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 7 questions in this calculation: for each question, the 4 derivative of x is calculated.
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/7]Find\ the\ 4th\ derivative\ of\ function\ sin(x) - arcsin(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( sin(x) - arcsin(x)\right)}{dx}\\=&cos(x) - (\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})\\=&cos(x) - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( cos(x) - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&-sin(x) - (\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})\\=&-sin(x) - \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( -sin(x) - \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&-cos(x) - (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})x - \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\\=&-cos(x) - \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( -cos(x) - \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&--sin(x) - 3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2} - \frac{3*2x}{(-x^{2} + 1)^{\frac{5}{2}}} - (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})\\=&sin(x) - \frac{15x^{3}}{(-x^{2} + 1)^{\frac{7}{2}}} - \frac{9x}{(-x^{2} + 1)^{\frac{5}{2}}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/7]Find\ the\ 4th\ derivative\ of\ function\ sin(x) - arccos(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( sin(x) - arccos(x)\right)}{dx}\\=&cos(x) - (\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})\\=&cos(x) + \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( cos(x) + \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&-sin(x) + (\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})\\=&-sin(x) + \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( -sin(x) + \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&-cos(x) + (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})x + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\\=&-cos(x) + \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( -cos(x) + \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&--sin(x) + 3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2} + \frac{3*2x}{(-x^{2} + 1)^{\frac{5}{2}}} + (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})\\=&sin(x) + \frac{15x^{3}}{(-x^{2} + 1)^{\frac{7}{2}}} + \frac{9x}{(-x^{2} + 1)^{\frac{5}{2}}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[3/7]Find\ the\ 4th\ derivative\ of\ function\ cos(x) - arcsin(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( cos(x) - arcsin(x)\right)}{dx}\\=&-sin(x) - (\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})\\=&-sin(x) - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( -sin(x) - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&-cos(x) - (\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})\\=&-cos(x) - \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( -cos(x) - \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&--sin(x) - (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})x - \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\\=&sin(x) - \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( sin(x) - \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&cos(x) - 3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2} - \frac{3*2x}{(-x^{2} + 1)^{\frac{5}{2}}} - (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})\\=&cos(x) - \frac{15x^{3}}{(-x^{2} + 1)^{\frac{7}{2}}} - \frac{9x}{(-x^{2} + 1)^{\frac{5}{2}}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[4/7]Find\ the\ 4th\ derivative\ of\ function\ cos(x) - arccos(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( cos(x) - arccos(x)\right)}{dx}\\=&-sin(x) - (\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})\\=&-sin(x) + \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( -sin(x) + \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&-cos(x) + (\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})\\=&-cos(x) + \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( -cos(x) + \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&--sin(x) + (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})x + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\\=&sin(x) + \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( sin(x) + \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&cos(x) + 3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2} + \frac{3*2x}{(-x^{2} + 1)^{\frac{5}{2}}} + (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})\\=&cos(x) + \frac{15x^{3}}{(-x^{2} + 1)^{\frac{7}{2}}} + \frac{9x}{(-x^{2} + 1)^{\frac{5}{2}}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[5/7]Find\ the\ 4th\ derivative\ of\ function\ cos(x) - arctan(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( cos(x) - arctan(x)\right)}{dx}\\=&-sin(x) - (\frac{(1)}{(1 + (x)^{2})})\\=&-sin(x) - \frac{1}{(x^{2} + 1)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( -sin(x) - \frac{1}{(x^{2} + 1)}\right)}{dx}\\=&-cos(x) - (\frac{-(2x + 0)}{(x^{2} + 1)^{2}})\\=&-cos(x) + \frac{2x}{(x^{2} + 1)^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( -cos(x) + \frac{2x}{(x^{2} + 1)^{2}}\right)}{dx}\\=&--sin(x) + 2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x + \frac{2}{(x^{2} + 1)^{2}}\\=&sin(x) - \frac{8x^{2}}{(x^{2} + 1)^{3}} + \frac{2}{(x^{2} + 1)^{2}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( sin(x) - \frac{8x^{2}}{(x^{2} + 1)^{3}} + \frac{2}{(x^{2} + 1)^{2}}\right)}{dx}\\=&cos(x) - 8(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{2} - \frac{8*2x}{(x^{2} + 1)^{3}} + 2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})\\=&cos(x) + \frac{48x^{3}}{(x^{2} + 1)^{4}} - \frac{24x}{(x^{2} + 1)^{3}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[6/7]Find\ the\ 4th\ derivative\ of\ function\ tan(x) - arccos(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( tan(x) - arccos(x)\right)}{dx}\\=&sec^{2}(x)(1) - (\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})\\=&sec^{2}(x) + \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( sec^{2}(x) + \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&2sec^{2}(x)tan(x) + (\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})\\=&2tan(x)sec^{2}(x) + \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 2tan(x)sec^{2}(x) + \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&2sec^{2}(x)(1)sec^{2}(x) + 2tan(x)*2sec^{2}(x)tan(x) + (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})x + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\\=&2sec^{4}(x) + 4tan^{2}(x)sec^{2}(x) + \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 2sec^{4}(x) + 4tan^{2}(x)sec^{2}(x) + \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&2*4sec^{4}(x)tan(x) + 4*2tan(x)sec^{2}(x)(1)sec^{2}(x) + 4tan^{2}(x)*2sec^{2}(x)tan(x) + 3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2} + \frac{3*2x}{(-x^{2} + 1)^{\frac{5}{2}}} + (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})\\=&16tan(x)sec^{4}(x) + 8tan^{3}(x)sec^{2}(x) + \frac{15x^{3}}{(-x^{2} + 1)^{\frac{7}{2}}} + \frac{9x}{(-x^{2} + 1)^{\frac{5}{2}}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[7/7]Find\ the\ 4th\ derivative\ of\ function\ tan(x) - arctan(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( tan(x) - arctan(x)\right)}{dx}\\=&sec^{2}(x)(1) - (\frac{(1)}{(1 + (x)^{2})})\\=&sec^{2}(x) - \frac{1}{(x^{2} + 1)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( sec^{2}(x) - \frac{1}{(x^{2} + 1)}\right)}{dx}\\=&2sec^{2}(x)tan(x) - (\frac{-(2x + 0)}{(x^{2} + 1)^{2}})\\=&2tan(x)sec^{2}(x) + \frac{2x}{(x^{2} + 1)^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 2tan(x)sec^{2}(x) + \frac{2x}{(x^{2} + 1)^{2}}\right)}{dx}\\=&2sec^{2}(x)(1)sec^{2}(x) + 2tan(x)*2sec^{2}(x)tan(x) + 2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x + \frac{2}{(x^{2} + 1)^{2}}\\=&2sec^{4}(x) + 4tan^{2}(x)sec^{2}(x) - \frac{8x^{2}}{(x^{2} + 1)^{3}} + \frac{2}{(x^{2} + 1)^{2}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 2sec^{4}(x) + 4tan^{2}(x)sec^{2}(x) - \frac{8x^{2}}{(x^{2} + 1)^{3}} + \frac{2}{(x^{2} + 1)^{2}}\right)}{dx}\\=&2*4sec^{4}(x)tan(x) + 4*2tan(x)sec^{2}(x)(1)sec^{2}(x) + 4tan^{2}(x)*2sec^{2}(x)tan(x) - 8(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{2} - \frac{8*2x}{(x^{2} + 1)^{3}} + 2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})\\=&16tan(x)sec^{4}(x) + 8tan^{3}(x)sec^{2}(x) + \frac{48x^{3}}{(x^{2} + 1)^{4}} - \frac{24x}{(x^{2} + 1)^{3}}\\ \end{split}\end{equation} \]
