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.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{(\frac{ln(x)}{sin(x)})}{(1 - cos(x))}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{ln(x)}{(-cos(x) + 1)sin(x)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{ln(x)}{(-cos(x) + 1)sin(x)}\right)}{dx}\\=&\frac{(\frac{-(--sin(x) + 0)}{(-cos(x) + 1)^{2}})ln(x)}{sin(x)} + \frac{1}{(-cos(x) + 1)(x)sin(x)} + \frac{ln(x)*-cos(x)}{(-cos(x) + 1)sin^{2}(x)}\\=&\frac{-ln(x)cos(x)}{(-cos(x) + 1)sin^{2}(x)} + \frac{1}{(-cos(x) + 1)xsin(x)} - \frac{ln(x)}{(-cos(x) + 1)^{2}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-ln(x)cos(x)}{(-cos(x) + 1)sin^{2}(x)} + \frac{1}{(-cos(x) + 1)xsin(x)} - \frac{ln(x)}{(-cos(x) + 1)^{2}}\right)}{dx}\\=&\frac{-(\frac{-(--sin(x) + 0)}{(-cos(x) + 1)^{2}})ln(x)cos(x)}{sin^{2}(x)} - \frac{cos(x)}{(-cos(x) + 1)(x)sin^{2}(x)} - \frac{ln(x)*-2cos(x)cos(x)}{(-cos(x) + 1)sin^{3}(x)} - \frac{ln(x)*-sin(x)}{(-cos(x) + 1)sin^{2}(x)} + \frac{(\frac{-(--sin(x) + 0)}{(-cos(x) + 1)^{2}})}{xsin(x)} + \frac{-1}{(-cos(x) + 1)x^{2}sin(x)} + \frac{-cos(x)}{(-cos(x) + 1)xsin^{2}(x)} - (\frac{-2(--sin(x) + 0)}{(-cos(x) + 1)^{3}})ln(x) - \frac{1}{(-cos(x) + 1)^{2}(x)}\\=&\frac{ln(x)cos(x)}{(-cos(x) + 1)^{2}sin(x)} - \frac{2cos(x)}{(-cos(x) + 1)xsin^{2}(x)} + \frac{2ln(x)cos^{2}(x)}{(-cos(x) + 1)sin^{3}(x)} + \frac{ln(x)}{(-cos(x) + 1)sin(x)} - \frac{1}{(-cos(x) + 1)x^{2}sin(x)} - \frac{2}{(-cos(x) + 1)^{2}x} + \frac{2ln(x)sin(x)}{(-cos(x) + 1)^{3}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{ln(x)cos(x)}{(-cos(x) + 1)^{2}sin(x)} - \frac{2cos(x)}{(-cos(x) + 1)xsin^{2}(x)} + \frac{2ln(x)cos^{2}(x)}{(-cos(x) + 1)sin^{3}(x)} + \frac{ln(x)}{(-cos(x) + 1)sin(x)} - \frac{1}{(-cos(x) + 1)x^{2}sin(x)} - \frac{2}{(-cos(x) + 1)^{2}x} + \frac{2ln(x)sin(x)}{(-cos(x) + 1)^{3}}\right)}{dx}\\=&\frac{(\frac{-2(--sin(x) + 0)}{(-cos(x) + 1)^{3}})ln(x)cos(x)}{sin(x)} + \frac{cos(x)}{(-cos(x) + 1)^{2}(x)sin(x)} + \frac{ln(x)*-cos(x)cos(x)}{(-cos(x) + 1)^{2}sin^{2}(x)} + \frac{ln(x)*-sin(x)}{(-cos(x) + 1)^{2}sin(x)} - \frac{2(\frac{-(--sin(x) + 0)}{(-cos(x) + 1)^{2}})cos(x)}{xsin^{2}(x)} - \frac{2*-cos(x)}{(-cos(x) + 1)x^{2}sin^{2}(x)} - \frac{2*-2cos(x)cos(x)}{(-cos(x) + 1)xsin^{3}(x)} - \frac{2*-sin(x)}{(-cos(x) + 1)xsin^{2}(x)} + \frac{2(\frac{-(--sin(x) + 0)}{(-cos(x) + 1)^{2}})ln(x)cos^{2}(x)}{sin^{3}(x)} + \frac{2cos^{2}(x)}{(-cos(x) + 1)(x)sin^{3}(x)} + \frac{2ln(x)*-3cos(x)cos^{2}(x)}{(-cos(x) + 