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.
\[ \begin{equation}\begin{split}[1/2]Find\ the\ 4th\ derivative\ of\ function\ sin(sin(sin(x)))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( sin(sin(sin(x)))\right)}{dx}\\=&cos(sin(sin(x)))cos(sin(x))cos(x)\\=&cos(sin(sin(x)))cos(x)cos(sin(x))\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( cos(sin(sin(x)))cos(x)cos(sin(x))\right)}{dx}\\=&-sin(sin(sin(x)))cos(sin(x))cos(x)cos(x)cos(sin(x)) + cos(sin(sin(x)))*-sin(x)cos(sin(x)) + cos(sin(sin(x)))cos(x)*-sin(sin(x))cos(x)\\=&-sin(sin(sin(x)))cos^{2}(x)cos^{2}(sin(x)) - sin(x)cos(sin(sin(x)))cos(sin(x)) - sin(sin(x))cos(sin(sin(x)))cos^{2}(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( -sin(sin(sin(x)))cos^{2}(x)cos^{2}(sin(x)) - sin(x)cos(sin(sin(x)))cos(sin(x)) - sin(sin(x))cos(sin(sin(x)))cos^{2}(x)\right)}{dx}\\=&-cos(sin(sin(x)))cos(sin(x))cos(x)cos^{2}(x)cos^{2}(sin(x)) - sin(sin(sin(x)))*-2cos(x)sin(x)cos^{2}(sin(x)) - sin(sin(sin(x)))cos^{2}(x)*-2cos(sin(x))sin(sin(x))cos(x) - cos(x)cos(sin(sin(x)))cos(sin(x)) - sin(x)*-sin(sin(sin(x)))cos(sin(x))cos(x)cos(sin(x)) - sin(x)cos(sin(sin(x)))*-sin(sin(x))cos(x) - cos(sin(x))cos(x)cos(sin(sin(x)))cos^{2}(x) - sin(sin(x))*-sin(sin(sin(x)))cos(sin(x))cos(x)cos^{2}(x) - sin(sin(x))cos(sin(sin(x)))*-2cos(x)sin(x)\\=&-cos(sin(sin(x)))cos^{3}(x)cos^{3}(sin(x)) + 2sin(x)sin(sin(sin(x)))cos(x)cos^{2}(sin(x)) + 2sin(sin(sin(x)))sin(sin(x))cos(sin(x))cos^{3}(x) - cos(x)cos(sin(sin(x)))cos(sin(x)) + sin(sin(sin(x)))sin(x)cos(x)cos^{2}(sin(x)) + sin(x)sin(sin(x))cos(x)cos(sin(sin(x))) - cos^{3}(x)cos(sin(x))cos(sin(sin(x))) + sin(sin(sin(x)))sin(sin(x))cos^{3}(x)cos(sin(x)) + 2sin(x)sin(sin(x))cos(sin(sin(x)))cos(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( -cos(sin(sin(x)))cos^{3}(x)cos^{3}(sin(x)) + 2sin(x)sin(sin(sin(x)))cos(x)cos^{2}(sin(x)) + 2sin(sin(sin(x)))sin(sin(x))cos(sin(x))cos^{3}(x) - cos(x)cos(sin(sin(x)))cos(sin(x)) + sin(sin(sin(x)))sin(x)cos(x)cos^{2}(sin(x)) + sin(x)sin(sin(x))cos(x)cos(sin(sin(x))) - cos^{3}(x)cos(sin(x))cos(sin(sin(x))) + sin(sin(sin(x)))sin(sin(x))cos^{3}(x)cos(sin(x)) + 2sin(x)sin(sin(x))cos(sin(sin(x)))cos(x)\right)}{dx}\\=&--sin(sin(sin(x)))cos(sin(x))cos(x)cos^{3}(x)cos^{3}(sin(x)) - cos(sin(sin(x)))*-3cos^{2}(x)sin(x)cos^{3}(sin(x)) - cos(sin(sin(x)))cos^{3}(x)*-3cos^{2}(sin(x))sin(sin(x))cos(x) + 2cos(x)sin(sin(sin(x)))cos(x)cos^{2}(sin(x)) + 2sin(x)cos(sin(sin(x)))cos(sin(x))cos(x)cos(x)cos^{2}(sin(x)) + 2sin(x)sin(sin(sin(x)))*-sin(x)cos^{2}(sin(x)) + 