x))^{2})}) + 0)csc^{6}(x)}{(cot^{2}(x) + 1)^{2}} + \frac{2cos(arctan(cot(x)) + 5)*-6csc^{6}(x)cot(x)}{(cot^{2}(x) + 1)^{2}} + 12(\frac{-2(-2cot(x)csc^{2}(x) + 0)}{(cot^{2}(x) + 1)^{3}})cos(arctan(cot(x)) + 5)cot^{2}(x)csc^{4}(x) + \frac{12*-sin(arctan(cot(x)) + 5)((\frac{(-csc^{2}(x))}{(1 + (cot(x))^{2})}) + 0)cot^{2}(x)csc^{4}(x)}{(cot^{2}(x) + 1)^{2}} + \frac{12cos(arctan(cot(x)) + 5)*-2cot(x)csc^{2}(x)csc^{4}(x)}{(cot^{2}(x) + 1)^{2}} + \frac{12cos(arctan(cot(x)) + 5)cot^{2}(x)*-4csc^{4}(x)cot(x)}{(cot^{2}(x) + 1)^{2}} + (\frac{-3(-2cot(x)csc^{2}(x) + 0)}{(cot^{2}(x) + 1)^{4}})cos(arctan(cot(x)) + 5)csc^{6}(x) + \frac{-sin(arctan(cot(x)) + 5)((\frac{(-csc^{2}(x))}{(1 + (cot(x))^{2})}) + 0)csc^{6}(x)}{(cot^{2}(x) + 1)^{3}} + \frac{cos(arctan(cot(x)) + 5)*-6csc^{6}(x)cot(x)}{(cot^{2}(x) + 1)^{3}} + 6(\frac{-2(-2cot(x)csc^{2}(x) + 0)}{(cot^{2}(x) + 1)^{3}})sin(arctan(cot(x)) + 5)cot(x)csc^{4}(x) + \frac{6cos(arctan(cot(x)) + 5)((\frac{(-csc^{2}(x))}{(1 + (cot(x))^{2})}) + 0)cot(x)csc^{4}(x)}{(cot^{2}(x) + 1)^{2}} + \frac{6sin(arctan(cot(x)) + 5)*-csc^{2}(x)csc^{4}(x)}{(cot^{2}(x) + 1)^{2}} + \frac{6sin(arctan(cot(x)) + 5)cot(x)*-4csc^{4}(x)cot(x)}{(cot^{2}(x) + 1)^{2}} - 2(\frac{-(-2cot(x)csc^{2}(x) + 0)}{(cot^{2}(x) + 1)^{2}})cos(arctan(cot(x)) + 5)csc^{4}(x) - \frac{2*-sin(arctan(cot(x)) + 5)((\frac{(-csc^{2}(x))}{(1 + (cot(x))^{2})}) + 0)csc^{4}(x)}{(cot^{2}(x) + 1)} - \frac{2cos(arctan(cot(x)) + 5)*-4csc^{4}(x)cot(x)}{(cot^{2}(x) + 1)} - 4(\frac{-(-2cot(x)csc^{2}(x) + 0)}{(cot^{2}(x) + 1)^{2}})cos(arctan(cot(x)) + 5)cot^{2}(x)csc^{2}(x) - \frac{4*-sin(arctan(cot(x)) + 5)((\frac{(-csc^{2}(x))}{(1 + (cot(x))^{2})}) + 0)cot^{2}(x)csc^{2}(x)}{(cot^{2}(x) + 1)} - \frac{4cos(arctan(cot(x)) + 5)*-2cot(x)csc^{2}(x)csc^{2}(x)}{(cot^{2}(x) + 1)} - \frac{4cos(arctan(cot(x)) + 5)cot^{2}(x)*-2csc^{2}(x)cot(x)}{(cot^{2}(x) + 1)}\\=&\frac{-48cos(arctan(cot(x)) + 5)cot^{3}(x)csc^{8}(x)}{(cot^{2}(x) + 1)^{4}} - \frac{44sin(arctan(cot(x)) + 5)cot^{2}(x)csc^{8}(x)}{(cot^{2}(x) + 1)^{4}} + \frac{24cos(arctan(cot(x)) + 5)cot(x)csc^{8}(x)}{(cot^{2}(x) + 1)^{3}} + \frac{96cos(arctan(cot(x)) + 5)cot^{3}(x)csc^{6}(x)}{(cot^{2}(x) + 1)^{3}} + \frac{12cos(arctan(cot(x)) + 5)cot(x)csc^{8}(x)}{(cot^{2}(x) + 1)^{4}} + \frac{8sin(arctan(cot(x)) + 5)csc^{8}(x)}{(cot^{2}(x) + 1)^{3}} + \frac{72sin(arctan(cot(x)) + 5)cot^{2}(x)csc^{6}(x)}{(cot^{2}(x) + 1)^{3}} - \frac{40cos(arctan(cot(x)) + 5)cot(x)csc^{6}(x)}{(cot^{2}(x) + 1)^{2}} + \frac{16cos(arctan(cot(x)) + 5)cot(x)csc^{4}(x)}{(cot^{2}(x) + 1)} - \frac{56cos(arctan(cot(x)) + 5)cot^{3}(x)csc^{4}(x)}{(cot^{2}(x) + 1)^{2}} + \frac{sin(arctan(cot(x)) + 5)csc^{8}(x)}{(cot^{2}(x) + 1)^{4}} - \frac{12cos(arctan(cot(x)) + 5)cot(x)csc^{6}(x)}{(cot^{2}(x) + 1)^{3}} - \frac{8sin(arctan(cot(x)) + 5)csc^{6}(x)}{(cot^{2}(x) + 1)^{2}} - \frac{28sin(arctan(cot(x)) + 5)cot^{2}(x)csc^{4}(x)}{(cot^{2}(x) + 1)^{2}} + \frac{8cos(arctan(cot(x)) + 5)cot^{3}(x)csc^{2}(x)}{(cot^{2}(x) + 1)}\\ \end{split}\end{equation}