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.
\[ \begin{equation}\begin{split}[1/3]Find\ the\ 4th\ derivative\ of\ function\ e^{arcsin(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( e^{arcsin(x)}\right)}{dx}\\=&e^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})\\=&\frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})e^{arcsin(x)} + \frac{e^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{e^{arcsin(x)}}{(-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}}})xe^{arcsin(x)} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{xe^{arcsin(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}}})e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{e^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{3x^{2}e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} + \frac{2xe^{arcsin(x)}}{(-x^{2} + 1)^{2}} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{3x^{2}e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} + \frac{2xe^{arcsin(x)}}{(-x^{2} + 1)^{2}} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2}e^{arcsin(x)} + \frac{3*2xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{3x^{2}e^{arcsin(x)}(\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}}})e^{arcsin(x)} + \frac{e^{arcsin(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}}})xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} + \frac{xe^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} + 2(\frac{-2(-2x + 0)}{(-x^{2} + 1)^{3}})xe^{arcsin(x)} + \frac{2e^{arcsin(x)}}{(-x^{2} + 1)^{2}} + \frac{2xe^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{2}} + \frac{(\frac{-(-2x + 0)}{(-x^{2} + 1)^{2}})e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})e^{arcsin(x)}}{(-x^{2} + 1)} + \frac{e^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{15x^{3}e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{7}{2}}} + \frac{12xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{3x^{2}e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{5}{2}}} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} + \frac{12x^{2}e^{arcsin(x)}}{(-x^{2} + 1)^{3}} + \frac{3e^{arcsin(x)}}{(-x^{2} + 1)^{2}} + \frac{xe^{arcsin(x)}}{(-x^{2} + 1)^{2}(-x^{2} + 1)^{\frac{1}{2}}} + \frac{2xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{2}} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/3]Find\ the\ 4th\ derivative\ of\ function\ e^{arccos(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( e^{arccos(x)}\right)}{dx}\\=&e^{arccos(x)}(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})\\=&\frac{-e^{arccos(x)}}{(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-e^{arccos(x)}}{(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&-(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})e^{arccos(x)} - \frac{e^{arccos(x)}(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{-xe^{arccos(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{e^{arccos(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-xe^{arccos(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{e^{arccos(x)}}{(-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}}})xe^{arccos(x)} - \frac{e^{arccos(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{xe^{arccos(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}}})e^{arccos(x)}}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})e^{arccos(x)}}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{e^{arccos(x)}(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{-3x^{2}e^{arccos(x)}}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{e^{arccos(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{xe^{arccos(x)}}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}} + \frac{2xe^{arccos(x)}}{(-x^{2} + 1)^{2}} - \frac{e^{arccos(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-3x^{2}e^{arccos(x)}}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{e^{arccos(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{xe^{arccos(x)}}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}} + \frac{2xe^{arccos(x)}}{(-x^{2} + 