2sin(x)sin(sin(sin(x)))cos(x)*-2cos(sin(x))sin(sin(x))cos(x) + 2cos(sin(sin(x)))cos(sin(x))cos(x)sin(sin(x))cos(sin(x))cos^{3}(x) + 2sin(sin(sin(x)))cos(sin(x))cos(x)cos(sin(x))cos^{3}(x) + 2sin(sin(sin(x)))sin(sin(x))*-sin(sin(x))cos(x)cos^{3}(x) + 2sin(sin(sin(x)))sin(sin(x))cos(sin(x))*-3cos^{2}(x)sin(x) - -sin(x)cos(sin(sin(x)))cos(sin(x)) - cos(x)*-sin(sin(sin(x)))cos(sin(x))cos(x)cos(sin(x)) - cos(x)cos(sin(sin(x)))*-sin(sin(x))cos(x) + cos(sin(sin(x)))cos(sin(x))cos(x)sin(x)cos(x)cos^{2}(sin(x)) + sin(sin(sin(x)))cos(x)cos(x)cos^{2}(sin(x)) + sin(sin(sin(x)))sin(x)*-sin(x)cos^{2}(sin(x)) + sin(sin(sin(x)))sin(x)cos(x)*-2cos(sin(x))sin(sin(x))cos(x) + cos(x)sin(sin(x))cos(x)cos(sin(sin(x))) + sin(x)cos(sin(x))cos(x)cos(x)cos(sin(sin(x))) + sin(x)sin(sin(x))*-sin(x)cos(sin(sin(x))) + sin(x)sin(sin(x))cos(x)*-sin(sin(sin(x)))cos(sin(x))cos(x) - -3cos^{2}(x)sin(x)cos(sin(x))cos(sin(sin(x))) - cos^{3}(x)*-sin(sin(x))cos(x)cos(sin(sin(x))) - cos^{3}(x)cos(sin(x))*-sin(sin(sin(x)))cos(sin(x))cos(x) + cos(sin(sin(x)))cos(sin(x))cos(x)sin(sin(x))cos^{3}(x)cos(sin(x)) + sin(sin(sin(x)))cos(sin(x))cos(x)cos^{3}(x)cos(sin(x)) + sin(sin(sin(x)))sin(sin(x))*-3cos^{2}(x)sin(x)cos(sin(x)) + sin(sin(sin(x)))sin(sin(x))cos^{3}(x)*-sin(sin(x))cos(x) + 2cos(x)sin(sin(x))cos(sin(sin(x)))cos(x) + 2sin(x)cos(sin(x))cos(x)cos(sin(sin(x)))cos(x) + 2sin(x)sin(sin(x))*-sin(sin(sin(x)))cos(sin(x))cos(x)cos(x) + 2sin(x)sin(sin(x))cos(sin(sin(x)))*-sin(x)\\=&5sin(x)cos(sin(sin(x)))cos^{2}(x)cos^{3}(sin(x)) + 3sin(sin(x))cos^{4}(x)cos^{2}(sin(x))cos(sin(sin(x))) + sin(x)cos^{2}(x)cos^{3}(sin(x))cos(sin(sin(x))) + 6sin(x)cos^{2}(x)cos(sin(x))cos(sin(sin(x))) + 3sin(sin(x))cos^{4}(x)cos(sin(sin(x)))cos^{2}(sin(x)) - 6sin(sin(sin(x)))sin(x)sin(sin(x))cos(sin(x))cos^{2}(x) - 4sin(sin(x))sin(sin(sin(x)))sin(x)cos(sin(x))cos^{2}(x) + sin(sin(sin(x)))cos^{4}(x)cos^{4}(sin(x)) + 4sin(sin(sin(x)))cos^{2}(x)cos^{2}(sin(x)) - 2sin(sin(x))sin(x)sin(sin(sin(x)))cos(sin(x))cos^{2}(x) - 3sin(sin(sin(x)))sin(x)sin(sin(x))cos^{2}(x)cos(sin(x)) + 3sin(sin(sin(x)))cos^{4}(x)cos^{2}(sin(x)) - sin(sin(sin(x)))sin^{2}(x)cos^{2}(sin(x)) - 3sin^{2}(sin(x))sin(sin(sin(x)))cos^{4}(x) + 4sin(sin(x))cos^{2}(x)cos(sin(sin(x))) - 3sin^{2}(x)sin(sin(x))cos(sin(sin(x))) - 3sin(sin(x))sin(x)sin(sin(sin(x)))cos^{2}(x)cos(sin(x)) + sin(sin(x))cos^{4}(x)cos(sin(sin(x))) + sin(sin(sin(x)))cos^{2}(sin(x))cos^{4}(x) - 2sin^{2}(x)sin(sin(sin(x)))cos^{2}(sin(x)) + sin(x)cos(sin(sin(x)))cos(sin(x))\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/2]Find\ the\ 