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 a is calculated.
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/2]Find\ the\ 4th\ derivative\ of\ function\ arcsin(a)(cos(a))\ with\ respect\ to\ a：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = cos(a)arcsin(a)\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( cos(a)arcsin(a)\right)}{da}\\=&-sin(a)arcsin(a) + cos(a)(\frac{(1)}{((1 - (a)^{2})^{\frac{1}{2}})})\\=&-sin(a)arcsin(a) + \frac{cos(a)}{(-a^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( -sin(a)arcsin(a) + \frac{cos(a)}{(-a^{2} + 1)^{\frac{1}{2}}}\right)}{da}\\=&-cos(a)arcsin(a) - sin(a)(\frac{(1)}{((1 - (a)^{2})^{\frac{1}{2}})}) + (\frac{\frac{-1}{2}(-2a + 0)}{(-a^{2} + 1)^{\frac{3}{2}}})cos(a) + \frac{-sin(a)}{(-a^{2} + 1)^{\frac{1}{2}}}\\=&-cos(a)arcsin(a) - \frac{sin(a)}{(-a^{2} + 1)^{\frac{1}{2}}} + \frac{acos(a)}{(-a^{2} + 1)^{\frac{3}{2}}} - \frac{sin(a)}{(-a^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( -cos(a)arcsin(a) - \frac{sin(a)}{(-a^{2} + 1)^{\frac{1}{2}}} + \frac{acos(a)}{(-a^{2} + 1)^{\frac{3}{2}}} - \frac{sin(a)}{(-a^{2} + 1)^{\frac{1}{2}}}\right)}{da}\\=&--sin(a)arcsin(a) - cos(a)(\frac{(1)}{((1 - (a)^{2})^{\frac{1}{2}})}) - (\frac{\frac{-1}{2}(-2a + 0)}{(-a^{2} + 1)^{\frac{3}{2}}})sin(a) - \frac{cos(a)}{(-a^{2} + 1)^{\frac{1}{2}}} + (\frac{\frac{-3}{2}(-2a + 0)}{(-a^{2} + 1)^{\frac{5}{2}}})acos(a) + \frac{cos(a)}{(-a^{2} + 1)^{\frac{3}{2}}} + \frac{a*-sin(a)}{(-a^{2} + 1)^{\frac{3}{2}}} - (\frac{\frac{-1}{2}(-2a + 0)}{(-a^{2} + 1)^{\frac{3}{2}}})sin(a) - \frac{cos(a)}{(-a^{2} + 1)^{\frac{1}{2}}}\\=&sin(a)arcsin(a) - \frac{cos(a)}{(-a^{2} + 1)^{\frac{1}{2}}} - \frac{3asin(a)}{(-a^{2} + 1)^{\frac{3}{2}}} - \frac{2cos(a)}{(-a^{2} + 1)^{\frac{1}{2}}} + \frac{3a^{2}cos(a)}{(-a^{2} + 1)^{\frac{5}{2}}} + \frac{cos(a)}{(-a^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( sin(a)arcsin(a) - \frac{cos(a)}{(-a^{2} + 1)^{\frac{1}{2}}} - \frac{3asin(a)}{(-a^{2} + 1)^{\frac{3}{2}}} - \frac{2cos(a)}{(-a^{2} + 1)^{\frac{1}{2}}} + \frac{3a^{2}cos(a)}{(-a^{2} + 1)^{\frac{5}{2}}} + \frac{cos(a)}{(-a^{2} + 1)^{\frac{3}{2}}}\right)}{da}\\=&cos(a)arcsin(a) + sin(a)(\frac{(1)}{((1 - (a)^{2})^{\frac{1}{2}})}) - (\frac{\frac{-1}{2}(-2a + 0)}{(-a^{2} + 1)^{\frac{3}{2}}})cos(a) - \frac{-sin(a)}{(-a^{2} + 1)^{\frac{1}{2}}} - 3(\frac{\frac{-3}{2}(-2a + 0)}{(-a^{2} + 1)^{\frac{5}{2}}})asin(a) - \frac{3sin(a)}{(-a^{2} + 1)^{\frac{3}{2}}} - \frac{3acos(a)}{(-a^{2} + 1)^{\frac{3}{2}}} - 2(\frac{\frac{-1}{2}(-2a + 0)}{(-a^{2} + 1)^{\frac{3}{2}}})cos(a) - \frac{2*-sin(a)}{(-a^{2} + 1)^{\frac{1}{2}}} + 3(\frac{\frac{-5}{2}(-2a + 0)}{(-a^{2} + 1)^{\frac{7}{2}}})a^{2}cos(a) + \frac{3*2acos(a)}{(-a^{2} + 1)^{\frac{5}{2}}} + \frac{3a^{2}*-sin(a)}{(-a^{2} + 1)^{\frac{5}{2}}} + (\frac{\frac{-3}{2}(-2a + 0)}{(-a^{2} + 1)^{\frac{5}{2}}})cos(a) + \frac{-sin(a)}{(-a^{2} + 1)^{\frac{3}{2}}}\\=&cos(a)arcsin(a) + \frac{sin(a)}{(-a^{2} + 