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.
\[ \begin{equation}\begin{split}[1/3]Find\ the\ 4th\ derivative\ of\ function\ {sin({sin(x)}^{2})}^{2}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = sin^{2}(sin^{2}(x))\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( sin^{2}(sin^{2}(x))\right)}{dx}\\=&2sin(sin^{2}(x))cos(sin^{2}(x))*2sin(x)cos(x)\\=&4sin(x)sin(sin^{2}(x))cos(x)cos(sin^{2}(x))\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( 4sin(x)sin(sin^{2}(x))cos(x)cos(sin^{2}(x))\right)}{dx}\\=&4cos(x)sin(sin^{2}(x))cos(x)cos(sin^{2}(x)) + 4sin(x)cos(sin^{2}(x))*2sin(x)cos(x)cos(x)cos(sin^{2}(x)) + 4sin(x)sin(sin^{2}(x))*-sin(x)cos(sin^{2}(x)) + 4sin(x)sin(sin^{2}(x))cos(x)*-sin(sin^{2}(x))*2sin(x)cos(x)\\=&4sin(sin^{2}(x))cos^{2}(x)cos(sin^{2}(x)) + 8sin^{2}(x)cos^{2}(x)cos^{2}(sin^{2}(x)) - 4sin^{2}(x)sin(sin^{2}(x))cos(sin^{2}(x)) - 8sin^{2}(sin^{2}(x))sin^{2}(x)cos^{2}(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 4sin(sin^{2}(x))cos^{2}(x)cos(sin^{2}(x)) + 8sin^{2}(x)cos^{2}(x)cos^{2}(sin^{2}(x)) - 4sin^{2}(x)sin(sin^{2}(x))cos(sin^{2}(x)) - 8sin^{2}(sin^{2}(x))sin^{2}(x)cos^{2}(x)\right)}{dx}\\=&4cos(sin^{2}(x))*2sin(x)cos(x)cos^{2}(x)cos(sin^{2}(x)) + 4sin(sin^{2}(x))*-2cos(x)sin(x)cos(sin^{2}(x)) + 4sin(sin^{2}(x))cos^{2}(x)*-sin(sin^{2}(x))*2sin(x)cos(x) + 8*2sin(x)cos(x)cos^{2}(x)cos^{2}(sin^{2}(x)) + 8sin^{2}(x)*-2cos(x)sin(x)cos^{2}(sin^{2}(x)) + 8sin^{2}(x)cos^{2}(x)*-2cos(sin^{2}(x))sin(sin^{2}(x))*2sin(x)cos(x) - 4*2sin(x)cos(x)sin(sin^{2}(x))cos(sin^{2}(x)) - 4sin^{2}(x)cos(sin^{2}(x))*2sin(x)cos(x)cos(sin^{2}(x)) - 4sin^{2}(x)sin(sin^{2}(x))*-sin(sin^{2}(x))*2sin(x)cos(x) - 8*2sin(sin^{2}(x))cos(sin^{2}(x))*2sin(x)cos(x)sin^{2}(x)cos^{2}(x) - 8sin^{2}(sin^{2}(x))*2sin(x)cos(x)cos^{2}(x) - 8sin^{2}(sin^{2}(x))sin^{2}(x)*-2cos(x)sin(x)\\=&24sin(x)cos^{3}(x)cos^{2}(sin^{2}(x)) - 16sin(x)sin(sin^{2}(x))cos(x)cos(sin^{2}(x)) - 32sin(sin^{2}(x))sin^{3}(x)cos(sin^{2}(x))cos^{3}(x) - 24sin^{3}(x)cos(x)cos^{2}(sin^{2}(x)) - 32sin^{3}(x)sin(sin^{2}(x))cos(sin^{2}(x))cos^{3}(x) + 24sin^{2}(sin^{2}(x))sin^{3}(x)cos(x) - 8sin^{2}(sin^{2}(x))sin(x)cos^{3}(x) - 16sin(x)sin^{2}(sin^{2}(x))cos^{3}(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 24sin(x)cos^{3}(x)cos^{2}(sin^{2}(x)) - 16sin(x)sin(sin^{2}(x))cos(x)cos(sin^{2}(x)) - 32sin(sin^{2}(x))sin^{3}(x)cos(sin^{2}(x))cos^{3}(x) - 24sin^{3}(x)cos(x)cos^{2}(sin^{2}(x)) - 32sin^{3}(x)sin(sin^{2}(x))cos(sin^{2}(x))cos^{3}(x) + 24sin^{2}(sin^{2}(x))sin^{3}(x)cos(x) - 8sin^{2}(sin^{2}(x))sin(x)cos^{3}(x) - 16sin(x)sin^{2}(sin^{2}(x))cos^{3}(x)\right)}{dx}\\=&24cos(x)cos^{3}(x)cos^{2}(sin^{2}(x)) + 24sin(x)*-3cos^{2}(x)sin(x)cos^{2}(sin^{2}(x)) + 24sin(x)cos^{3}(x)*-2cos(sin^{2}(x))sin(sin^{2}(x))*2sin(x)cos(x) - 16cos(x)sin(sin^{2}(x))cos(x)cos(sin^{2}(x)) - 16sin(x)cos(sin^{2}(x))*2sin(x)cos(x)cos(x)cos(sin^{2}(x)) - 16sin(x)sin(sin^{2}(x))*-sin(x)cos(sin^{2}(x)) - 16sin(x)sin(sin^{2}(x))cos(x)*-sin(sin^{2}(x))*2sin(x)cos(x) - 32cos(sin^{2}(x))*2sin(x)cos(x)sin^{3}(x)cos(sin^{2}(x))cos^{3}(x) - 32sin(sin^{2}(x))*3sin^{2}(x)cos(x)cos(sin^{2}(x))cos^{3}(x) - 32sin(sin^{2}(x))sin^{3}(x)*-sin(sin^{2}(x))*2sin(x)cos(x)cos^{3}(x) - 32sin(sin^{2}(x))sin^{3}(x)cos(sin^{2}(x))*-3cos^{2}(x)sin(x) - 24*3sin^{2}(x)cos(x)cos(x)cos^{2}(sin^{2}(x)) - 24sin^{3}(x)*-sin(x)cos^{2}(sin^{2}(x)) - 24sin^{3}(x)cos(x)*-2cos(sin^{2}(x))sin(sin^{2}(x))*2sin(x)cos(x) - 32*3sin^{2}(x)cos(x)sin(sin^{2}(x))cos(sin^{2}(x))cos^{3}(x) - 32sin^{3}(x)cos(sin^{2}(x))*2sin(x)cos(x)cos(sin^{2}(x))cos^{3}(x) - 32sin^{3}(x)sin(sin^{2}(x))*-sin(sin^{2}(x))*2sin(x)cos(x)cos^{3}(x) - 32sin^{3}(x)sin(sin^{2}(x))cos(sin^{2}(x))*-3cos^{2}(x)sin(x) + 24*2sin(sin^{2}(x))cos(sin^{2}(x))*2sin(x)cos(x)sin^{3}(x)cos(x) + 24sin^{2}(sin^{2}(x))*3sin^{2}(x)cos(x)cos(x) + 24sin^{2}(sin^{2}(x))sin^{3}(x)*-sin(x) - 8*2sin(sin^{2}(x))cos(sin^{2}(x))*2sin(x)cos(x)sin(x)cos^{3}(x) - 8sin^{2}(sin^{2}(x))cos(x)cos^{3}(x) - 8sin^{2}(sin^{2}(x))sin(x)*-3cos^{2}(x)sin(x) - 16cos(x)sin^{2}(sin^{2}(x))cos^{3}(x) - 16sin(x)*2sin(sin^{2}(x))cos(sin^{2}(x))*2sin(x)cos(x)cos^{3}(x) - 16sin(x)sin^{2}(sin^{2}(x))*-3cos^{2}(x)sin(x)\\=&24cos^{4}(x)cos^{2}(sin^{2}(x)) - 192sin^{2}(x)sin(sin^{2}(x))cos^{4}(x)cos(sin^{2}(x)) - 96sin(sin^{2}(x))sin^{2}(x)cos(sin^{2}(x))cos^{4}(x) - 16sin(sin^{2}(x))cos^{2}(x)cos(sin^{2}(x)) - 176sin^{2}(x)cos^{2}(x)cos^{2}(sin^{2}(x)) + 192sin(sin^{2}(x))sin^{4}(x)cos(sin^{2}(x))cos^{2}(x) + 192sin^{4}(x)sin(sin^{2}(x))cos(sin^{2}(x))cos^{2}(x) - 64sin^{4}(x)cos^{2}(sin^{2}(x))cos^{4}(x) - 