4th\ derivative\ of\ function\ cos(cos(cos(x)))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( cos(cos(cos(x)))\right)}{dx}\\=&-sin(cos(cos(x)))*-sin(cos(x))*-sin(x)\\=&-sin(cos(x))sin(x)sin(cos(cos(x)))\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( -sin(cos(x))sin(x)sin(cos(cos(x)))\right)}{dx}\\=&-cos(cos(x))*-sin(x)sin(x)sin(cos(cos(x))) - sin(cos(x))cos(x)sin(cos(cos(x))) - sin(cos(x))sin(x)cos(cos(cos(x)))*-sin(cos(x))*-sin(x)\\=&sin^{2}(x)sin(cos(cos(x)))cos(cos(x)) - sin(cos(x))sin(cos(cos(x)))cos(x) - sin^{2}(cos(x))sin^{2}(x)cos(cos(cos(x)))\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( sin^{2}(x)sin(cos(cos(x)))cos(cos(x)) - sin(cos(x))sin(cos(cos(x)))cos(x) - sin^{2}(cos(x))sin^{2}(x)cos(cos(cos(x)))\right)}{dx}\\=&2sin(x)cos(x)sin(cos(cos(x)))cos(cos(x)) + sin^{2}(x)cos(cos(cos(x)))*-sin(cos(x))*-sin(x)cos(cos(x)) + sin^{2}(x)sin(cos(cos(x)))*-sin(cos(x))*-sin(x) - cos(cos(x))*-sin(x)sin(cos(cos(x)))cos(x) - sin(cos(x))cos(cos(cos(x)))*-sin(cos(x))*-sin(x)cos(x) - sin(cos(x))sin(cos(cos(x)))*-sin(x) - 2sin(cos(x))cos(cos(x))*-sin(x)sin^{2}(x)cos(cos(cos(x))) - sin^{2}(cos(x))*2sin(x)cos(x)cos(cos(cos(x))) - sin^{2}(cos(x))sin^{2}(x)*-sin(cos(cos(x)))*-sin(cos(x))*-sin(x)\\=&2sin(x)sin(cos(cos(x)))cos(x)cos(cos(x)) + sin(cos(x))sin^{3}(x)cos(cos(cos(x)))cos(cos(x)) + sin(cos(cos(x)))sin^{3}(x)sin(cos(x)) + sin(x)sin(cos(cos(x)))cos(cos(x))cos(x) - sin^{2}(cos(x))sin(x)cos(cos(cos(x)))cos(x) + sin(cos(x))sin(x)sin(cos(cos(x))) + 2sin^{3}(x)sin(cos(x))cos(cos(x))cos(cos(cos(x))) - 2sin(x)sin^{2}(cos(x))cos(x)cos(cos(cos(x))) + sin^{3}(cos(x))sin(cos(cos(x)))sin^{3}(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 2sin(x)sin(cos(cos(x)))cos(x)cos(cos(x)) + sin(cos(x))sin^{3}(x)cos(cos(cos(x)))cos(cos(x)) + sin(cos(cos(x)))sin^{3}(x)sin(cos(x)) + sin(x)sin(cos(cos(x)))cos(cos(x))cos(x) - sin^{2}(cos(x))sin(x)cos(cos(cos(x)))cos(x) + sin(cos(x))sin(x)sin(cos(cos(x))) + 2sin^{3}(x)sin(cos(x))cos(cos(x))cos(cos(cos(x))) - 2sin(x)sin^{2}(cos(x))cos(x)cos(cos(cos(x))) + sin^{3}(cos(x))sin(cos(cos(x)))sin^{3}(x)\right)}{dx}\\=&2cos(x)sin(cos(cos(x)))cos(x)cos(cos(x)) + 2sin(x)cos(cos(cos(x)))*-sin(cos(x))*-sin(x)cos(x)cos(cos(x)) + 2sin(x)sin(cos(cos(x)))*-sin(x)cos(cos(x)) + 