1)^{\frac{1}{2}}} - \frac{6acos(a)}{(-a^{2} + 1)^{\frac{3}{2}}} + \frac{3sin(a)}{(-a^{2} + 1)^{\frac{1}{2}}} - \frac{12a^{2}sin(a)}{(-a^{2} + 1)^{\frac{5}{2}}} - \frac{4sin(a)}{(-a^{2} + 1)^{\frac{3}{2}}} + \frac{15a^{3}cos(a)}{(-a^{2} + 1)^{\frac{7}{2}}} + \frac{9acos(a)}{(-a^{2} + 1)^{\frac{5}{2}}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/2]Find\ the\ 4th\ derivative\ of\ function\ a(sin(a))cos(a)\ with\ respect\ to\ a：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = asin(a)cos(a)\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( asin(a)cos(a)\right)}{da}\\=&sin(a)cos(a) + acos(a)cos(a) + asin(a)*-sin(a)\\=&sin(a)cos(a) + acos^{2}(a) - asin^{2}(a)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( sin(a)cos(a) + acos^{2}(a) - asin^{2}(a)\right)}{da}\\=&cos(a)cos(a) + sin(a)*-sin(a) + cos^{2}(a) + a*-2cos(a)sin(a) - sin^{2}(a) - a*2sin(a)cos(a)\\=&2cos^{2}(a) - 2sin^{2}(a) - 4asin(a)cos(a)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 2cos^{2}(a) - 2sin^{2}(a) - 4asin(a)cos(a)\right)}{da}\\=&2*-2cos(a)sin(a) - 2*2sin(a)cos(a) - 4sin(a)cos(a) - 4acos(a)cos(a) - 4asin(a)*-sin(a)\\=&-12sin(a)cos(a) - 4acos^{2}(a) + 4asin^{2}(a)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( -12sin(a)cos(a) - 4acos^{2}(a) + 4asin^{2}(a)\right)}{da}\\=&-12cos(a)cos(a) - 12sin(a)*-sin(a) - 4cos^{2}(a) - 4a*-2cos(a)sin(a) + 4sin^{2}(a) + 4a*2sin(a)cos(a)\\=&-16cos^{2}(a) + 16sin^{2}(a) + 16asin(a)cos(a)\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/2]Find\ the\ 4th\ derivative\ of\ function\ arcsin(a)(cos(a))\ with\ respect\ to\ a：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = cos(a)arcsin(a)\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( cos(a)arcsin(a)\right)}{da}\\=&-sin(a)arcsin(a) + cos(a)(\frac{(1)}{((1 - (a)^{2})^{\frac{1}{2}})})\\=&-sin(a)arcsin(a) + \frac{cos(a)}{(-a^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( -sin(a)arcsin(a) + \frac{cos(a)}{(-a^{2} + 1)^{\frac{1}{2}}}\right)}{da}\\=&-cos(a)arcsin(a) - sin(a)(\frac{(1)}{((1 - (a)^{2})^{\frac{1}{2}})}) + (\frac{\frac{-1}{2}(-2a + 0)}{(-a^{2} + 1)^{\frac{3}{2}}})cos(a) + \frac{-sin(a)}{(-a^{2} + 1)^{\frac{1}{2}}}\\=&-cos(a)arcsin(a) - \frac{sin(a)}{(-a^{2} + 1)^{\frac{1}{2}}} + \frac{acos(a)}{(-a^{2} + 1)^{\frac{3}{2}}} - \frac{sin(a)}{(-a^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( -cos(a)arcsin(a) - \frac{sin(a)}{(-a^{2} + 1)^{\frac{1}{2}}} + \frac{acos(a)}{(-a^{2} + 1)^{\frac{3}{2}}} - \frac{sin(a)}{(-a^{2} + 1)^{\frac{1}{2}}}\right)}{da}\\=&--sin(a)arcsin(a) - cos(a)(\frac{(1)}{((1 - (a)^{2})^{\frac{1}{2}})}) - (\frac{\frac{-1}{2}(-2a + 0)}{(-a^{2} + 1)^{\frac{3}{2}}})sin(a) - \frac{cos(a)}{(-a^{2} + 1)^{\frac{1}{2}}} + (\frac{\frac{-3}{2}(-2a + 0)}{(-a^{2} + 1)^{\frac{5}{2}}})acos(a) + \frac{cos(a)}{(-a^{2} + 1)^{\frac{3}{2}}} + \frac{a*-sin(a)}{(-a^{2} + 1)^{\frac{3}{2}}} - (\frac{\frac{-1}{2}(-2a + 0)}{(-a^{2} + 1)^{\frac{3}{2}}})sin(a) - \frac{cos(a)}{(-a^{2} + 1)^{\frac{1}{2}}}\\=&sin(a)arcsin(a) - \frac{cos(a)}{(-a^{2} + 1)^{\frac{1}{2}}} - \frac{3asin(a)}{(-a^{2} + 1)^{\frac{3}{2}}} - \frac{2cos(a)}{(-a^{2} + 1)^{\frac{1}{2}}} + \frac{3a^{2}cos(a)}{(-a^{2} + 1)^{\frac{5}{2}}} + \frac{cos(a)}{(-a^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( sin(a)arcsin(a) - \frac{cos(a)}{(-a^{2} + 1)^{\frac{1}{2}}} - \frac{3asin(a)}{(-a^{2} + 1)^{\frac{3}{2}}} - \frac{2cos(a)}{(-a^{2} + 1)^{\frac{1}{2}}} + \frac{3a^{2}cos(a)}{(-a^{2} + 1)^{\frac{5}{2}}} + \frac{cos(a)}{(-a^{2} + 1)^{\frac{3}{2}}}\right)}{da}\\=&cos(a)arcsin(a) + sin(a)(\frac{(1)}{((1 - (a)^{2})^{\frac{1}{2}})}) - (\frac{\frac{-1}{2}(-2a + 0)}{(-a^{2} + 1)^{\frac{3}{2}}})cos(a) - \frac{-sin(a)}{(-a^{2} + 1)^{\frac{1}{2}}} - 3(\frac{\frac{-3}{2}(-2a + 0)}{(-a^{2} + 1)^{\frac{5}{2}}})asin(a) - \frac{3sin(a)}{(-a^{2} + 1)^{\frac{3}{2}}} - \frac{3acos(a)}{(-a^{2} + 1)^{\frac{3}{2}}} - 2(\frac{\frac{-1}{2}(-2a + 0)}{(-a^{2} + 1)^{\frac{3}{2}}})cos(a) - \frac{2*-sin(a)}{(-a^{2} + 1)^{\frac{1}{2}}} + 3(\frac{\frac{-5}{2}(-2a + 0)}{(-a^{2} + 1)^{\frac{7}{2}}})a^{2}cos(a) + \frac{3*2acos(a)}{(-a^{2} + 1)^{\frac{5}{2}}} + \frac{3a^{2}*-sin(a)}{(-a^{2} + 1)^{\frac{5}{2}}} + (\frac{\frac{-3}{2}(-2a + 0)}{(-a^{2} + 1)^{\frac{5}{2}}})cos(a) + \frac{-sin(a)}{(-a^{2} + 1)^{\frac{3}{2}}}\\=&cos(a)arcsin(a) + \frac{sin(a)}{(-a^{2} + 1)^{\frac{1}{2}}} - \frac{6acos(a)}{(-a^{2} + 1)^{\frac{3}{2}}} + \frac{3sin(a)}{(-a^{2} + 1)^{\frac{1}{2}}} - \frac{12a^{2}sin(a)}{(-a^{2} + 1)^{\frac{5}{2}}} - \frac{4sin(a)}{(-a^{2} + 1)^{\frac{3}{2}}} + \frac{15a^{3}cos(a)}{(-a^{2} + 1)^{\frac{7}{2}}} + \frac{9acos(a)}{(-a^{2} + 1)^{\frac{5}{2}}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/2]Find\ the\ 4th\ derivative\ of\ function\ a(sin(a))cos(a)\ with\ respect\ to\ a：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = asin(a)cos(a)\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( asin(a)cos(a)\right)}{da}\\=&sin(a)cos(a) + acos(a)cos(a) + asin(a)*-sin(a)\\=&sin(a)cos(a) + acos^{2}(a) - asin^{2}(a)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( sin(a)cos(a) + acos^{2}(a) - asin^{2}(a)\right)}{da}\\=&cos(a)cos(a) + sin(a)*-sin(a) + cos^{2}(a) + a*-2cos(a)sin(a) - sin^{2}(a) - a*2sin(a)cos(a)\\=&2cos^{2}(a) - 2sin^{2}(a) - 4asin(a)cos(a)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 2cos^{2}(a) - 2sin^{2}(a) - 4asin(a)cos(a)\right)}{da}\\=&2*-2cos(a)sin(a) - 2*2sin(a)cos(a) - 4sin(a)cos(a) - 4acos(a)cos(a) - 4asin(a)*-sin(a)\\=&-12sin(a)cos(a) - 4acos^{2}(a) + 4asin^{2}(a)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( -12sin(a)cos(a) - 4acos^{2}(a) + 4asin^{2}(a)\right)}{da}\\=&-12cos(a)cos(a) - 12sin(a)*-sin(a) - 4cos^{2}(a) - 4a*-2cos(a)sin(a) + 4sin^{2}(a) + 4a*2sin(a)cos(a)\\=&-16cos^{2}(a) + 16sin^{2}(a) + 16asin(a)cos(a)\\ \end{split}\end{equation} \]