32sin^{2}(x)sin(sin^{2}(x))cos(sin^{2}(x))cos^{4}(x) - 64sin(sin^{2}(x))sin^{2}(x)cos^{4}(x)cos(sin^{2}(x)) + 120sin^{2}(x)sin^{2}(sin^{2}(x))cos^{2}(x) - 64sin^{4}(x)cos^{4}(x)cos^{2}(sin^{2}(x)) + 64sin^{2}(sin^{2}(x))sin^{4}(x)cos^{4}(x) + 16sin^{2}(x)sin(sin^{2}(x))cos(sin^{2}(x)) + 24sin^{4}(x)cos^{2}(sin^{2}(x)) + 64sin^{4}(x)sin^{2}(sin^{2}(x))cos^{4}(x) - 24sin^{2}(sin^{2}(x))cos^{4}(x) + 56sin^{2}(sin^{2}(x))sin^{2}(x)cos^{2}(x) - 24sin^{2}(sin^{2}(x))sin^{4}(x)\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/3]Find\ the\ 4th\ derivative\ of\ function\ {cos({cos(x)}^{2})}^{2}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = cos^{2}(cos^{2}(x))\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( cos^{2}(cos^{2}(x))\right)}{dx}\\=&-2cos(cos^{2}(x))sin(cos^{2}(x))*-2cos(x)sin(x)\\=&4sin(x)sin(cos^{2}(x))cos(cos^{2}(x))cos(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( 4sin(x)sin(cos^{2}(x))cos(cos^{2}(x))cos(x)\right)}{dx}\\=&4cos(x)sin(cos^{2}(x))cos(cos^{2}(x))cos(x) + 4sin(x)cos(cos^{2}(x))*-2cos(x)sin(x)cos(cos^{2}(x))cos(x) + 4sin(x)sin(cos^{2}(x))*-sin(cos^{2}(x))*-2cos(x)sin(x)cos(x) + 4sin(x)sin(cos^{2}(x))cos(cos^{2}(x))*-sin(x)\\=&4sin(cos^{2}(x))cos^{2}(x)cos(cos^{2}(x)) - 8sin^{2}(x)cos^{2}(cos^{2}(x))cos^{2}(x) + 8sin^{2}(cos^{2}(x))sin^{2}(x)cos^{2}(x) - 4sin^{2}(x)sin(cos^{2}(x))cos(cos^{2}(x))\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 4sin(cos^{2}(x))cos^{2}(x)cos(cos^{2}(x)) - 8sin^{2}(x)cos^{2}(cos^{2}(x))cos^{2}(x) + 8sin^{2}(cos^{2}(x))sin^{2}(x)cos^{2}(x) - 4sin^{2}(x)sin(cos^{2}(x))cos(cos^{2}(x))\right)}{dx}\\=&4cos(cos^{2}(x))*-2cos(x)sin(x)cos^{2}(x)cos(cos^{2}(x)) + 4sin(cos^{2}(x))*-2cos(x)sin(x)cos(cos^{2}(x)) + 4sin(cos^{2}(x))cos^{2}(x)*-sin(cos^{2}(x))*-2cos(x)sin(x) - 8*2sin(x)cos(x)cos^{2}(cos^{2}(x))cos^{2}(x) - 8sin^{2}(x)*-2cos(cos^{2}(x))sin(cos^{2}(x))*-2cos(x)sin(x)cos^{2}(x) - 8sin^{2}(x)cos^{2}(cos^{2}(x))*-2cos(x)sin(x) + 8*2sin(cos^{2}(x))cos(cos^{2}(x))*-2cos(x)sin(x)sin^{2}(x)cos^{2}(x) + 8sin^{2}(cos^{2}(x))*2sin(x)cos(x)cos^{2}(x) + 8sin^{2}(cos^{2}(x))sin^{2}(x)*-2cos(x)sin(x) - 4*2sin(x)cos(x)sin(cos^{2}(x))cos(cos^{2}(x)) - 4sin^{2}(x)cos(cos^{2}(x))*-2cos(x)sin(x)cos(cos^{2}(x)) - 4sin^{2}(x)sin(cos^{2}(x))*-sin(cos^{2}(x))*-2cos(x)sin(x)\\=&-8sin(x)cos^{2}(cos^{2}(x))cos^{3}(x) - 16sin(x)sin(cos^{2}(x))cos(x)cos(cos^{2}(x)) - 32sin(cos^{2}(x))sin^{3}(x)cos(cos^{2}(x))cos^{3}(x) - 16sin(x)cos^{3}(x)cos^{2}(cos^{2}(x)) - 32sin^{3}(x)sin(cos^{2}(x))cos^{3}(x)cos(cos^{2}(x)) + 24sin^{3}(x)cos^{2}(cos^{2}(x))cos(x) + 8sin^{2}(cos^{2}(x))sin(x)cos^{3}(x) + 16sin(x)sin^{2}(cos^{2}(x))cos^{3}(x) - 24sin^{2}(cos^{2}(x))sin^{3}(x)cos(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( -8sin(x)cos^{2}(cos^{2}(x))cos^{3}(x) - 16sin(x)sin(cos^{2}(x))cos(x)cos(cos^{2}(x)) - 32sin(cos^{2}(x))sin^{3}(x)cos(cos^{2}(x))cos^{3}(x) - 16sin(x)cos^{3}(x)cos^{2}(cos^{2}(x)) - 32sin^{3}(x)sin(cos^{2}(x))cos^{3}(x)cos(cos^{2}(x)) + 24sin^{3}(x)cos^{2}(cos^{2}(x))cos(x) + 8sin^{2}(cos^{2}(x))sin(x)cos^{3}(x) + 16sin(x)sin^{2}(cos^{2}(x))cos^{3}(x) - 24sin^{2}(cos^{2}(x))sin^{3}(x)cos(x)\right)}{dx}\\=&-8cos(x)cos^{2}(cos^{2}(x))cos^{3}(x) - 8sin(x)*-2cos(cos^{2}(x))sin(cos^{2}(x))*-2cos(x)sin(x)cos^{3}(x) - 8sin(x)cos^{2}(cos^{2}(x))*-3cos^{2}(x)sin(x) - 16cos(x)sin(cos^{2}(x))cos(x)cos(cos^{2}(x)) - 16sin(x)cos(cos^{2}(x))*-2cos(x)sin(x)cos(x)cos(cos^{2}(x)) - 16sin(x)sin(cos^{2}(x))*-sin(x)cos(cos^{2}(x)) - 16sin(x)sin(cos^{2}(x))cos(x)*-sin(cos^{2}(x))*-2cos(x)sin(x) - 32cos(cos^{2}(x))*-2cos(x)sin(x)sin^{3}(x)cos(cos^{2}(x))cos^{3}(x) - 32sin(cos^{2}(x))*3sin^{2}(x)cos(x)cos(cos^{2}(x))cos^{3}(x) - 32sin(cos^{2}(x))sin^{3}(x)*-sin(cos^{2}(x))*-2cos(x)sin(x)cos^{3}(x) - 32sin(cos^{2}(x))sin^{3}(x)cos(cos^{2}(x))*-3cos^{2}(x)sin(x) - 16cos(x)cos^{3}(x)cos^{2}(cos^{2}(x)) - 16sin(x)*-3cos^{2}(x)sin(x)cos^{2}(cos^{2}(x)) - 16sin(x)cos^{3}(x)*-2cos(cos^{2}(x))sin(cos^{2}(x))*-2cos(x)sin(x) - 32*3sin^{2}(x)cos(x)sin(cos^{2}(x))cos^{3}(x)cos(cos^{2}(x)) - 32sin^{3}(x)cos(cos^{2}(x))*-2cos(x)sin(x)cos^{3}(x)cos(cos^{2}(x)) - 32sin^{3}(x)sin(cos^{2}(x))*-3cos^{2}(x)sin(x)cos(cos^{2}(x)) - 32sin^{3}(x)sin(cos^{2}(x))cos^{3}(x)*-sin(cos^{2}(x))*-2cos(x)sin(x) + 24*3sin^{2}(x)cos(x)cos^{2}(cos^{2}(x))cos(x) + 24sin^{3}(x)*-2cos(cos^{2}(x))sin(cos^{2}(x))*-2cos(x)sin(x)cos(x) + 24sin^{3}(x)cos^{2}(cos^{2}(x))*-sin(x) + 8*2sin(cos^{2}(x))cos(cos^{2}(x))*-2cos(x)sin(x)sin(x)cos^{3}(x) + 8sin^{2}(cos^{2}(x))cos(x)cos^{3}(x) + 8sin^{2}(cos^{2}(x))sin(x)*-3cos^{2}(x)sin(x) + 16cos(x)sin^{2}(cos^{2}(x))cos^{3}(x) + 