>>注：本次最多计算 7 道题。
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/7]Find\ the\ 4th\ derivative\ of\ function\ sin(x) - arcsin(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( sin(x) - arcsin(x)\right)}{dx}\\=&cos(x) - (\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})\\=&cos(x) - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( cos(x) - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&-sin(x) - (\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})\\=&-sin(x) - \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( -sin(x) - \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&-cos(x) - (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})x - \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\\=&-cos(x) - \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( -cos(x) - \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&--sin(x) - 3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2} - \frac{3*2x}{(-x^{2} + 1)^{\frac{5}{2}}} - (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})\\=&sin(x) - \frac{15x^{3}}{(-x^{2} + 1)^{\frac{7}{2}}} - \frac{9x}{(-x^{2} + 1)^{\frac{5}{2}}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/7]Find\ the\ 4th\ derivative\ of\ function\ sin(x) - arccos(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( sin(x) - arccos(x)\right)}{dx}\\=&cos(x) - (\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})\\=&cos(x) + \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( cos(x) + \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&-sin(x) + (\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})\\=&-sin(x) + \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( -sin(x) + \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&-cos(x) + (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})x + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\\=&-cos(x) + \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( -cos(x) + \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&--sin(x) + 3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2} + \frac{3*2x}{(-x^{2} + 1)^{\frac{5}{2}}} + (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})\\=&sin(x) + \frac{15x^{3}}{(-x^{2} + 1)^{\frac{7}{2}}} + \frac{9x}{(-x^{2} + 1)^{\frac{5}{2}}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[3/7]Find\ the\ 4th\ derivative\ of\ function\ cos(x) - arcsin(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( cos(x) - arcsin(x)\right)}{dx}\\=&-sin(x) - (\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})\\=&-sin(x) - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( -sin(x) - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&-cos(x) - (\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})\\=&-cos(x) - \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( -cos(x) - \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&--sin(x) - (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})x - \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\\=&sin(x) - \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( sin(x) - \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&cos(x) - 3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2} - \frac{3*2x}{(-x^{2} + 1)^{\frac{5}{2}}} - (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})\\=&cos(x) - \frac{15x^{3}}{(-x^{2} + 1)^{\frac{7}{2}}} - \frac{9x}{(-x^{2} + 1)^{\frac{5}{2}}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[4/7]Find\ the\ 4th\ derivative\ of\ function\ cos(x) - arccos(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( cos(x) - arccos(x)\right)}{dx}\\=&-sin(x) - (\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})\\=&-sin(x) + \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( -sin(x) + \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&-cos(x) + (\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})\\=&-cos(x) + \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( -cos(x) + \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&--sin(x) + (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})x + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\\=&sin(x) + \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( sin(x) + \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&cos(x) + 