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 2 questions in this calculation: for each question, the 4 derivative of x is calculated.
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/2]Find\ the\ 4th\ derivative\ of\ function\ arctan(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( arctan(x)\right)}{dx}\\=&(\frac{(1)}{(1 + (x)^{2})})\\=&\frac{1}{(x^{2} + 1)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{1}{(x^{2} + 1)}\right)}{dx}\\=&(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})\\=&\frac{-2x}{(x^{2} + 1)^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2x}{(x^{2} + 1)^{2}}\right)}{dx}\\=&-2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x - \frac{2}{(x^{2} + 1)^{2}}\\=&\frac{8x^{2}}{(x^{2} + 1)^{3}} - \frac{2}{(x^{2} + 1)^{2}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{8x^{2}}{(x^{2} + 1)^{3}} - \frac{2}{(x^{2} + 1)^{2}}\right)}{dx}\\=&8(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{2} + \frac{8*2x}{(x^{2} + 1)^{3}} - 2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})\\=&\frac{-48x^{3}}{(x^{2} + 1)^{4}} + \frac{24x}{(x^{2} + 1)^{3}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/2]Find\ the\ 4th\ derivative\ of\ function\ arcsin(x)arccos(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( arcsin(x)arccos(x)\right)}{dx}\\=&(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})arccos(x) + arcsin(x)(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})\\=&\frac{arccos(x)}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{arcsin(x)}{(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{arccos(x)}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{arcsin(x)}{(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})arccos(x) + \frac{(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}} - (\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})arcsin(x) - \frac{(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{xarccos(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{xarcsin(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{xarccos(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{xarcsin(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})xarccos(x) + \frac{arccos(x)}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{x(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{(-x^{2} + 1)^{\frac{1}{2}}} - (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})xarcsin(x) - \frac{arcsin(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{x(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{3x^{2}arccos(x)}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{arccos(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{3x^{2}arcsin(x)}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{4x}{(-x^{2} + 1)^{2}} - \frac{arcsin(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{x}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{3x^{2}arccos(x)}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{arccos(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{3x^{2}arcsin(x)}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{4x}{(-x^{2} + 1)^{2}} - \frac{arcsin(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{x}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2}arccos(x) + \frac{3*2xarccos(x)}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{3x^{2}(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{5}{2}}} + (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})arccos(x) + \frac{(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})x}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})x}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}} - 3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2}arcsin(x) - \frac{3*2xarcsin(x)}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{3x^{2}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{5}{2}}} - 4(\frac{-2(-2x + 0)}{(-x^{2} + 1)^{3}})x - \frac{4}{(-x^{2} + 1)^{2}} - (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})arcsin(x) - \frac{(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})x}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})x}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}}\\=&\frac{15x^{3}arccos(x)}{(-x^{2} + 1)^{\frac{7}{2}}} + \frac{9xarccos(x)}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{15x^{3}arcsin(x)}{(-x^{2} + 1)^{\frac{7}{2}}} - \frac{3x^{2}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{5}{2}}} - \frac{9xarcsin(x)}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{24x^{2}}{(-x^{2} + 1)^{3}} - \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} - \frac{6}{(-x^{2} + 1)^{2}}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/2]Find\ the\ 4th\ derivative\ of\ function\ arctan(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( arctan(x)\right)}{dx}\\=&(\frac{(1)}{(1 + (x)^{2})})\\=&\frac{1}{(x^{2} + 1)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{1}{(x^{2} + 1)}\right)}{dx}\\=&(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})\\=&\frac{-2x}{(x^{2} + 1)^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2x}{(x^{2} + 1)^{2}}\right)}{dx}\\=&-2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x - \frac{2}{(x^{2} + 1)^{2}}\\=&\frac{8x^{2}}{(x^{2} + 1)^{3}} - \frac{2}{(x^{2} + 1)^{2}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{8x^{2}}{(x^{2} + 1)^{3}} - \frac{2}{(x^{2} + 1)^{2}}\right)}{dx}\\=&8(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{2} + \frac{8*2x}{(x^{2} + 1)^{3}} - 2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})\\=&\frac{-48x^{3}}{(x^{2} + 1)^{4}} + \frac{24x}{(x^{2} + 1)^{3}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/2]Find\ the\ 4th\ derivative\ of\ function\ arcsin(x)arccos(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( arcsin(x)arccos(x)\right)}{dx}\\=&(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})arccos(x) + arcsin(x)(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})\\=&\frac{arccos(x)}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{arcsin(x)}{(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{arccos(x)}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{arcsin(x)}{(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})arccos(x) + \frac{(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}} - (\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})arcsin(x) - \frac{(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{xarccos(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{xarcsin(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{xarccos(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{xarcsin(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})xarccos(x) + \frac{arccos(x)}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{x(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{(-x^{2} + 1)^{\frac{1}{2}}} - (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})xarcsin(x) - \frac{arcsin(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{x(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{3x^{2}arccos(x)}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{arccos(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{3x^{2}arcsin(x)}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{4x}{(-x^{2} + 1)^{2}} - \frac{arcsin(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{x}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{3x^{2}arccos(x)}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{arccos(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{3x^{2}arcsin(x)}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{4x}{(-x^{2} + 1)^{2}} - \frac{arcsin(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{x}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2}arccos(x) + \frac{3*2xarccos(x)}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{3x^{2}(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{5}{2}}} + (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})arccos(x) + \frac{(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})x}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})x}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}} - 3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2}arcsin(x) - \frac{3*2xarcsin(x)}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{3x^{2}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{5}{2}}} - 4(\frac{-2(-2x + 0)}{(-x^{2} + 1)^{3}})x - \frac{4}{(-x^{2} + 1)^{2}} - (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})arcsin(x) - \frac{(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})x}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})x}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}}\\=&\frac{15x^{3}arccos(x)}{(-x^{2} + 1)^{\frac{7}{2}}} + \frac{9xarccos(x)}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{15x^{3}arcsin(x)}{(-x^{2} + 1)^{\frac{7}{2}}} - \frac{3x^{2}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{5}{2}}} - \frac{9xarcsin(x)}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{24x^{2}}{(-x^{2} + 1)^{3}} - \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} - \frac{6}{(-x^{2} + 1)^{2}}\\ \end{split}\end{equation} \]