16sin(x)*2sin(cos^{2}(x))cos(cos^{2}(x))*-2cos(x)sin(x)cos^{3}(x) + 16sin(x)sin^{2}(cos^{2}(x))*-3cos^{2}(x)sin(x) - 24*2sin(cos^{2}(x))cos(cos^{2}(x))*-2cos(x)sin(x)sin^{3}(x)cos(x) - 24sin^{2}(cos^{2}(x))*3sin^{2}(x)cos(x)cos(x) - 24sin^{2}(cos^{2}(x))sin^{3}(x)*-sin(x)\\=&-24cos^{4}(x)cos^{2}(cos^{2}(x)) - 96sin(cos^{2}(x))sin^{2}(x)cos(cos^{2}(x))cos^{4}(x) - 224sin^{2}(x)sin(cos^{2}(x))cos^{4}(x)cos(cos^{2}(x)) - 16sin(cos^{2}(x))cos^{2}(x)cos(cos^{2}(x)) + 56sin^{2}(x)cos^{2}(cos^{2}(x))cos^{2}(x) + 192sin(cos^{2}(x))sin^{4}(x)cos(cos^{2}(x))cos^{2}(x) + 192sin^{4}(x)sin(cos^{2}(x))cos^{2}(x)cos(cos^{2}(x)) + 64sin^{4}(x)cos^{4}(x)cos^{2}(cos^{2}(x)) - 64sin(cos^{2}(x))sin^{2}(x)cos^{4}(x)cos(cos^{2}(x)) - 56sin^{2}(cos^{2}(x))sin^{2}(x)cos^{2}(x) + 120sin^{2}(x)cos^{2}(x)cos^{2}(cos^{2}(x)) - 64sin^{4}(x)sin^{2}(cos^{2}(x))cos^{4}(x) + 64sin^{4}(x)cos^{2}(cos^{2}(x))cos^{4}(x) + 16sin^{2}(x)sin(cos^{2}(x))cos(cos^{2}(x)) - 64sin^{2}(cos^{2}(x))sin^{4}(x)cos^{4}(x) - 120sin^{2}(x)sin^{2}(cos^{2}(x))cos^{2}(x) + 24sin^{2}(cos^{2}(x))cos^{4}(x) - 24sin^{4}(x)cos^{2}(cos^{2}(x)) + 24sin^{2}(cos^{2}(x))sin^{4}(x)\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[3/3]Find\ the\ 4th\ derivative\ of\ function\ {tan({tan(x)}^{2})}^{2}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = tan^{2}(tan^{2}(x))\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( tan^{2}(tan^{2}(x))\right)}{dx}\\=&2tan(tan^{2}(x))sec^{2}(tan^{2}(x))(2tan(x)sec^{2}(x)(1))\\=&4tan(x)tan(tan^{2}(x))sec^{2}(x)sec^{2}(tan^{2}(x))\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( 4tan(x)tan(tan^{2}(x))sec^{2}(x)sec^{2}(tan^{2}(x))\right)}{dx}\\=&4sec^{2}(x)(1)tan(tan^{2}(x))sec^{2}(x)sec^{2}(tan^{2}(x)) + 4tan(x)sec^{2}(tan^{2}(x))(2tan(x)sec^{2}(x)(1))sec^{2}(x)sec^{2}(tan^{2}(x)) + 4tan(x)tan(tan^{2}(x))*2sec^{2}(x)tan(x)sec^{2}(tan^{2}(x)) + 4tan(x)tan(tan^{2}(x))sec^{2}(x)*2sec^{2}(tan^{2}(x))tan(tan^{2}(x))*2tan(x)sec^{2}(x)(1)\\=&4tan(tan^{2}(x))sec^{4}(x)sec^{2}(tan^{2}(x)) + 8tan^{2}(x)sec^{4}(x)sec^{4}(tan^{2}(x)) + 8tan^{2}(x)tan(tan^{2}(x))sec^{2}(x)sec^{2}(tan^{2}(x)) + 16tan^{2}(tan^{2}(x))tan^{2}(x)sec^{2}(tan^{2}(x))sec^{4}(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 4tan(tan^{2}(x))sec^{4}(x)sec^{2}(tan^{2}(x)) + 8tan^{2}(x)sec^{4}(x)sec^{4}(tan^{2}(x)) + 8tan^{2}(x)tan(tan^{2}(x))sec^{2}(x)sec^{2}(tan^{2}(x)) + 16tan^{2}(tan^{2}(x))tan^{2}(x)sec^{2}(tan^{2}(x))sec^{4}(x)\right)}{dx}\\=&4sec^{2}(tan^{2}(x))(2tan(x)sec^{2}(x)(1))sec^{4}(x)sec^{2}(tan^{2}(x)) + 4tan(tan^{2}(x))*4sec^{4}(x)tan(x)sec^{2}(tan^{2}(x)) + 4tan(tan^{2}(x))sec^{4}(x)*2sec^{2}(tan^{2}(x))tan(tan^{2}(x))*2tan(x)sec^{2}(x)(1) + 8*2tan(x)sec^{2}(x)(1)sec^{4}(x)sec^{4}(tan^{2}(x)) + 8tan^{2}(x)*4sec^{4}(x)tan(x)sec^{4}(tan^{2}(x)) + 8tan^{2}(x)sec^{4}(x)*4sec^{4}(tan^{2}(x))tan(tan^{2}(x))*2tan(x)sec^{2}(x)(1) + 8*2tan(x)sec^{2}(x)(1)tan(tan^{2}(x))sec^{2}(x)sec^{2}(tan^{2}(x)) + 8tan^{2}(x)sec^{2}(tan^{2}(x))(2tan(x)sec^{2}(x)(1))sec^{2}(x)sec^{2}(tan^{2}(x)) + 8tan^{2}(x)tan(tan^{2}(x))*2sec^{2}(x)tan(x)sec^{2}(tan^{2}(x)) + 8tan^{2}(x)tan(tan^{2}(x))sec^{2}(x)*2sec^{2}(tan^{2}(x))tan(tan^{2}(x))*2tan(x)sec^{2}(x)(1) + 16*2tan(tan^{2}(x))sec^{2}(tan^{2}(x))(2tan(x)sec^{2}(x)(1))tan^{2}(x)sec^{2}(tan^{2}(x))sec^{4}(x) + 16tan^{2}(tan^{2}(x))*2tan(x)sec^{2}(x)(1)sec^{2}(tan^{2}(x))sec^{4}(x) + 16tan^{2}(tan^{2}(x))tan^{2}(x)*2sec^{2}(tan^{2}(x))tan(tan^{2}(x))*2tan(x)sec^{2}(x)(1)sec^{4}(x) + 16tan^{2}(tan^{2}(x))tan^{2}(x)sec^{2}(tan^{2}(x))*4sec^{4}(x)tan(x)\\=&24tan(x)sec^{6}(x)sec^{4}(tan^{2}(x)) + 32tan(x)tan(tan^{2}(x))sec^{4}(x)sec^{2}(tan^{2}(x)) + 64tan^{3}(x)tan(tan^{2}(x))sec^{4}(tan^{2}(x))sec^{6}(x) + 32tan(x)tan^{2}(tan^{2}(x))sec^{6}(x)sec^{2}(tan^{2}(x)) + 64tan(tan^{2}(x))tan^{3}(x)sec^{4}(tan^{2}(x))sec^{6}(x) + 48tan^{3}(x)sec^{4}(x)sec^{4}(tan^{2}(x)) + 16tan^{3}(x)tan(tan^{2}(x))sec^{2}(x)sec^{2}(tan^{2}(x)) + 96tan^{2}(tan^{2}(x))tan^{3}(x)sec^{2}(tan^{2}(x))sec^{4}(x) + 16tan^{2}(tan^{2}(x))tan(x)sec^{2}(tan^{2}(x))sec^{6}(x) + 64tan^{3}(x)tan^{3}(tan^{2}(x))sec^{6}(x)sec^{2}(tan^{2}(x))\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 24tan(x)sec^{6}(x)sec^{4}(tan^{2}(x)) + 32tan(x)tan(tan^{2}(x))sec^{4}(x)sec^{2}(tan^{2}(x)) + 64tan^{3}(x)tan(tan^{2}(x))sec^{4}(tan^{2}(x))sec^{6}(x) + 32tan(x)tan^{2}(tan^{2}(x))sec^{6}(x)sec^{2}(tan^{2}(x)) + 64tan(tan^{2}(x))tan^{3}(x)sec^{4}(tan^{2}(x))sec^{6}(x) + 48tan^{3}(x)sec^{4}(x)sec^{4}(tan^{2}(x)) + 16tan^{3}(x)tan(tan^{2}(x))sec^{2}(x)sec^{2}(tan^{2}(x)) + 