2sin(x)sin(cos(cos(x)))cos(x)*-sin(cos(x))*-sin(x) + cos(cos(x))*-sin(x)sin^{3}(x)cos(cos(cos(x)))cos(cos(x)) + sin(cos(x))*3sin^{2}(x)cos(x)cos(cos(cos(x)))cos(cos(x)) + sin(cos(x))sin^{3}(x)*-sin(cos(cos(x)))*-sin(cos(x))*-sin(x)cos(cos(x)) + sin(cos(x))sin^{3}(x)cos(cos(cos(x)))*-sin(cos(x))*-sin(x) + cos(cos(cos(x)))*-sin(cos(x))*-sin(x)sin^{3}(x)sin(cos(x)) + sin(cos(cos(x)))*3sin^{2}(x)cos(x)sin(cos(x)) + sin(cos(cos(x)))sin^{3}(x)cos(cos(x))*-sin(x) + cos(x)sin(cos(cos(x)))cos(cos(x))cos(x) + sin(x)cos(cos(cos(x)))*-sin(cos(x))*-sin(x)cos(cos(x))cos(x) + sin(x)sin(cos(cos(x)))*-sin(cos(x))*-sin(x)cos(x) + sin(x)sin(cos(cos(x)))cos(cos(x))*-sin(x) - 2sin(cos(x))cos(cos(x))*-sin(x)sin(x)cos(cos(cos(x)))cos(x) - sin^{2}(cos(x))cos(x)cos(cos(cos(x)))cos(x) - sin^{2}(cos(x))sin(x)*-sin(cos(cos(x)))*-sin(cos(x))*-sin(x)cos(x) - sin^{2}(cos(x))sin(x)cos(cos(cos(x)))*-sin(x) + cos(cos(x))*-sin(x)sin(x)sin(cos(cos(x))) + sin(cos(x))cos(x)sin(cos(cos(x))) + sin(cos(x))sin(x)cos(cos(cos(x)))*-sin(cos(x))*-sin(x) + 2*3sin^{2}(x)cos(x)sin(cos(x))cos(cos(x))cos(cos(cos(x))) + 2sin^{3}(x)cos(cos(x))*-sin(x)cos(cos(x))cos(cos(cos(x))) + 2sin^{3}(x)sin(cos(x))*-sin(cos(x))*-sin(x)cos(cos(cos(x))) + 2sin^{3}(x)sin(cos(x))cos(cos(x))*-sin(cos(cos(x)))*-sin(cos(x))*-sin(x) - 2cos(x)sin^{2}(cos(x))cos(x)cos(cos(cos(x))) - 2sin(x)*2sin(cos(x))cos(cos(x))*-sin(x)cos(x)cos(cos(cos(x))) - 2sin(x)sin^{2}(cos(x))*-sin(x)cos(cos(cos(x))) - 2sin(x)sin^{2}(cos(x))cos(x)*-sin(cos(cos(x)))*-sin(cos(x))*-sin(x) + 3sin^{2}(cos(x))cos(cos(x))*-sin(x)sin(cos(cos(x)))sin^{3}(x) + sin^{3}(cos(x))cos(cos(cos(x)))*-sin(cos(x))*-sin(x)sin^{3}(x) + sin^{3}(cos(x))sin(cos(cos(x)))*3sin^{2}(x)cos(x)\\=&3sin(cos(cos(x)))cos^{2}(x)cos(cos(x)) + 2sin(cos(x))sin^{2}(x)cos(cos(cos(x)))cos(x)cos(cos(x)) + 3sin^{2}(x)sin(cos(x))cos(x)cos(cos(cos(x)))cos(cos(x)) + 3sin(cos(cos(x)))sin^{2}(x)sin(cos(x))cos(x) - 3sin^{4}(x)cos^{2}(cos(x))cos(cos(cos(x))) + sin(cos(x))sin^{2}(x)cos(cos(cos(x)))cos(cos(x))cos(x) - 3sin^{2}(cos(x))sin(cos(cos(x)))sin^{4}(x)cos(cos(x)) + 2sin^{2}(x)sin(cos(x))cos(cos(x))cos(cos(cos(x)))cos(x) + 3sin^{2}(x)sin(cos(cos(x)))sin(cos(x))cos(x) + 6sin^{2}(x)sin(cos(x))cos(x)cos(cos(x))cos(cos(cos(x))) + 4sin(cos(x))sin^{2}(x)cos(cos(x))cos(x)cos(cos(cos(x))) + 2sin^{4}(x)sin^{2}(cos(x))cos(cos(cos(x))) - 3sin^{2}(cos(x))cos^{2}(x)cos(cos(cos(x))) + 3sin^{3}(cos(x))sin(cos(cos(x)))sin^{2}(x)cos(x) + 2sin^{2}(cos(x))sin^{2}(x)cos(cos(cos(x))) + sin(cos(x))sin(cos(cos(x)))cos(x) - sin^{4}(x)sin(cos(cos(x)))cos(cos(x)) + 2sin^{2}(cos(x))sin^{4}(x)cos(cos(cos(x))) - 4sin^{2}(x)sin(cos(cos(x)))cos(cos(x)) + 2sin^{2}(x)sin^{2}(cos(x))cos(cos(cos(x))) - 3sin^{4}(x)sin^{2}(cos(x))sin(cos(cos(x)))cos(cos(x)) + sin^{4}(cos(x))sin^{4}(x)cos(cos(cos(x))) + 