3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2} + \frac{3*2x}{(-x^{2} + 1)^{\frac{5}{2}}} + (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})\\=&cos(x) + \frac{15x^{3}}{(-x^{2} + 1)^{\frac{7}{2}}} + \frac{9x}{(-x^{2} + 1)^{\frac{5}{2}}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[5/7]Find\ the\ 4th\ derivative\ of\ function\ cos(x) - arctan(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( cos(x) - arctan(x)\right)}{dx}\\=&-sin(x) - (\frac{(1)}{(1 + (x)^{2})})\\=&-sin(x) - \frac{1}{(x^{2} + 1)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( -sin(x) - \frac{1}{(x^{2} + 1)}\right)}{dx}\\=&-cos(x) - (\frac{-(2x + 0)}{(x^{2} + 1)^{2}})\\=&-cos(x) + \frac{2x}{(x^{2} + 1)^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( -cos(x) + \frac{2x}{(x^{2} + 1)^{2}}\right)}{dx}\\=&--sin(x) + 2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x + \frac{2}{(x^{2} + 1)^{2}}\\=&sin(x) - \frac{8x^{2}}{(x^{2} + 1)^{3}} + \frac{2}{(x^{2} + 1)^{2}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( sin(x) - \frac{8x^{2}}{(x^{2} + 1)^{3}} + \frac{2}{(x^{2} + 1)^{2}}\right)}{dx}\\=&cos(x) - 8(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{2} - \frac{8*2x}{(x^{2} + 1)^{3}} + 2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})\\=&cos(x) + \frac{48x^{3}}{(x^{2} + 1)^{4}} - \frac{24x}{(x^{2} + 1)^{3}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[6/7]Find\ the\ 4th\ derivative\ of\ function\ tan(x) - arccos(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( tan(x) - arccos(x)\right)}{dx}\\=&sec^{2}(x)(1) - (\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})\\=&sec^{2}(x) + \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( sec^{2}(x) + \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&2sec^{2}(x)tan(x) + (\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})\\=&2tan(x)sec^{2}(x) + \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 2tan(x)sec^{2}(x) + \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&2sec^{2}(x)(1)sec^{2}(x) + 2tan(x)*2sec^{2}(x)tan(x) + (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})x + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\\=&2sec^{4}(x) + 4tan^{2}(x)sec^{2}(x) + \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 2sec^{4}(x) + 4tan^{2}(x)sec^{2}(x) + \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&2*4sec^{4}(x)tan(x) + 4*2tan(x)sec^{2}(x)(1)sec^{2}(x) + 4tan^{2}(x)*2sec^{2}(x)tan(x) + 3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2} + \frac{3*2x}{(-x^{2} + 1)^{\frac{5}{2}}} + (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})\\=&16tan(x)sec^{4}(x) + 8tan^{3}(x)sec^{2}(x) + \frac{15x^{3}}{(-x^{2} + 1)^{\frac{7}{2}}} + \frac{9x}{(-x^{2} + 1)^{\frac{5}{2}}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[7/7]Find\ the\ 4th\ derivative\ of\ function\ tan(x) - arctan(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( tan(x) - arctan(x)\right)}{dx}\\=&sec^{2}(x)(1) - (\frac{(1)}{(1 + (x)^{2})})\\=&sec^{2}(x) - \frac{1}{(x^{2} + 1)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( sec^{2}(x) - \frac{1}{(x^{2} + 1)}\right)}{dx}\\=&2sec^{2}(x)tan(x) - (\frac{-(2x + 0)}{(x^{2} + 1)^{2}})\\=&2tan(x)sec^{2}(x) + \frac{2x}{(x^{2} + 1)^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 2tan(x)sec^{2}(x) + \frac{2x}{(x^{2} + 1)^{2}}\right)}{dx}\\=&2sec^{2}(x)(1)sec^{2}(x) + 2tan(x)*2sec^{2}(x)tan(x) + 2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x + \frac{2}{(x^{2} + 1)^{2}}\\=&2sec^{4}(x) + 4tan^{2}(x)sec^{2}(x) - \frac{8x^{2}}{(x^{2} + 1)^{3}} + \frac{2}{(x^{2} + 1)^{2}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 2sec^{4}(x) + 4tan^{2}(x)sec^{2}(x) - \frac{8x^{2}}{(x^{2} + 1)^{3}} + \frac{2}{(x^{2} + 1)^{2}}\right)}{dx}\\=&2*4sec^{4}(x)tan(x) + 4*2tan(x)sec^{2}(x)(1)sec^{2}(x) + 4tan^{2}(x)*2sec^{2}(x)tan(x) - 8(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{2} - \frac{8*2x}{(x^{2} + 1)^{3}} + 2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})\\=&16tan(x)sec^{4}(x) + 8tan^{3}(x)sec^{2}(x) + \frac{48x^{3}}{(x^{2} + 1)^{4}} - \frac{24x}{(x^{2} + 1)^{3}}\\ \end{split}\end{equation} \]
>>注：本次最多计算 7 道题。