96tan^{2}(tan^{2}(x))tan^{3}(x)sec^{2}(tan^{2}(x))sec^{4}(x) + 16tan^{2}(tan^{2}(x))tan(x)sec^{2}(tan^{2}(x))sec^{6}(x) + 64tan^{3}(x)tan^{3}(tan^{2}(x))sec^{6}(x)sec^{2}(tan^{2}(x))\right)}{dx}\\=&24sec^{2}(x)(1)sec^{6}(x)sec^{4}(tan^{2}(x)) + 24tan(x)*6sec^{6}(x)tan(x)sec^{4}(tan^{2}(x)) + 24tan(x)sec^{6}(x)*4sec^{4}(tan^{2}(x))tan(tan^{2}(x))*2tan(x)sec^{2}(x)(1) + 32sec^{2}(x)(1)tan(tan^{2}(x))sec^{4}(x)sec^{2}(tan^{2}(x)) + 32tan(x)sec^{2}(tan^{2}(x))(2tan(x)sec^{2}(x)(1))sec^{4}(x)sec^{2}(tan^{2}(x)) + 32tan(x)tan(tan^{2}(x))*4sec^{4}(x)tan(x)sec^{2}(tan^{2}(x)) + 32tan(x)tan(tan^{2}(x))sec^{4}(x)*2sec^{2}(tan^{2}(x))tan(tan^{2}(x))*2tan(x)sec^{2}(x)(1) + 64*3tan^{2}(x)sec^{2}(x)(1)tan(tan^{2}(x))sec^{4}(tan^{2}(x))sec^{6}(x) + 64tan^{3}(x)sec^{2}(tan^{2}(x))(2tan(x)sec^{2}(x)(1))sec^{4}(tan^{2}(x))sec^{6}(x) + 64tan^{3}(x)tan(tan^{2}(x))*4sec^{4}(tan^{2}(x))tan(tan^{2}(x))*2tan(x)sec^{2}(x)(1)sec^{6}(x) + 64tan^{3}(x)tan(tan^{2}(x))sec^{4}(tan^{2}(x))*6sec^{6}(x)tan(x) + 32sec^{2}(x)(1)tan^{2}(tan^{2}(x))sec^{6}(x)sec^{2}(tan^{2}(x)) + 32tan(x)*2tan(tan^{2}(x))sec^{2}(tan^{2}(x))(2tan(x)sec^{2}(x)(1))sec^{6}(x)sec^{2}(tan^{2}(x)) + 32tan(x)tan^{2}(tan^{2}(x))*6sec^{6}(x)tan(x)sec^{2}(tan^{2}(x)) + 32tan(x)tan^{2}(tan^{2}(x))sec^{6}(x)*2sec^{2}(tan^{2}(x))tan(tan^{2}(x))*2tan(x)sec^{2}(x)(1) + 64sec^{2}(tan^{2}(x))(2tan(x)sec^{2}(x)(1))tan^{3}(x)sec^{4}(tan^{2}(x))sec^{6}(x) + 64tan(tan^{2}(x))*3tan^{2}(x)sec^{2}(x)(1)sec^{4}(tan^{2}(x))sec^{6}(x) + 64tan(tan^{2}(x))tan^{3}(x)*4sec^{4}(tan^{2}(x))tan(tan^{2}(x))*2tan(x)sec^{2}(x)(1)sec^{6}(x) + 64tan(tan^{2}(x))tan^{3}(x)sec^{4}(tan^{2}(x))*6sec^{6}(x)tan(x) + 48*3tan^{2}(x)sec^{2}(x)(1)sec^{4}(x)sec^{4}(tan^{2}(x)) + 48tan^{3}(x)*4sec^{4}(x)tan(x)sec^{4}(tan^{2}(x)) + 48tan^{3}(x)sec^{4}(x)*4sec^{4}(tan^{2}(x))tan(tan^{2}(x))*2tan(x)sec^{2}(x)(1) + 16*3tan^{2}(x)sec^{2}(x)(1)tan(tan^{2}(x))sec^{2}(x)sec^{2}(tan^{2}(x)) + 16tan^{3}(x)sec^{2}(tan^{2}(x))(2tan(x)sec^{2}(x)(1))sec^{2}(x)sec^{2}(tan^{2}(x)) + 16tan^{3}(x)tan(tan^{2}(x))*2sec^{2}(x)tan(x)sec^{2}(tan^{2}(x)) + 16tan^{3}(x)tan(tan^{2}(x))sec^{2}(x)*2sec^{2}(tan^{2}(x))tan(tan^{2}(x))*2tan(x)sec^{2}(x)(1) + 96*2tan(tan^{2}(x))sec^{2}(tan^{2}(x))(2tan(x)sec^{2}(x)(1))tan^{3}(x)sec^{2}(tan^{2}(x))sec^{4}(x) + 96tan^{2}(tan^{2}(x))*3tan^{2}(x)sec^{2}(x)(1)sec^{2}(tan^{2}(x))sec^{4}(x) + 96tan^{2}(tan^{2}(x))tan^{3}(x)*2sec^{2}(tan^{2}(x))tan(tan^{2}(x))*2tan(x)sec^{2}(x)(1)sec^{4}(x) + 96tan^{2}(tan^{2}(x))tan^{3}(x)sec^{2}(tan^{2}(x))*4sec^{4}(x)tan(x) + 16*2tan(tan^{2}(x))sec^{2}(tan^{2}(x))(2tan(x)sec^{2}(x)(1))tan(x)sec^{2}(tan^{2}(x))sec^{6}(x) + 16tan^{2}(tan^{2}(x))sec^{2}(x)(1)sec^{2}(tan^{2}(x))sec^{6}(x) + 16tan^{2}(tan^{2}(x))tan(x)*2sec^{2}(tan^{2}(x))tan(tan^{2}(x))*2tan(x)sec^{2}(x)(1)sec^{6}(x) + 16tan^{2}(tan^{2}(x))tan(x)sec^{2}(tan^{2}(x))*6sec^{6}(x)tan(x) + 64*3tan^{2}(x)sec^{2}(x)(1)tan^{3}(tan^{2}(x))sec^{6}(x)sec^{2}(tan^{2}(x)) + 64tan^{3}(x)*3tan^{2}(tan^{2}(x))sec^{2}(tan^{2}(x))(2tan(x)sec^{2}(x)(1))sec^{6}(x)sec^{2}(tan^{2}(x)) + 64tan^{3}(x)tan^{3}(tan^{2}(x))*6sec^{6}(x)tan(x)sec^{2}(tan^{2}(x)) + 64tan^{3}(x)tan^{3}(tan^{2}(x))sec^{6}(x)*2sec^{2}(tan^{2}(x))tan(tan^{2}(x))*2tan(x)sec^{2}(x)(1)\\=&24sec^{8}(x)sec^{4}(tan^{2}(x)) + 384tan^{2}(x)tan(tan^{2}(x))sec^{8}(x)sec^{4}(tan^{2}(x)) + 128tan(tan^{2}(x))tan^{2}(x)sec^{8}(x)sec^{4}(tan^{2}(x)) + 32tan(tan^{2}(x))sec^{6}(x)sec^{2}(tan^{2}(x)) + 352tan^{2}(x)sec^{6}(x)sec^{4}(tan^{2}(x)) + 176tan^{2}(x)tan(tan^{2}(x))sec^{4}(x)sec^{2}(tan^{2}(x)) + 224tan^{2}(tan^{2}(x))tan^{2}(x)sec^{2}(tan^{2}(x))sec^{6}(x) + 128tan^{4}(x)sec^{8}(x)sec^{6}(tan^{2}(x)) + 896tan^{2}(tan^{2}(x))tan^{4}(x)sec^{8}(x)sec^{4}(tan^{2}(x)) + 768tan^{4}(x)tan(tan^{2}(x))sec^{4}(tan^{2}(x))sec^{6}(x) + 48tan^{2}(tan^{2}(x))sec^{8}(x)sec^{2}(tan^{2}(x)) + 480tan^{2}(x)tan^{2}(tan^{2}(x))sec^{6}(x)sec^{2}(tan^{2}(x)) + 128tan^{3}(tan^{2}(x))tan^{2}(x)sec^{2}(tan^{2}(x))sec^{8}(x) + 128tan^{4}(x)sec^{6}(tan^{2}(x))sec^{8}(x) + 512tan^{4}(x)tan^{2}(tan^{2}(x))sec^{8}(x)sec^{4}(tan^{2}(x)) + 768tan(tan^{2}(x))tan^{4}(x)sec^{4}(tan^{2}(x))sec^{6}(x) + 64tan^{2}(x)tan(tan^{2}(x))sec^{4}(tan^{2}(x))sec^{8}(x) + 256tan^{2}(x)tan^{3}(tan^{2}(x))sec^{8}(x)sec^{2}(tan^{2}(x)) + 224tan^{4}(x)sec^{4}(x)sec^{4}(tan^{2}(x)) + 32tan^{4}(x)tan(tan^{2}(x))sec^{2}(x)sec^{2}(tan^{2}(x)) + 448tan^{2}(tan^{2}(x))tan^{4}(x)sec^{2}(tan^{2}(x))sec^{4}(x) + 768tan^{4}(x)tan^{3}(tan^{2}(x))sec^{6}(x)sec^{2}(tan^{2}(x)) + 192tan(tan^{2}(x))tan^{2}(x)sec^{4}(tan^{2}(x))sec^{8}(x) + 