3sin^{3}(cos(x))sin^{2}(x)sin(cos(cos(x)))cos(x)\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/2]Find\ the\ 4th\ derivative\ of\ function\ sin(sin(sin(x)))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( sin(sin(sin(x)))\right)}{dx}\\=&cos(sin(sin(x)))cos(sin(x))cos(x)\\=&cos(sin(sin(x)))cos(x)cos(sin(x))\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( cos(sin(sin(x)))cos(x)cos(sin(x))\right)}{dx}\\=&-sin(sin(sin(x)))cos(sin(x))cos(x)cos(x)cos(sin(x)) + cos(sin(sin(x)))*-sin(x)cos(sin(x)) + cos(sin(sin(x)))cos(x)*-sin(sin(x))cos(x)\\=&-sin(sin(sin(x)))cos^{2}(x)cos^{2}(sin(x)) - sin(x)cos(sin(sin(x)))cos(sin(x)) - sin(sin(x))cos(sin(sin(x)))cos^{2}(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( -sin(sin(sin(x)))cos^{2}(x)cos^{2}(sin(x)) - sin(x)cos(sin(sin(x)))cos(sin(x)) - sin(sin(x))cos(sin(sin(x)))cos^{2}(x)\right)}{dx}\\=&-cos(sin(sin(x)))cos(sin(x))cos(x)cos^{2}(x)cos^{2}(sin(x)) - sin(sin(sin(x)))*-2cos(x)sin(x)cos^{2}(sin(x)) - sin(sin(sin(x)))cos^{2}(x)*-2cos(sin(x))sin(sin(x))cos(x) - cos(x)cos(sin(sin(x)))cos(sin(x)) - sin(x)*-sin(sin(sin(x)))cos(sin(x))cos(x)cos(sin(x)) - sin(x)cos(sin(sin(x)))*-sin(sin(x))cos(x) - cos(sin(x))cos(x)cos(sin(sin(x)))cos^{2}(x) - sin(sin(x))*-sin(sin(sin(x)))cos(sin(x))cos(x)cos^{2}(x) - sin(sin(x))cos(sin(sin(x)))*-2cos(x)sin(x)\\=&-cos(sin(sin(x)))cos^{3}(x)cos^{3}(sin(x)) + 2sin(x)sin(sin(sin(x)))cos(x)cos^{2}(sin(x)) + 2sin(sin(sin(x)))sin(sin(x))cos(sin(x))cos^{3}(x) - cos(x)cos(sin(sin(x)))cos(sin(x)) + sin(sin(sin(x)))sin(x)cos(x)cos^{2}(sin(x)) + sin(x)sin(sin(x))cos(x)cos(sin(sin(x))) - cos^{3}(x)cos(sin(x))cos(sin(sin(x))) + sin(sin(sin(x)))sin(sin(x))cos^{3}(x)cos(sin(x)) + 2sin(x)sin(sin(x))cos(sin(sin(x)))cos(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( -cos(sin(sin(x)))cos^{3}(x)cos^{3}(sin(x)) + 2sin(x)sin(sin(sin(x)))cos(x)cos^{2}(sin(x)) + 2sin(sin(sin(x)))sin(sin(x))cos(sin(x))cos^{3}(x) - cos(x)cos(sin(sin(x)))cos(sin(x)) + sin(sin(sin(x)))sin(x)cos(x)cos^{2}(sin(x)) + sin(x)sin(sin(x))cos(x)cos(sin(sin(x))) - cos^{3}(x)cos(sin(x))cos(sin(sin(x))) + sin(sin(sin(x)))sin(sin(x))cos^{3}(x)cos(sin(x)) + 2sin(x)sin(sin(x))cos(sin(sin(x)))cos(x)\right)}{dx}\\=&--sin(sin(sin(x)))cos(sin(x))cos(x)cos^{3}(x)cos^{3}(sin(x)) - cos(sin(sin(x)))*-3cos^{2}(x)sin(x)cos^{3}(sin(x)) - cos(sin(sin(x)))cos^{3}(x)*-3cos^{2}(sin(x))sin(sin(x))cos(x) + 2cos(x)sin(sin(sin(x)))cos(x)cos^{2}(sin(x)) + 2sin(x)cos(sin(sin(x)))cos(sin(x))cos(x)cos(x)cos^{2}(sin(x)) + 2sin(x)sin(sin(sin(x)))*-sin(x)cos^{2}(sin(x)) + 2sin(x)sin(sin(sin(x)))cos(x)*-2cos(sin(x))sin(sin(x))cos(x) + 