1)^{2}} - \frac{e^{arccos(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)}\right)}{dx}\\=&-3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2}e^{arccos(x)} - \frac{3*2xe^{arccos(x)}}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{3x^{2}e^{arccos(x)}(\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}}})e^{arccos(x)} - \frac{e^{arccos(x)}(\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}}})xe^{arccos(x)}}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})xe^{arccos(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{e^{arccos(x)}}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}} + \frac{xe^{arccos(x)}(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}} + 2(\frac{-2(-2x + 0)}{(-x^{2} + 1)^{3}})xe^{arccos(x)} + \frac{2e^{arccos(x)}}{(-x^{2} + 1)^{2}} + \frac{2xe^{arccos(x)}(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{2}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})e^{arccos(x)}}{(-x^{2} + 1)} - \frac{(\frac{-(-2x + 0)}{(-x^{2} + 1)^{2}})e^{arccos(x)}}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{e^{arccos(x)}(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)}\\=&\frac{-15x^{3}e^{arccos(x)}}{(-x^{2} + 1)^{\frac{7}{2}}} - \frac{12xe^{arccos(x)}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{3x^{2}e^{arccos(x)}}{(-x^{2} + 1)^{\frac{5}{2}}(-x^{2} + 1)^{\frac{1}{2}}} + \frac{e^{arccos(x)}}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}} + \frac{12x^{2}e^{arccos(x)}}{(-x^{2} + 1)^{3}} + \frac{3e^{arccos(x)}}{(-x^{2} + 1)^{2}} - \frac{xe^{arccos(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{2}} - \frac{2xe^{arccos(x)}}{(-x^{2} + 1)^{2}(-x^{2} + 1)^{\frac{1}{2}}} + \frac{e^{arccos(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[3/3]Find\ the\ 4th\ derivative\ of\ function\ e^{arctan(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( e^{arctan(x)}\right)}{dx}\\=&e^{arctan(x)}(\frac{(1)}{(1 + (x)^{2})})\\=&\frac{e^{arctan(x)}}{(x^{2} + 1)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{e^{arctan(x)}}{(x^{2} + 1)}\right)}{dx}\\=&(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})e^{arctan(x)} + \frac{e^{arctan(x)}(\frac{(1)}{(1 + (x)^{2})})}{(x^{2} + 1)}\\=&\frac{-2xe^{arctan(x)}}{(x^{2} + 1)^{2}} + \frac{e^{arctan(x)}}{(x^{2} + 1)^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2xe^{arctan(x)}}{(x^{2} + 1)^{2}} + \frac{e^{arctan(x)}}{(x^{2} + 1)^{2}}\right)}{dx}\\=&-2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})xe^{arctan(x)} - \frac{2e^{arctan(x)}}{(x^{2} + 1)^{2}} - \frac{2xe^{arctan(x)}(\frac{(1)}{(1 + (x)^{2})})}{(x^{2} + 1)^{2}} + (\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})e^{arctan(x)} + \frac{e^{arctan(x)}(\frac{(1)}{(1 + (x)^{2})})}{(x^{2} + 1)^{2}}\\=&\frac{8x^{2}e^{arctan(x)}}{(x^{2} + 1)^{3}} - \frac{2e^{arctan(x)}}{(x^{2} + 1)^{2}} - \frac{6xe^{arctan(x)}}{(x^{2} + 1)^{3}} + \frac{e^{arctan(x)}}{(x^{2} + 1)^{3}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{8x^{2}e^{arctan(x)}}{(x^{2} + 1)^{3}} - \frac{2e^{arctan(x)}}{(x^{2} + 1)^{2}} - \frac{6xe^{arctan(x)}}{(x^{2} + 1)^{3}} + \frac{e^{arctan(x)}}{(x^{2} + 1)^{3}}\right)}{dx}\\=&8(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{2}e^{arctan(x)} + \frac{8*2xe^{arctan(x)}}{(x^{2} + 1)^{3}} + \frac{8x^{2}e^{arctan(x)}(\frac{(1)}{(1 + (x)^{2})})}{(x^{2} + 1)^{3}} - 2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})e^{arctan(x)} - \frac{2e^{arctan(x)}(\frac{(1)}{(1 + (x)^{2})})}{(x^{2} + 1)^{2}} - 6(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})xe^{arctan(x)} - \frac{6e^{arctan(x)}}{(x^{2} + 1)^{3}} - \frac{6xe^{arctan(x)}(\frac{(1)}{(1 + (x)^{2})})}{(x^{2} + 1)^{3}} + (\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})e^{arctan(x)} + \frac{e^{arctan(x)}(\frac{(1)}{(1 + (x)^{2})})}{(x^{2} + 1)^{3}}\\=&\frac{-48x^{3}e^{arctan(x)}}{(x^{2} + 1)^{4}} + \frac{24xe^{arctan(x)}}{(x^{2} + 1)^{3}} + \frac{44x^{2}e^{arctan(x)}}{(x^{2} + 1)^{4}} - \frac{8e^{arctan(x)}}{(x^{2} + 1)^{3}} - \frac{12xe^{arctan(x)}}{(x^{2} + 1)^{4}} + \frac{e^{arctan(x)}}{(x^{2} + 