256tan^{4}(tan^{2}(x))tan^{4}(x)sec^{2}(tan^{2}(x))sec^{8}(x)\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/3]Find\ the\ 4th\ derivative\ of\ function\ {sin({sin(x)}^{2})}^{2}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = sin^{2}(sin^{2}(x))\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( sin^{2}(sin^{2}(x))\right)}{dx}\\=&2sin(sin^{2}(x))cos(sin^{2}(x))*2sin(x)cos(x)\\=&4sin(x)sin(sin^{2}(x))cos(x)cos(sin^{2}(x))\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( 4sin(x)sin(sin^{2}(x))cos(x)cos(sin^{2}(x))\right)}{dx}\\=&4cos(x)sin(sin^{2}(x))cos(x)cos(sin^{2}(x)) + 4sin(x)cos(sin^{2}(x))*2sin(x)cos(x)cos(x)cos(sin^{2}(x)) + 4sin(x)sin(sin^{2}(x))*-sin(x)cos(sin^{2}(x)) + 4sin(x)sin(sin^{2}(x))cos(x)*-sin(sin^{2}(x))*2sin(x)cos(x)\\=&4sin(sin^{2}(x))cos^{2}(x)cos(sin^{2}(x)) + 8sin^{2}(x)cos^{2}(x)cos^{2}(sin^{2}(x)) - 4sin^{2}(x)sin(sin^{2}(x))cos(sin^{2}(x)) - 8sin^{2}(sin^{2}(x))sin^{2}(x)cos^{2}(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 4sin(sin^{2}(x))cos^{2}(x)cos(sin^{2}(x)) + 8sin^{2}(x)cos^{2}(x)cos^{2}(sin^{2}(x)) - 4sin^{2}(x)sin(sin^{2}(x))cos(sin^{2}(x)) - 8sin^{2}(sin^{2}(x))sin^{2}(x)cos^{2}(x)\right)}{dx}\\=&4cos(sin^{2}(x))*2sin(x)cos(x)cos^{2}(x)cos(sin^{2}(x)) + 4sin(sin^{2}(x))*-2cos(x)sin(x)cos(sin^{2}(x)) + 4sin(sin^{2}(x))cos^{2}(x)*-sin(sin^{2}(x))*2sin(x)cos(x) + 8*2sin(x)cos(x)cos^{2}(x)cos^{2}(sin^{2}(x)) + 8sin^{2}(x)*-2cos(x)sin(x)cos^{2}(sin^{2}(x)) + 8sin^{2}(x)cos^{2}(x)*-2cos(sin^{2}(x))sin(sin^{2}(x))*2sin(x)cos(x) - 4*2sin(x)cos(x)sin(sin^{2}(x))cos(sin^{2}(x)) - 4sin^{2}(x)cos(sin^{2}(x))*2sin(x)cos(x)cos(sin^{2}(x)) - 4sin^{2}(x)sin(sin^{2}(x))*-sin(sin^{2}(x))*2sin(x)cos(x) - 8*2sin(sin^{2}(x))cos(sin^{2}(x))*2sin(x)cos(x)sin^{2}(x)cos^{2}(x) - 8sin^{2}(sin^{2}(x))*2sin(x)cos(x)cos^{2}(x) - 8sin^{2}(sin^{2}(x))sin^{2}(x)*-2cos(x)sin(x)\\=&24sin(x)cos^{3}(x)cos^{2}(sin^{2}(x)) - 16sin(x)sin(sin^{2}(x))cos(x)cos(sin^{2}(x)) - 32sin(sin^{2}(x))sin^{3}(x)cos(sin^{2}(x))cos^{3}(x) - 24sin^{3}(x)cos(x)cos^{2}(sin^{2}(x)) - 32sin^{3}(x)sin(sin^{2}(x))cos(sin^{2}(x))cos^{3}(x) + 24sin^{2}(sin^{2}(x))sin^{3}(x)cos(x) - 8sin^{2}(sin^{2}(x))sin(x)cos^{3}(x) - 16sin(x)sin^{2}(sin^{2}(x))cos^{3}(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 24sin(x)cos^{3}(x)cos^{2}(sin^{2}(x)) - 16sin(x)sin(sin^{2}(x))cos(x)cos(sin^{2}(x)) - 32sin(sin^{2}(x))sin^{3}(x)cos(sin^{2}(x))cos^{3}(x) - 24sin^{3}(x)cos(x)cos^{2}(sin^{2}(x)) - 32sin^{3}(x)sin(sin^{2}(x))cos(sin^{2}(x))cos^{3}(x) + 24sin^{2}(sin^{2}(x))sin^{3}(x)cos(x) - 8sin^{2}(sin^{2}(x))sin(x)cos^{3}(x) - 16sin(x)sin^{2}(sin^{2}(x))cos^{3}(x)\right)}{dx}\\=&24cos(x)cos^{3}(x)cos^{2}(sin^{2}(x)) + 24sin(x)*-3cos^{2}(x)sin(x)cos^{2}(sin^{2}(x)) + 24sin(x)cos^{3}(x)*-2cos(sin^{2}(x))sin(sin^{2}(x))*2sin(x)cos(x) - 16cos(x)sin(sin^{2}(x))cos(x)cos(sin^{2}(x)) - 16sin(x)cos(sin^{2}(x))*2sin(x)cos(x)cos(x)cos(sin^{2}(x)) - 16sin(x)sin(sin^{2}(x))*-sin(x)cos(sin^{2}(x)) - 16sin(x)sin(sin^{2}(x))cos(x)*-sin(sin^{2}(x))*2sin(x)cos(x) - 32cos(sin^{2}(x))*2sin(x)cos(x)sin^{3}(x)cos(sin^{2}(x))cos^{3}(x) - 32sin(sin^{2}(x))*3sin^{2}(x)cos(x)cos(sin^{2}(x))cos^{3}(x) - 32sin(sin^{2}(x))sin^{3}(x)*-sin(sin^{2}(x))*2sin(x)cos(x)cos^{3}(x) - 32sin(sin^{2}(x))sin^{3}(x)cos(sin^{2}(x))*-3cos^{2}(x)sin(x) - 24*3sin^{2}(x)cos(x)cos(x)cos^{2}(sin^{2}(x)) - 24sin^{3}(x)*-sin(x)cos^{2}(sin^{2}(x)) - 24sin^{3}(x)cos(x)*-2cos(sin^{2}(x))sin(sin^{2}(x))*2sin(x)cos(x) - 32*3sin^{2}(x)cos(x)sin(sin^{2}(x))cos(sin^{2}(x))cos^{3}(x) - 32sin^{3}(x)cos(sin^{2}(x))*2sin(x)cos(x)cos(sin^{2}(x))cos^{3}(x) - 32sin^{3}(x)sin(sin^{2}(x))*-sin(sin^{2}(x))*2sin(x)cos(x)cos^{3}(x) - 32sin^{3}(x)sin(sin^{2}(x))cos(sin^{2}(x))*-3cos^{2}(x)sin(x) + 24*2sin(sin^{2}(x))cos(sin^{2}(x))*2sin(x)cos(x)sin^{3}(x)cos(x) + 24sin^{2}(sin^{2}(x))*3sin^{2}(x)cos(x)cos(x) + 24sin^{2}(sin^{2}(x))sin^{3}(x)*-sin(x) - 8*2sin(sin^{2}(x))cos(sin^{2}(x))*2sin(x)cos(x)sin(x)cos^{3}(x) - 8sin^{2}(sin^{2}(x))cos(x)cos^{3}(x) - 8sin^{2}(sin^{2}(x))sin(x)*-3cos^{2}(x)sin(x) - 16cos(x)sin^{2}(sin^{2}(x))cos^{3}(x) - 16sin(x)*2sin(sin^{2}(x))cos(sin^{2}(x))*2sin(x)cos(x)cos^{3}(x) - 16sin(x)sin^{2}(sin^{2}(x))*-3cos^{2}(x)sin(x)\\=&24cos^{4}(x)cos^{2}(sin^{2}(x)) - 192sin^{2}(x)sin(sin^{2}(x))cos^{4}(x)cos(sin^{2}(x)) - 96sin(sin^{2}(x))sin^{2}(x)cos(sin^{2}(x))cos^{4}(x) - 16sin(sin^{2}(x))cos^{2}(x)cos(sin^{2}(x)) - 176sin^{2}(x)cos^{2}(x)cos^{2}(sin^{2}(x)) + 192sin(sin^{2}(x))sin^{4}(x)cos(sin^{2}(x))cos^{2}(x) + 192sin^{4}(x)sin(sin^{2}(x))cos(sin^{2}(x))cos^{2}(x) - 64sin^{4}(x)cos^{2}(sin^{2}(x))cos^{4}(x) - 32sin^{2}(x)sin(sin^{2}(x))cos(sin^{2}(x))cos^{4}(x) - 