2cos(sin(sin(x)))cos(sin(x))cos(x)sin(sin(x))cos(sin(x))cos^{3}(x) + 2sin(sin(sin(x)))cos(sin(x))cos(x)cos(sin(x))cos^{3}(x) + 2sin(sin(sin(x)))sin(sin(x))*-sin(sin(x))cos(x)cos^{3}(x) + 2sin(sin(sin(x)))sin(sin(x))cos(sin(x))*-3cos^{2}(x)sin(x) - -sin(x)cos(sin(sin(x)))cos(sin(x)) - cos(x)*-sin(sin(sin(x)))cos(sin(x))cos(x)cos(sin(x)) - cos(x)cos(sin(sin(x)))*-sin(sin(x))cos(x) + cos(sin(sin(x)))cos(sin(x))cos(x)sin(x)cos(x)cos^{2}(sin(x)) + sin(sin(sin(x)))cos(x)cos(x)cos^{2}(sin(x)) + sin(sin(sin(x)))sin(x)*-sin(x)cos^{2}(sin(x)) + sin(sin(sin(x)))sin(x)cos(x)*-2cos(sin(x))sin(sin(x))cos(x) + cos(x)sin(sin(x))cos(x)cos(sin(sin(x))) + sin(x)cos(sin(x))cos(x)cos(x)cos(sin(sin(x))) + sin(x)sin(sin(x))*-sin(x)cos(sin(sin(x))) + sin(x)sin(sin(x))cos(x)*-sin(sin(sin(x)))cos(sin(x))cos(x) - -3cos^{2}(x)sin(x)cos(sin(x))cos(sin(sin(x))) - cos^{3}(x)*-sin(sin(x))cos(x)cos(sin(sin(x))) - cos^{3}(x)cos(sin(x))*-sin(sin(sin(x)))cos(sin(x))cos(x) + cos(sin(sin(x)))cos(sin(x))cos(x)sin(sin(x))cos^{3}(x)cos(sin(x)) + sin(sin(sin(x)))cos(sin(x))cos(x)cos^{3}(x)cos(sin(x)) + sin(sin(sin(x)))sin(sin(x))*-3cos^{2}(x)sin(x)cos(sin(x)) + sin(sin(sin(x)))sin(sin(x))cos^{3}(x)*-sin(sin(x))cos(x) + 2cos(x)sin(sin(x))cos(sin(sin(x)))cos(x) + 2sin(x)cos(sin(x))cos(x)cos(sin(sin(x)))cos(x) + 2sin(x)sin(sin(x))*-sin(sin(sin(x)))cos(sin(x))cos(x)cos(x) + 2sin(x)sin(sin(x))cos(sin(sin(x)))*-sin(x)\\=&5sin(x)cos(sin(sin(x)))cos^{2}(x)cos^{3}(sin(x)) + 3sin(sin(x))cos^{4}(x)cos^{2}(sin(x))cos(sin(sin(x))) + sin(x)cos^{2}(x)cos^{3}(sin(x))cos(sin(sin(x))) + 6sin(x)cos^{2}(x)cos(sin(x))cos(sin(sin(x))) + 3sin(sin(x))cos^{4}(x)cos(sin(sin(x)))cos^{2}(sin(x)) - 6sin(sin(sin(x)))sin(x)sin(sin(x))cos(sin(x))cos^{2}(x) - 4sin(sin(x))sin(sin(sin(x)))sin(x)cos(sin(x))cos^{2}(x) + sin(sin(sin(x)))cos^{4}(x)cos^{4}(sin(x)) + 4sin(sin(sin(x)))cos^{2}(x)cos^{2}(sin(x)) - 2sin(sin(x))sin(x)sin(sin(sin(x)))cos(sin(x))cos^{2}(x) - 3sin(sin(sin(x)))sin(x)sin(sin(x))cos^{2}(x)cos(sin(x)) + 3sin(sin(sin(x)))cos^{4}(x)cos^{2}(sin(x)) - sin(sin(sin(x)))sin^{2}(x)cos^{2}(sin(x)) - 3sin^{2}(sin(x))sin(sin(sin(x)))cos^{4}(x) + 4sin(sin(x))cos^{2}(x)cos(sin(sin(x))) - 3sin^{2}(x)sin(sin(x))cos(sin(sin(x))) - 3sin(sin(x))sin(x)sin(sin(sin(x)))cos^{2}(x)cos(sin(x)) + sin(sin(x))cos^{4}(x)cos(sin(sin(x))) + sin(sin(sin(x)))cos^{2}(sin(x))cos^{4}(x) - 2sin^{2}(x)sin(sin(sin(x)))cos^{2}(sin(x)) + sin(x)cos(sin(sin(x)))cos(sin(x))\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/2]Find\ the\ 4th\ derivative\ of\ function\ cos(cos(cos(x)))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( cos(cos(cos(x)))\right)}{dx}\\=&-sin(cos(cos(x)))*-sin(cos(x))*-sin(x)\\=&-sin(cos(x))sin(x)sin(cos(cos(x)))\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( -sin(cos(x))sin(x)sin(cos(cos(x)))\right)}{dx}\\=&-cos(cos(x))*-sin(x)sin(x)sin(cos(cos(x))) - sin(cos(x))cos(x)sin(cos(cos(x))) - sin(cos(x))sin(x)cos(cos(cos(x)))*-sin(cos(x))*-sin(x)\\=&sin^{2}(x)sin(cos(cos(x)))cos(cos(x)) - sin(cos(x))sin(cos(cos(x)))cos(x) - sin^{2}(cos(x))sin^{2}(x)cos(cos(cos(x)))\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( sin^{2}(x)sin(cos(cos(x)))cos(cos(x)) - sin(cos(x))sin(cos(cos(x)))cos(x) - sin^{2}(cos(x))sin^{2}(x)cos(cos(cos(x)))\right)}{dx}\\=&2sin(x)cos(x)sin(cos(cos(x)))cos(cos(x)) + sin^{2}(x)cos(cos(cos(x)))*-sin(cos(x))*-sin(x)cos(cos(x)) + sin^{2}(x)sin(cos(cos(x)))*-sin(cos(x))*-sin(x) - cos(cos(x))*-sin(x)sin(cos(cos(x)))cos(x) - sin(cos(x))cos(cos(cos(x)))*-sin(cos(x))*-sin(x)cos(x) - sin(cos(x))sin(cos(cos(x)))*-sin(x) - 2sin(cos(x))cos(cos(x))*-sin(x)sin^{2}(x)cos(cos(cos(x))) - sin^{2}(cos(x))*2sin(x)cos(x)cos(cos(cos(x))) - sin^{2}(cos(x))sin^{2}(x)*-sin(cos(cos(x)))*-sin(cos(x))*-sin(x)\\=&2sin(x)sin(cos(cos(x)))cos(x)cos(cos(x)) + sin(cos(x))sin^{3}(x)cos(cos(cos(x)))cos(cos(x)) + sin(cos(cos(x)))sin^{3}(x)sin(cos(x)) + sin(x)sin(cos(cos(x)))cos(cos(x))cos(x) - sin^{2}(cos(x))sin(x)cos(cos(cos(x)))cos(x) + sin(cos(x))sin(x)sin(cos(cos(x))) + 2sin^{3}(x)sin(cos(x))cos(cos(x))cos(cos(cos(x))) - 2sin(x)sin^{2}(cos(x))cos(x)cos(cos(cos(x))) + sin^{3}(cos(x))sin(cos(cos(x)))sin^{3}(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 2sin(x)sin(cos(cos(x)))cos(x)cos(cos(x)) + sin(cos(x))sin^{3}(x)cos(cos(cos(x)))cos(cos(x)) + sin(cos(cos(x)))sin^{3}(x)sin(cos(x)) + sin(x)sin(cos(cos(x)))cos(cos(x))cos(x) - sin^{2}(cos(x))sin(x)cos(cos(cos(x)))cos(x) + sin(cos(x))sin(x)sin(cos(cos(x))) + 2sin^{3}(x)sin(cos(x))cos(cos(x))cos(cos(cos(x))) - 2sin(x)sin^{2}(cos(x))cos(x)cos(cos(cos(x))) + sin^{3}(cos(x))sin(cos(cos(x)))sin^{3}(x)\right)}{dx}\\=&2cos(x)sin(cos(cos(x)))cos(x)cos(cos(x)) + 2sin(x)cos(cos(cos(x)))*-sin(cos(x))*-sin(x)cos(x)cos(cos(x)) + 2sin(x)sin(cos(cos(x)))*-sin(x)cos(cos(x)) + 