1)^{4}}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/3]Find\ the\ 4th\ derivative\ of\ function\ e^{arcsin(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( e^{arcsin(x)}\right)}{dx}\\=&e^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})\\=&\frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})e^{arcsin(x)} + \frac{e^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{e^{arcsin(x)}}{(-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}}})xe^{arcsin(x)} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{xe^{arcsin(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}}})e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{e^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{3x^{2}e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} + \frac{2xe^{arcsin(x)}}{(-x^{2} + 1)^{2}} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{3x^{2}e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} + \frac{2xe^{arcsin(x)}}{(-x^{2} + 1)^{2}} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2}e^{arcsin(x)} + \frac{3*2xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{3x^{2}e^{arcsin(x)}(\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}}})e^{arcsin(x)} + \frac{e^{arcsin(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}}})xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} + \frac{xe^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} + 2(\frac{-2(-2x + 0)}{(-x^{2} + 1)^{3}})xe^{arcsin(x)} + \frac{2e^{arcsin(x)}}{(-x^{2} + 1)^{2}} + \frac{2xe^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{2}} + \frac{(\frac{-(-2x + 0)}{(-x^{2} + 1)^{2}})e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})e^{arcsin(x)}}{(-x^{2} + 1)} + \frac{e^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{15x^{3}e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{7}{2}}} + \frac{12xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{3x^{2}e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{5}{2}}} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} + \frac{12x^{2}e^{arcsin(x)}}{(-x^{2} + 1)^{3}} + \frac{3e^{arcsin(x)}}{(-x^{2} + 1)^{2}} + \frac{xe^{arcsin(x)}}{(-x^{2} + 1)^{2}(-x^{2} + 1)^{\frac{1}{2}}} + \frac{2xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{2}} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/3]Find\ the\ 4th\ derivative\ of\ function\ e^{arccos(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( e^{arccos(x)}\right)}{dx}\\=&e^{arccos(x)}(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})\\=&\frac{-e^{arccos(x)}}{(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-e^{arccos(x)}}{(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&-(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})e^{arccos(x)} - \frac{e^{arccos(x)}(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{-xe^{arccos(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{e^{arccos(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-xe^{arccos(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{e^{arccos(x)}}{(-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}}})xe^{arccos(x)} - \frac{e^{arccos(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{xe^{arccos(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}}})e^{arccos(x)}}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})e^{arccos(x)}}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{e^{arccos(x)}(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{-3x^{2}e^{arccos(x)}}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{e^{arccos(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{xe^{arccos(x)}}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}} + \frac{2xe^{arccos(x)}}{(-x^{2} + 1)^{2}} - \frac{e^{arccos(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-3x^{2}e^{arccos(x)}}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{e^{arccos(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{xe^{arccos(x)}}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}} + \frac{2xe^{arccos(x)}}{(-x^{2} + 