64sin(sin^{2}(x))sin^{2}(x)cos^{4}(x)cos(sin^{2}(x)) + 120sin^{2}(x)sin^{2}(sin^{2}(x))cos^{2}(x) - 64sin^{4}(x)cos^{4}(x)cos^{2}(sin^{2}(x)) + 64sin^{2}(sin^{2}(x))sin^{4}(x)cos^{4}(x) + 16sin^{2}(x)sin(sin^{2}(x))cos(sin^{2}(x)) + 24sin^{4}(x)cos^{2}(sin^{2}(x)) + 64sin^{4}(x)sin^{2}(sin^{2}(x))cos^{4}(x) - 24sin^{2}(sin^{2}(x))cos^{4}(x) + 56sin^{2}(sin^{2}(x))sin^{2}(x)cos^{2}(x) - 24sin^{2}(sin^{2}(x))sin^{4}(x)\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/3]Find\ the\ 4th\ derivative\ of\ function\ {cos({cos(x)}^{2})}^{2}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = cos^{2}(cos^{2}(x))\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( cos^{2}(cos^{2}(x))\right)}{dx}\\=&-2cos(cos^{2}(x))sin(cos^{2}(x))*-2cos(x)sin(x)\\=&4sin(x)sin(cos^{2}(x))cos(cos^{2}(x))cos(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( 4sin(x)sin(cos^{2}(x))cos(cos^{2}(x))cos(x)\right)}{dx}\\=&4cos(x)sin(cos^{2}(x))cos(cos^{2}(x))cos(x) + 4sin(x)cos(cos^{2}(x))*-2cos(x)sin(x)cos(cos^{2}(x))cos(x) + 4sin(x)sin(cos^{2}(x))*-sin(cos^{2}(x))*-2cos(x)sin(x)cos(x) + 4sin(x)sin(cos^{2}(x))cos(cos^{2}(x))*-sin(x)\\=&4sin(cos^{2}(x))cos^{2}(x)cos(cos^{2}(x)) - 8sin^{2}(x)cos^{2}(cos^{2}(x))cos^{2}(x) + 8sin^{2}(cos^{2}(x))sin^{2}(x)cos^{2}(x) - 4sin^{2}(x)sin(cos^{2}(x))cos(cos^{2}(x))\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 4sin(cos^{2}(x))cos^{2}(x)cos(cos^{2}(x)) - 8sin^{2}(x)cos^{2}(cos^{2}(x))cos^{2}(x) + 8sin^{2}(cos^{2}(x))sin^{2}(x)cos^{2}(x) - 4sin^{2}(x)sin(cos^{2}(x))cos(cos^{2}(x))\right)}{dx}\\=&4cos(cos^{2}(x))*-2cos(x)sin(x)cos^{2}(x)cos(cos^{2}(x)) + 4sin(cos^{2}(x))*-2cos(x)sin(x)cos(cos^{2}(x)) + 4sin(cos^{2}(x))cos^{2}(x)*-sin(cos^{2}(x))*-2cos(x)sin(x) - 8*2sin(x)cos(x)cos^{2}(cos^{2}(x))cos^{2}(x) - 8sin^{2}(x)*-2cos(cos^{2}(x))sin(cos^{2}(x))*-2cos(x)sin(x)cos^{2}(x) - 8sin^{2}(x)cos^{2}(cos^{2}(x))*-2cos(x)sin(x) + 8*2sin(cos^{2}(x))cos(cos^{2}(x))*-2cos(x)sin(x)sin^{2}(x)cos^{2}(x) + 8sin^{2}(cos^{2}(x))*2sin(x)cos(x)cos^{2}(x) + 8sin^{2}(cos^{2}(x))sin^{2}(x)*-2cos(x)sin(x) - 4*2sin(x)cos(x)sin(cos^{2}(x))cos(cos^{2}(x)) - 4sin^{2}(x)cos(cos^{2}(x))*-2cos(x)sin(x)cos(cos^{2}(x)) - 4sin^{2}(x)sin(cos^{2}(x))*-sin(cos^{2}(x))*-2cos(x)sin(x)\\=&-8sin(x)cos^{2}(cos^{2}(x))cos^{3}(x) - 16sin(x)sin(cos^{2}(x))cos(x)cos(cos^{2}(x)) - 32sin(cos^{2}(x))sin^{3}(x)cos(cos^{2}(x))cos^{3}(x) - 16sin(x)cos^{3}(x)cos^{2}(cos^{2}(x)) - 32sin^{3}(x)sin(cos^{2}(x))cos^{3}(x)cos(cos^{2}(x)) + 24sin^{3}(x)cos^{2}(cos^{2}(x))cos(x) + 8sin^{2}(cos^{2}(x))sin(x)cos^{3}(x) + 16sin(x)sin^{2}(cos^{2}(x))cos^{3}(x) - 24sin^{2}(cos^{2}(x))sin^{3}(x)cos(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( -8sin(x)cos^{2}(cos^{2}(x))cos^{3}(x) - 16sin(x)sin(cos^{2}(x))cos(x)cos(cos^{2}(x)) - 32sin(cos^{2}(x))sin^{3}(x)cos(cos^{2}(x))cos^{3}(x) - 16sin(x)cos^{3}(x)cos^{2}(cos^{2}(x)) - 32sin^{3}(x)sin(cos^{2}(x))cos^{3}(x)cos(cos^{2}(x)) + 24sin^{3}(x)cos^{2}(cos^{2}(x))cos(x) + 8sin^{2}(cos^{2}(x))sin(x)cos^{3}(x) + 16sin(x)sin^{2}(cos^{2}(x))cos^{3}(x) - 24sin^{2}(cos^{2}(x))sin^{3}(x)cos(x)\right)}{dx}\\=&-8cos(x)cos^{2}(cos^{2}(x))cos^{3}(x) - 8sin(x)*-2cos(cos^{2}(x))sin(cos^{2}(x))*-2cos(x)sin(x)cos^{3}(x) - 8sin(x)cos^{2}(cos^{2}(x))*-3cos^{2}(x)sin(x) - 16cos(x)sin(cos^{2}(x))cos(x)cos(cos^{2}(x)) - 16sin(x)cos(cos^{2}(x))*-2cos(x)sin(x)cos(x)cos(cos^{2}(x)) - 16sin(x)sin(cos^{2}(x))*-sin(x)cos(cos^{2}(x)) - 16sin(x)sin(cos^{2}(x))cos(x)*-sin(cos^{2}(x))*-2cos(x)sin(x) - 32cos(cos^{2}(x))*-2cos(x)sin(x)sin^{3}(x)cos(cos^{2}(x))cos^{3}(x) - 32sin(cos^{2}(x))*3sin^{2}(x)cos(x)cos(cos^{2}(x))cos^{3}(x) - 32sin(cos^{2}(x))sin^{3}(x)*-sin(cos^{2}(x))*-2cos(x)sin(x)cos^{3}(x) - 32sin(cos^{2}(x))sin^{3}(x)cos(cos^{2}(x))*-3cos^{2}(x)sin(x) - 16cos(x)cos^{3}(x)cos^{2}(cos^{2}(x)) - 16sin(x)*-3cos^{2}(x)sin(x)cos^{2}(cos^{2}(x)) - 16sin(x)cos^{3}(x)*-2cos(cos^{2}(x))sin(cos^{2}(x))*-2cos(x)sin(x) - 32*3sin^{2}(x)cos(x)sin(cos^{2}(x))cos^{3}(x)cos(cos^{2}(x)) - 32sin^{3}(x)cos(cos^{2}(x))*-2cos(x)sin(x)cos^{3}(x)cos(cos^{2}(x)) - 32sin^{3}(x)sin(cos^{2}(x))*-3cos^{2}(x)sin(x)cos(cos^{2}(x)) - 32sin^{3}(x)sin(cos^{2}(x))cos^{3}(x)*-sin(cos^{2}(x))*-2cos(x)sin(x) + 24*3sin^{2}(x)cos(x)cos^{2}(cos^{2}(x))cos(x) + 24sin^{3}(x)*-2cos(cos^{2}(x))sin(cos^{2}(x))*-2cos(x)sin(x)cos(x) + 24sin^{3}(x)cos^{2}(cos^{2}(x))*-sin(x) + 8*2sin(cos^{2}(x))cos(cos^{2}(x))*-2cos(x)sin(x)sin(x)cos^{3}(x) + 8sin^{2}(cos^{2}(x))cos(x)cos^{3}(x) + 8sin^{2}(cos^{2}(x))sin(x)*-3cos^{2}(x)sin(x) + 16cos(x)sin^{2}(cos^{2}(x))cos^{3}(x) + 