2sin(x)sin(cos(cos(x)))cos(x)*-sin(cos(x))*-sin(x) + cos(cos(x))*-sin(x)sin^{3}(x)cos(cos(cos(x)))cos(cos(x)) + sin(cos(x))*3sin^{2}(x)cos(x)cos(cos(cos(x)))cos(cos(x)) + sin(cos(x))sin^{3}(x)*-sin(cos(cos(x)))*-sin(cos(x))*-sin(x)cos(cos(x)) + sin(cos(x))sin^{3}(x)cos(cos(cos(x)))*-sin(cos(x))*-sin(x) + cos(cos(cos(x)))*-sin(cos(x))*-sin(x)sin^{3}(x)sin(cos(x)) + sin(cos(cos(x)))*3sin^{2}(x)cos(x)sin(cos(x)) + sin(cos(cos(x)))sin^{3}(x)cos(cos(x))*-sin(x) + cos(x)sin(cos(cos(x)))cos(cos(x))cos(x) + sin(x)cos(cos(cos(x)))*-sin(cos(x))*-sin(x)cos(cos(x))cos(x) + sin(x)sin(cos(cos(x)))*-sin(cos(x))*-sin(x)cos(x) + sin(x)sin(cos(cos(x)))cos(cos(x))*-sin(x) - 2sin(cos(x))cos(cos(x))*-sin(x)sin(x)cos(cos(cos(x)))cos(x) - sin^{2}(cos(x))cos(x)cos(cos(cos(x)))cos(x) - sin^{2}(cos(x))sin(x)*-sin(cos(cos(x)))*-sin(cos(x))*-sin(x)cos(x) - sin^{2}(cos(x))sin(x)cos(cos(cos(x)))*-sin(x) + cos(cos(x))*-sin(x)sin(x)sin(cos(cos(x))) + sin(cos(x))cos(x)sin(cos(cos(x))) + sin(cos(x))sin(x)cos(cos(cos(x)))*-sin(cos(x))*-sin(x) + 2*3sin^{2}(x)cos(x)sin(cos(x))cos(cos(x))cos(cos(cos(x))) + 2sin^{3}(x)cos(cos(x))*-sin(x)cos(cos(x))cos(cos(cos(x))) + 2sin^{3}(x)sin(cos(x))*-sin(cos(x))*-sin(x)cos(cos(cos(x))) + 2sin^{3}(x)sin(cos(x))cos(cos(x))*-sin(cos(cos(x)))*-sin(cos(x))*-sin(x) - 2cos(x)sin^{2}(cos(x))cos(x)cos(cos(cos(x))) - 2sin(x)*2sin(cos(x))cos(cos(x))*-sin(x)cos(x)cos(cos(cos(x))) - 2sin(x)sin^{2}(cos(x))*-sin(x)cos(cos(cos(x))) - 2sin(x)sin^{2}(cos(x))cos(x)*-sin(cos(cos(x)))*-sin(cos(x))*-sin(x) + 3sin^{2}(cos(x))cos(cos(x))*-sin(x)sin(cos(cos(x)))sin^{3}(x) + sin^{3}(cos(x))cos(cos(cos(x)))*-sin(cos(x))*-sin(x)sin^{3}(x) + sin^{3}(cos(x))sin(cos(cos(x)))*3sin^{2}(x)cos(x)\\=&3sin(cos(cos(x)))cos^{2}(x)cos(cos(x)) + 2sin(cos(x))sin^{2}(x)cos(cos(cos(x)))cos(x)cos(cos(x)) + 3sin^{2}(x)sin(cos(x))cos(x)cos(cos(cos(x)))cos(cos(x)) + 3sin(cos(cos(x)))sin^{2}(x)sin(cos(x))cos(x) - 3sin^{4}(x)cos^{2}(cos(x))cos(cos(cos(x))) + sin(cos(x))sin^{2}(x)cos(cos(cos(x)))cos(cos(x))cos(x) - 3sin^{2}(cos(x))sin(cos(cos(x)))sin^{4}(x)cos(cos(x)) + 2sin^{2}(x)sin(cos(x))cos(cos(x))cos(cos(cos(x)))cos(x) + 3sin^{2}(x)sin(cos(cos(x)))sin(cos(x))cos(x) + 6sin^{2}(x)sin(cos(x))cos(x)cos(cos(x))cos(cos(cos(x))) + 4sin(cos(x))sin^{2}(x)cos(cos(x))cos(x)cos(cos(cos(x))) + 2sin^{4}(x)sin^{2}(cos(x))cos(cos(cos(x))) - 3sin^{2}(cos(x))cos^{2}(x)cos(cos(cos(x))) + 3sin^{3}(cos(x))sin(cos(cos(x)))sin^{2}(x)cos(x) + 2sin^{2}(cos(x))sin^{2}(x)cos(cos(cos(x))) + sin(cos(x))sin(cos(cos(x)))cos(x) - sin^{4}(x)sin(cos(cos(x)))cos(cos(x)) + 2sin^{2}(cos(x))sin^{4}(x)cos(cos(cos(x))) - 4sin^{2}(x)sin(cos(cos(x)))cos(cos(x)) + 2sin^{2}(x)sin^{2}(cos(x))cos(cos(cos(x))) - 3sin^{4}(x)sin^{2}(cos(x))sin(cos(cos(x)))cos(cos(x)) + sin^{4}(cos(x))sin^{4}(x)cos(cos(cos(x))) + 3sin^{3}(cos(x))sin^{2}(x)sin(cos(cos(x)))cos(x)\\ \end{split}\end{equation} \]