1)^{2}} - \frac{e^{arccos(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)}\right)}{dx}\\=&-3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2}e^{arccos(x)} - \frac{3*2xe^{arccos(x)}}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{3x^{2}e^{arccos(x)}(\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}}})e^{arccos(x)} - \frac{e^{arccos(x)}(\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}}})xe^{arccos(x)}}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})xe^{arccos(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{e^{arccos(x)}}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}} + \frac{xe^{arccos(x)}(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}} + 2(\frac{-2(-2x + 0)}{(-x^{2} + 1)^{3}})xe^{arccos(x)} + \frac{2e^{arccos(x)}}{(-x^{2} + 1)^{2}} + \frac{2xe^{arccos(x)}(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{2}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})e^{arccos(x)}}{(-x^{2} + 1)} - \frac{(\frac{-(-2x + 0)}{(-x^{2} + 1)^{2}})e^{arccos(x)}}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{e^{arccos(x)}(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)}\\=&\frac{-15x^{3}e^{arccos(x)}}{(-x^{2} + 1)^{\frac{7}{2}}} - \frac{12xe^{arccos(x)}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{3x^{2}e^{arccos(x)}}{(-x^{2} + 1)^{\frac{5}{2}}(-x^{2} + 1)^{\frac{1}{2}}} + \frac{e^{arccos(x)}}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}} + \frac{12x^{2}e^{arccos(x)}}{(-x^{2} + 1)^{3}} + \frac{3e^{arccos(x)}}{(-x^{2} + 1)^{2}} - \frac{xe^{arccos(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{2}} - \frac{2xe^{arccos(x)}}{(-x^{2} + 1)^{2}(-x^{2} + 1)^{\frac{1}{2}}} + \frac{e^{arccos(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[3/3]Find\ the\ 4th\ derivative\ of\ function\ e^{arctan(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( e^{arctan(x)}\right)}{dx}\\=&e^{arctan(x)}(\frac{(1)}{(1 + (x)^{2})})\\=&\frac{e^{arctan(x)}}{(x^{2} + 1)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{e^{arctan(x)}}{(x^{2} + 1)}\right)}{dx}\\=&(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})e^{arctan(x)} + \frac{e^{arctan(x)}(\frac{(1)}{(1 + (x)^{2})})}{(x^{2} + 1)}\\=&\frac{-2xe^{arctan(x)}}{(x^{2} + 1)^{2}} + \frac{e^{arctan(x)}}{(x^{2} + 1)^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2xe^{arctan(x)}}{(x^{2} + 1)^{2}} + \frac{e^{arctan(x)}}{(x^{2} + 1)^{2}}\right)}{dx}\\=&-2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})xe^{arctan(x)} - \frac{2e^{arctan(x)}}{(x^{2} + 1)^{2}} - \frac{2xe^{arctan(x)}(\frac{(1)}{(1 + (x)^{2})})}{(x^{2} + 1)^{2}} + (\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})e^{arctan(x)} + \frac{e^{arctan(x)}(\frac{(1)}{(1 + (x)^{2})})}{(x^{2} + 1)^{2}}\\=&\frac{8x^{2}e^{arctan(x)}}{(x^{2} + 1)^{3}} - \frac{2e^{arctan(x)}}{(x^{2} + 1)^{2}} - \frac{6xe^{arctan(x)}}{(x^{2} + 1)^{3}} + \frac{e^{arctan(x)}}{(x^{2} + 1)^{3}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{8x^{2}e^{arctan(x)}}{(x^{2} + 1)^{3}} - \frac{2e^{arctan(x)}}{(x^{2} + 1)^{2}} - \frac{6xe^{arctan(x)}}{(x^{2} + 1)^{3}} + \frac{e^{arctan(x)}}{(x^{2} + 1)^{3}}\right)}{dx}\\=&8(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{2}e^{arctan(x)} + \frac{8*2xe^{arctan(x)}}{(x^{2} + 1)^{3}} + \frac{8x^{2}e^{arctan(x)}(\frac{(1)}{(1 + (x)^{2})})}{(x^{2} + 1)^{3}} - 2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})e^{arctan(x)} - \frac{2e^{arctan(x)}(\frac{(1)}{(1 + (x)^{2})})}{(x^{2} + 1)^{2}} - 6(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})xe^{arctan(x)} - \frac{6e^{arctan(x)}}{(x^{2} + 1)^{3}} - \frac{6xe^{arctan(x)}(\frac{(1)}{(1 + (x)^{2})})}{(x^{2} + 1)^{3}} + (\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})e^{arctan(x)} + \frac{e^{arctan(x)}(\frac{(1)}{(1 + (x)^{2})})}{(x^{2} + 1)^{3}}\\=&\frac{-48x^{3}e^{arctan(x)}}{(x^{2} + 1)^{4}} + \frac{24xe^{arctan(x)}}{(x^{2} + 1)^{3}} + \frac{44x^{2}e^{arctan(x)}}{(x^{2} + 1)^{4}} - \frac{8e^{arctan(x)}}{(x^{2} + 1)^{3}} - \frac{12xe^{arctan(x)}}{(x^{2} + 1)^{4}} + \frac{e^{arctan(x)}}{(x^{2} + 1)^{4}}\\ \end{split}\end{equation} \]