16sin(x)*2sin(cos^{2}(x))cos(cos^{2}(x))*-2cos(x)sin(x)cos^{3}(x) + 16sin(x)sin^{2}(cos^{2}(x))*-3cos^{2}(x)sin(x) - 24*2sin(cos^{2}(x))cos(cos^{2}(x))*-2cos(x)sin(x)sin^{3}(x)cos(x) - 24sin^{2}(cos^{2}(x))*3sin^{2}(x)cos(x)cos(x) - 24sin^{2}(cos^{2}(x))sin^{3}(x)*-sin(x)\\=&-24cos^{4}(x)cos^{2}(cos^{2}(x)) - 96sin(cos^{2}(x))sin^{2}(x)cos(cos^{2}(x))cos^{4}(x) - 224sin^{2}(x)sin(cos^{2}(x))cos^{4}(x)cos(cos^{2}(x)) - 16sin(cos^{2}(x))cos^{2}(x)cos(cos^{2}(x)) + 56sin^{2}(x)cos^{2}(cos^{2}(x))cos^{2}(x) + 192sin(cos^{2}(x))sin^{4}(x)cos(cos^{2}(x))cos^{2}(x) + 192sin^{4}(x)sin(cos^{2}(x))cos^{2}(x)cos(cos^{2}(x)) + 64sin^{4}(x)cos^{4}(x)cos^{2}(cos^{2}(x)) - 64sin(cos^{2}(x))sin^{2}(x)cos^{4}(x)cos(cos^{2}(x)) - 56sin^{2}(cos^{2}(x))sin^{2}(x)cos^{2}(x) + 120sin^{2}(x)cos^{2}(x)cos^{2}(cos^{2}(x)) - 64sin^{4}(x)sin^{2}(cos^{2}(x))cos^{4}(x) + 64sin^{4}(x)cos^{2}(cos^{2}(x))cos^{4}(x) + 16sin^{2}(x)sin(cos^{2}(x))cos(cos^{2}(x)) - 64sin^{2}(cos^{2}(x))sin^{4}(x)cos^{4}(x) - 120sin^{2}(x)sin^{2}(cos^{2}(x))cos^{2}(x) + 24sin^{2}(cos^{2}(x))cos^{4}(x) - 24sin^{4}(x)cos^{2}(cos^{2}(x)) + 24sin^{2}(cos^{2}(x))sin^{4}(x)\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[3/3]Find\ the\ 4th\ derivative\ of\ function\ {tan({tan(x)}^{2})}^{2}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = tan^{2}(tan^{2}(x))\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( tan^{2}(tan^{2}(x))\right)}{dx}\\=&2tan(tan^{2}(x))sec^{2}(tan^{2}(x))(2tan(x)sec^{2}(x)(1))\\=&4tan(x)tan(tan^{2}(x))sec^{2}(x)sec^{2}(tan^{2}(x))\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( 4tan(x)tan(tan^{2}(x))sec^{2}(x)sec^{2}(tan^{2}(x))\right)}{dx}\\=&4sec^{2}(x)(1)tan(tan^{2}(x))sec^{2}(x)sec^{2}(tan^{2}(x)) + 4tan(x)sec^{2}(tan^{2}(x))(2tan(x)sec^{2}(x)(1))sec^{2}(x)sec^{2}(tan^{2}(x)) + 4tan(x)tan(tan^{2}(x))*2sec^{2}(x)tan(x)sec^{2}(tan^{2}(x)) + 4tan(x)tan(tan^{2}(x))sec^{2}(x)*2sec^{2}(tan^{2}(x))tan(tan^{2}(x))*2tan(x)sec^{2}(x)(1)\\=&4tan(tan^{2}(x))sec^{4}(x)sec^{2}(tan^{2}(x)) + 8tan^{2}(x)sec^{4}(x)sec^{4}(tan^{2}(x)) + 8tan^{2}(x)tan(tan^{2}(x))sec^{2}(x)sec^{2}(tan^{2}(x)) + 16tan^{2}(tan^{2}(x))tan^{2}(x)sec^{2}(tan^{2}(x))sec^{4}(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 4tan(tan^{2}(x))sec^{4}(x)sec^{2}(tan^{2}(x)) + 8tan^{2}(x)sec^{4}(x)sec^{4}(tan^{2}(x)) + 8tan^{2}(x)tan(tan^{2}(x))sec^{2}(x)sec^{2}(tan^{2}(x)) + 16tan^{2}(tan^{2}(x))tan^{2}(x)sec^{2}(tan^{2}(x))sec^{4}(x)\right)}{dx}\\=&4sec^{2}(tan^{2}(x))(2tan(x)sec^{2}(x)(1))sec^{4}(x)sec^{2}(tan^{2}(x)) + 4tan(tan^{2}(x))*4sec^{4}(x)tan(x)sec^{2}(tan^{2}(x)) + 4tan(tan^{2}(x))sec^{4}(x)*2sec^{2}(tan^{2}(x))tan(tan^{2}(x))*2tan(x)sec^{2}(x)(1) + 8*2tan(x)sec^{2}(x)(1)sec^{4}(x)sec^{4}(tan^{2}(x)) + 8tan^{2}(x)*4sec^{4}(x)tan(x)sec^{4}(tan^{2}(x)) + 8tan^{2}(x)sec^{4}(x)*4sec^{4}(tan^{2}(x))tan(tan^{2}(x))*2tan(x)sec^{2}(x)(1) + 8*2tan(x)sec^{2}(x)(1)tan(tan^{2}(x))sec^{2}(x)sec^{2}(tan^{2}(x)) + 8tan^{2}(x)sec^{2}(tan^{2}(x))(2tan(x)sec^{2}(x)(1))sec^{2}(x)sec^{2}(tan^{2}(x)) + 8tan^{2}(x)tan(tan^{2}(x))*2sec^{2}(x)tan(x)sec^{2}(tan^{2}(x)) + 8tan^{2}(x)tan(tan^{2}(x))sec^{2}(x)*2sec^{2}(tan^{2}(x))tan(tan^{2}(x))*2tan(x)sec^{2}(x)(1) + 16*2tan(tan^{2}(x))sec^{2}(tan^{2}(x))(2tan(x)sec^{2}(x)(1))tan^{2}(x)sec^{2}(tan^{2}(x))sec^{4}(x) + 16tan^{2}(tan^{2}(x))*2tan(x)sec^{2}(x)(1)sec^{2}(tan^{2}(x))sec^{4}(x) + 16tan^{2}(tan^{2}(x))tan^{2}(x)*2sec^{2}(tan^{2}(x))tan(tan^{2}(x))*2tan(x)sec^{2}(x)(1)sec^{4}(x) + 16tan^{2}(tan^{2}(x))tan^{2}(x)sec^{2}(tan^{2}(x))*4sec^{4}(x)tan(x)\\=&24tan(x)sec^{6}(x)sec^{4}(tan^{2}(x)) + 32tan(x)tan(tan^{2}(x))sec^{4}(x)sec^{2}(tan^{2}(x)) + 64tan^{3}(x)tan(tan^{2}(x))sec^{4}(tan^{2}(x))sec^{6}(x) + 32tan(x)tan^{2}(tan^{2}(x))sec^{6}(x)sec^{2}(tan^{2}(x)) + 64tan(tan^{2}(x))tan^{3}(x)sec^{4}(tan^{2}(x))sec^{6}(x) + 48tan^{3}(x)sec^{4}(x)sec^{4}(tan^{2}(x)) + 16tan^{3}(x)tan(tan^{2}(x))sec^{2}(x)sec^{2}(tan^{2}(x)) + 96tan^{2}(tan^{2}(x))tan^{3}(x)sec^{2}(tan^{2}(x))sec^{4}(x) + 16tan^{2}(tan^{2}(x))tan(x)sec^{2}(tan^{2}(x))sec^{6}(x) + 64tan^{3}(x)tan^{3}(tan^{2}(x))sec^{6}(x)sec^{2}(tan^{2}(x))\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 24tan(x)sec^{6}(x)sec^{4}(tan^{2}(x)) + 32tan(x)tan(tan^{2}(x))sec^{4}(x)sec^{2}(tan^{2}(x)) + 64tan^{3}(x)tan(tan^{2}(x))sec^{4}(tan^{2}(x))sec^{6}(x) + 32tan(x)tan^{2}(tan^{2}(x))sec^{6}(x)sec^{2}(tan^{2}(x)) + 64tan(tan^{2}(x))tan^{3}(x)sec^{4}(tan^{2}(x))sec^{6}(x) + 48tan^{3}(x)sec^{4}(x)sec^{4}(tan^{2}(x)) + 16tan^{3}(x)tan(tan^{2}(x))sec^{2}(x)sec^{2}(tan^{2}(x)) + 96tan^{2}(tan^{2}(x))tan^{3}(x)sec^{2}(tan^{2}(x))sec^{4}(x) + 