1)sin^{4}(x)} + \frac{2ln(x)*-2cos(x)sin(x)}{(-cos(x) + 1)sin^{3}(x)} + \frac{(\frac{-(--sin(x) + 0)}{(-cos(x) + 1)^{2}})ln(x)}{sin(x)} + \frac{1}{(-cos(x) + 1)(x)sin(x)} + \frac{ln(x)*-cos(x)}{(-cos(x) + 1)sin^{2}(x)} - \frac{(\frac{-(--sin(x) + 0)}{(-cos(x) + 1)^{2}})}{x^{2}sin(x)} - \frac{-2}{(-cos(x) + 1)x^{3}sin(x)} - \frac{-cos(x)}{(-cos(x) + 1)x^{2}sin^{2}(x)} - \frac{2(\frac{-2(--sin(x) + 0)}{(-cos(x) + 1)^{3}})}{x} - \frac{2*-1}{(-cos(x) + 1)^{2}x^{2}} + 2(\frac{-3(--sin(x) + 0)}{(-cos(x) + 1)^{4}})ln(x)sin(x) + \frac{2sin(x)}{(-cos(x) + 1)^{3}(x)} + \frac{2ln(x)cos(x)}{(-cos(x) + 1)^{3}}\\=&\frac{6cos^{2}(x)}{(-cos(x) + 1)xsin^{3}(x)} - \frac{3ln(x)cos^{2}(x)}{(-cos(x) + 1)^{2}sin^{2}(x)} - \frac{5ln(x)cos(x)}{(-cos(x) + 1)sin^{2}(x)} + \frac{3cos(x)}{(-cos(x) + 1)^{2}xsin(x)} + \frac{3cos(x)}{(-cos(x) + 1)x^{2}sin^{2}(x)} + \frac{3}{(-cos(x) + 1)xsin(x)} - \frac{6ln(x)cos^{3}(x)}{(-cos(x) + 1)sin^{4}(x)} - \frac{6ln(x)sin^{2}(x)}{(-cos(x) + 1)^{4}} + \frac{2}{(-cos(x) + 1)x^{3}sin(x)} + \frac{6sin(x)}{(-cos(x) + 1)^{3}x} + \frac{3}{(-cos(x) + 1)^{2}x^{2}} - \frac{2ln(x)}{(-cos(x) + 1)^{2}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{6cos^{2}(x)}{(-cos(x) + 1)xsin^{3}(x)} - \frac{3ln(x)cos^{2}(x)}{(-cos(x) + 1)^{2}sin^{2}(x)} - \frac{5ln(x)cos(x)}{(-cos(x) + 1)sin^{2}(x)} + \frac{3cos(x)}{(-cos(x) + 1)^{2}xsin(x)} + \frac{3cos(x)}{(-cos(x) + 1)x^{2}sin^{2}(x)} + \frac{3}{(-cos(x) + 1)xsin(x)} - \frac{6ln(x)cos^{3}(x)}{(-cos(x) + 1)sin^{4}(x)} - \frac{6ln(x)sin^{2}(x)}{(-cos(x) + 1)^{4}} + \frac{2}{(-cos(x) + 1)x^{3}sin(x)} + \frac{6sin(x)}{(-cos(x) + 1)^{3}x} + \frac{3}{(-cos(x) + 1)^{2}x^{2}} - \frac{2ln(x)}{(-cos(x) + 1)^{2}}\right)}{dx}\\=&\frac{6(\frac{-(--sin(x) + 0)}{(-cos(x) + 1)^{2}})cos^{2}(x)}{xsin^{3}(x)} + \frac{6*-cos^{2}(x)}{(-cos(x) + 1)x^{2}sin^{3}(x)} + \frac{6*-3cos(x)cos^{2}(x)}{(-cos(x) + 1)xsin^{4}(x)} + \frac{6*-2cos(x)sin(x)}{(-cos(x) + 1)xsin^{3}(x)} - \frac{3(\frac{-2(--sin(x) + 0)}{(-cos(x) + 1)^{3}})ln(x)cos^{2}(x)}{sin^{2}(x)} - \frac{3cos^{2}(x)}{(-cos(x) + 1)^{2}(x)sin^{2}(x)} - \frac{3ln(x)*-2cos(x)cos^{2}(x)}{(-cos(x) + 1)^{2}sin^{3}(x)} - \frac{3ln(x)*-2cos(x)sin(x)}{(-cos(x) + 1)^{2}sin^{2}(x)} - \frac{5(\frac{-(--sin(x) + 0)}{(-cos(x) + 1)^{2}})ln(x)cos(x)}{sin^{2}(x)} - \frac{5cos(x)}{(-cos(x) + 1)(x)sin^{2}(x)} - \frac{5ln(x)*-2cos(x)cos(x)}{(-cos(x) + 1)sin^{3}(x)} - \frac{5ln(x)*-sin(x)}{(-cos(x) + 1)sin^{2}(x)} + \frac{3(\frac{-2(--sin(x) + 