16tan^{2}(tan^{2}(x))tan(x)sec^{2}(tan^{2}(x))sec^{6}(x) + 64tan^{3}(x)tan^{3}(tan^{2}(x))sec^{6}(x)sec^{2}(tan^{2}(x))\right)}{dx}\\=&24sec^{2}(x)(1)sec^{6}(x)sec^{4}(tan^{2}(x)) + 24tan(x)*6sec^{6}(x)tan(x)sec^{4}(tan^{2}(x)) + 24tan(x)sec^{6}(x)*4sec^{4}(tan^{2}(x))tan(tan^{2}(x))*2tan(x)sec^{2}(x)(1) + 32sec^{2}(x)(1)tan(tan^{2}(x))sec^{4}(x)sec^{2}(tan^{2}(x)) + 32tan(x)sec^{2}(tan^{2}(x))(2tan(x)sec^{2}(x)(1))sec^{4}(x)sec^{2}(tan^{2}(x)) + 32tan(x)tan(tan^{2}(x))*4sec^{4}(x)tan(x)sec^{2}(tan^{2}(x)) + 32tan(x)tan(tan^{2}(x))sec^{4}(x)*2sec^{2}(tan^{2}(x))tan(tan^{2}(x))*2tan(x)sec^{2}(x)(1) + 64*3tan^{2}(x)sec^{2}(x)(1)tan(tan^{2}(x))sec^{4}(tan^{2}(x))sec^{6}(x) + 64tan^{3}(x)sec^{2}(tan^{2}(x))(2tan(x)sec^{2}(x)(1))sec^{4}(tan^{2}(x))sec^{6}(x) + 64tan^{3}(x)tan(tan^{2}(x))*4sec^{4}(tan^{2}(x))tan(tan^{2}(x))*2tan(x)sec^{2}(x)(1)sec^{6}(x) + 64tan^{3}(x)tan(tan^{2}(x))sec^{4}(tan^{2}(x))*6sec^{6}(x)tan(x) + 32sec^{2}(x)(1)tan^{2}(tan^{2}(x))sec^{6}(x)sec^{2}(tan^{2}(x)) + 32tan(x)*2tan(tan^{2}(x))sec^{2}(tan^{2}(x))(2tan(x)sec^{2}(x)(1))sec^{6}(x)sec^{2}(tan^{2}(x)) + 32tan(x)tan^{2}(tan^{2}(x))*6sec^{6}(x)tan(x)sec^{2}(tan^{2}(x)) + 32tan(x)tan^{2}(tan^{2}(x))sec^{6}(x)*2sec^{2}(tan^{2}(x))tan(tan^{2}(x))*2tan(x)sec^{2}(x)(1) + 64sec^{2}(tan^{2}(x))(2tan(x)sec^{2}(x)(1))tan^{3}(x)sec^{4}(tan^{2}(x))sec^{6}(x) + 64tan(tan^{2}(x))*3tan^{2}(x)sec^{2}(x)(1)sec^{4}(tan^{2}(x))sec^{6}(x) + 64tan(tan^{2}(x))tan^{3}(x)*4sec^{4}(tan^{2}(x))tan(tan^{2}(x))*2tan(x)sec^{2}(x)(1)sec^{6}(x) + 64tan(tan^{2}(x))tan^{3}(x)sec^{4}(tan^{2}(x))*6sec^{6}(x)tan(x) + 48*3tan^{2}(x)sec^{2}(x)(1)sec^{4}(x)sec^{4}(tan^{2}(x)) + 48tan^{3}(x)*4sec^{4}(x)tan(x)sec^{4}(tan^{2}(x)) + 48tan^{3}(x)sec^{4}(x)*4sec^{4}(tan^{2}(x))tan(tan^{2}(x))*2tan(x)sec^{2}(x)(1) + 16*3tan^{2}(x)sec^{2}(x)(1)tan(tan^{2}(x))sec^{2}(x)sec^{2}(tan^{2}(x)) + 16tan^{3}(x)sec^{2}(tan^{2}(x))(2tan(x)sec^{2}(x)(1))sec^{2}(x)sec^{2}(tan^{2}(x)) + 16tan^{3}(x)tan(tan^{2}(x))*2sec^{2}(x)tan(x)sec^{2}(tan^{2}(x)) + 16tan^{3}(x)tan(tan^{2}(x))sec^{2}(x)*2sec^{2}(tan^{2}(x))tan(tan^{2}(x))*2tan(x)sec^{2}(x)(1) + 96*2tan(tan^{2}(x))sec^{2}(tan^{2}(x))(2tan(x)sec^{2}(x)(1))tan^{3}(x)sec^{2}(tan^{2}(x))sec^{4}(x) + 96tan^{2}(tan^{2}(x))*3tan^{2}(x)sec^{2}(x)(1)sec^{2}(tan^{2}(x))sec^{4}(x) + 96tan^{2}(tan^{2}(x))tan^{3}(x)*2sec^{2}(tan^{2}(x))tan(tan^{2}(x))*2tan(x)sec^{2}(x)(1)sec^{4}(x) + 96tan^{2}(tan^{2}(x))tan^{3}(x)sec^{2}(tan^{2}(x))*4sec^{4}(x)tan(x) + 16*2tan(tan^{2}(x))sec^{2}(tan^{2}(x))(2tan(x)sec^{2}(x)(1))tan(x)sec^{2}(tan^{2}(x))sec^{6}(x) + 16tan^{2}(tan^{2}(x))sec^{2}(x)(1)sec^{2}(tan^{2}(x))sec^{6}(x) + 16tan^{2}(tan^{2}(x))tan(x)*2sec^{2}(tan^{2}(x))tan(tan^{2}(x))*2tan(x)sec^{2}(x)(1)sec^{6}(x) + 16tan^{2}(tan^{2}(x))tan(x)sec^{2}(tan^{2}(x))*6sec^{6}(x)tan(x) + 64*3tan^{2}(x)sec^{2}(x)(1)tan^{3}(tan^{2}(x))sec^{6}(x)sec^{2}(tan^{2}(x)) + 64tan^{3}(x)*3tan^{2}(tan^{2}(x))sec^{2}(tan^{2}(x))(2tan(x)sec^{2}(x)(1))sec^{6}(x)sec^{2}(tan^{2}(x)) + 64tan^{3}(x)tan^{3}(tan^{2}(x))*6sec^{6}(x)tan(x)sec^{2}(tan^{2}(x)) + 64tan^{3}(x)tan^{3}(tan^{2}(x))sec^{6}(x)*2sec^{2}(tan^{2}(x))tan(tan^{2}(x))*2tan(x)sec^{2}(x)(1)\\=&24sec^{8}(x)sec^{4}(tan^{2}(x)) + 384tan^{2}(x)tan(tan^{2}(x))sec^{8}(x)sec^{4}(tan^{2}(x)) + 128tan(tan^{2}(x))tan^{2}(x)sec^{8}(x)sec^{4}(tan^{2}(x)) + 32tan(tan^{2}(x))sec^{6}(x)sec^{2}(tan^{2}(x)) + 352tan^{2}(x)sec^{6}(x)sec^{4}(tan^{2}(x)) + 176tan^{2}(x)tan(tan^{2}(x))sec^{4}(x)sec^{2}(tan^{2}(x)) + 224tan^{2}(tan^{2}(x))tan^{2}(x)sec^{2}(tan^{2}(x))sec^{6}(x) + 128tan^{4}(x)sec^{8}(x)sec^{6}(tan^{2}(x)) + 896tan^{2}(tan^{2}(x))tan^{4}(x)sec^{8}(x)sec^{4}(tan^{2}(x)) + 768tan^{4}(x)tan(tan^{2}(x))sec^{4}(tan^{2}(x))sec^{6}(x) + 48tan^{2}(tan^{2}(x))sec^{8}(x)sec^{2}(tan^{2}(x)) + 480tan^{2}(x)tan^{2}(tan^{2}(x))sec^{6}(x)sec^{2}(tan^{2}(x)) + 128tan^{3}(tan^{2}(x))tan^{2}(x)sec^{2}(tan^{2}(x))sec^{8}(x) + 128tan^{4}(x)sec^{6}(tan^{2}(x))sec^{8}(x) + 512tan^{4}(x)tan^{2}(tan^{2}(x))sec^{8}(x)sec^{4}(tan^{2}(x)) + 768tan(tan^{2}(x))tan^{4}(x)sec^{4}(tan^{2}(x))sec^{6}(x) + 64tan^{2}(x)tan(tan^{2}(x))sec^{4}(tan^{2}(x))sec^{8}(x) + 256tan^{2}(x)tan^{3}(tan^{2}(x))sec^{8}(x)sec^{2}(tan^{2}(x)) + 224tan^{4}(x)sec^{4}(x)sec^{4}(tan^{2}(x)) + 32tan^{4}(x)tan(tan^{2}(x))sec^{2}(x)sec^{2}(tan^{2}(x)) + 448tan^{2}(tan^{2}(x))tan^{4}(x)sec^{2}(tan^{2}(x))sec^{4}(x) + 768tan^{4}(x)tan^{3}(tan^{2}(x))sec^{6}(x)sec^{2}(tan^{2}(x)) + 192tan(tan^{2}(x))tan^{2}(x)sec^{4}(tan^{2}(x))sec^{8}(x) + 256tan^{4}(tan^{2}(x))tan^{4}(x)sec^{2}(tan^{2}(x))sec^{8}(x)\\ \end{split}\end{equation} \]