0)}{(-cos(x) + 1)^{3}})cos(x)}{xsin(x)} + \frac{3*-cos(x)}{(-cos(x) + 1)^{2}x^{2}sin(x)} + \frac{3*-cos(x)cos(x)}{(-cos(x) + 1)^{2}xsin^{2}(x)} + \frac{3*-sin(x)}{(-cos(x) + 1)^{2}xsin(x)} + \frac{3(\frac{-(--sin(x) + 0)}{(-cos(x) + 1)^{2}})cos(x)}{x^{2}sin^{2}(x)} + \frac{3*-2cos(x)}{(-cos(x) + 1)x^{3}sin^{2}(x)} + \frac{3*-2cos(x)cos(x)}{(-cos(x) + 1)x^{2}sin^{3}(x)} + \frac{3*-sin(x)}{(-cos(x) + 1)x^{2}sin^{2}(x)} + \frac{3(\frac{-(--sin(x) + 0)}{(-cos(x) + 1)^{2}})}{xsin(x)} + \frac{3*-1}{(-cos(x) + 1)x^{2}sin(x)} + \frac{3*-cos(x)}{(-cos(x) + 1)xsin^{2}(x)} - \frac{6(\frac{-(--sin(x) + 0)}{(-cos(x) + 1)^{2}})ln(x)cos^{3}(x)}{sin^{4}(x)} - \frac{6cos^{3}(x)}{(-cos(x) + 1)(x)sin^{4}(x)} - \frac{6ln(x)*-4cos(x)cos^{3}(x)}{(-cos(x) + 1)sin^{5}(x)} - \frac{6ln(x)*-3cos^{2}(x)sin(x)}{(-cos(x) + 1)sin^{4}(x)} - 6(\frac{-4(--sin(x) + 0)}{(-cos(x) + 1)^{5}})ln(x)sin^{2}(x) - \frac{6sin^{2}(x)}{(-cos(x) + 1)^{4}(x)} - \frac{6ln(x)*2sin(x)cos(x)}{(-cos(x) + 1)^{4}} + \frac{2(\frac{-(--sin(x) + 0)}{(-cos(x) + 1)^{2}})}{x^{3}sin(x)} + \frac{2*-3}{(-cos(x) + 1)x^{4}sin(x)} + \frac{2*-cos(x)}{(-cos(x) + 1)x^{3}sin^{2}(x)} + \frac{6(\frac{-3(--sin(x) + 0)}{(-cos(x) + 1)^{4}})sin(x)}{x} + \frac{6*-sin(x)}{(-cos(x) + 1)^{3}x^{2}} + \frac{6cos(x)}{(-cos(x) + 1)^{3}x} + \frac{3(\frac{-2(--sin(x) + 0)}{(-cos(x) + 1)^{3}})}{x^{2}} + \frac{3*-2}{(-cos(x) + 1)^{2}x^{3}} - 2(\frac{-2(--sin(x) + 0)}{(-cos(x) + 1)^{3}})ln(x) - \frac{2}{(-cos(x) + 1)^{2}(x)}\\=& - \frac{12cos^{2}(x)}{(-cos(x) + 1)^{2}xsin^{2}(x)} - \frac{12cos^{2}(x)}{(-cos(x) + 1)x^{2}sin^{3}(x)} - \frac{24cos^{3}(x)}{(-cos(x) + 1)xsin^{4}(x)} - \frac{20cos(x)}{(-cos(x) + 1)xsin^{2}(x)} + \frac{6ln(x)cos^{2}(x)}{(-cos(x) + 1)^{3}sin(x)} + \frac{12ln(x)cos^{3}(x)}{(-cos(x) + 1)^{2}sin^{3}(x)} + \frac{11ln(x)cos(x)}{(-cos(x) + 1)^{2}sin(x)} + \frac{28ln(x)cos^{2}(x)}{(-cos(x) + 1)sin^{3}(x)} - \frac{6cos(x)}{(-cos(x) + 1)^{2}x^{2}sin(x)} - \frac{8cos(x)}{(-cos(x) + 1)x^{3}sin^{2}(x)} - \frac{6}{(-cos(x) + 1)x^{2}sin(x)} - \frac{6}{(-cos(x) + 1)x^{4}sin(x)} + \frac{24ln(x)cos^{4}(x)}{(-cos(x) + 1)sin^{5}(x)} - \frac{12ln(x)sin(x)cos(x)}{(-cos(x) + 1)^{4}} + \frac{24ln(x)sin^{3}(x)}{(-cos(x) + 1)^{5}} + \frac{5ln(x)}{(-cos(x) + 1)sin(x)} - \frac{24sin^{2}(x)}{(-cos(x) + 1)^{4}x} - \frac{12sin(x)}{(-cos(x) + 1)^{3}x^{2}} - \frac{8}{(-cos(x) + 1)^{2}x^{3}} - \frac{8}{(-cos(x) + 1)^{2}x} + \frac{4ln(x)sin(x)}{(-cos(x) + 1)^{3}}\\ \end{split}\end{equation} \]