There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ sin(\frac{sin(x)}{2})\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = sin(\frac{1}{2}sin(x))\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( sin(\frac{1}{2}sin(x))\right)}{dx}\\=&cos(\frac{1}{2}sin(x))*\frac{1}{2}cos(x)\\=&\frac{cos(x)cos(\frac{1}{2}sin(x))}{2}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{cos(x)cos(\frac{1}{2}sin(x))}{2}\right)}{dx}\\=&\frac{-sin(x)cos(\frac{1}{2}sin(x))}{2} + \frac{cos(x)*-sin(\frac{1}{2}sin(x))*\frac{1}{2}cos(x)}{2}\\=&\frac{-sin(x)cos(\frac{1}{2}sin(x))}{2} - \frac{sin(\frac{1}{2}sin(x))cos^{2}(x)}{4}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-sin(x)cos(\frac{1}{2}sin(x))}{2} - \frac{sin(\frac{1}{2}sin(x))cos^{2}(x)}{4}\right)}{dx}\\=&\frac{-cos(x)cos(\frac{1}{2}sin(x))}{2} - \frac{sin(x)*-sin(\frac{1}{2}sin(x))*\frac{1}{2}cos(x)}{2} - \frac{cos(\frac{1}{2}sin(x))*\frac{1}{2}cos(x)cos^{2}(x)}{4} - \frac{sin(\frac{1}{2}sin(x))*-2cos(x)sin(x)}{4}\\=&\frac{-cos(x)cos(\frac{1}{2}sin(x))}{2} + \frac{3sin(x)sin(\frac{1}{2}sin(x))cos(x)}{4} - \frac{cos^{3}(x)cos(\frac{1}{2}sin(x))}{8}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-cos(x)cos(\frac{1}{2}sin(x))}{2} + \frac{3sin(x)sin(\frac{1}{2}sin(x))cos(x)}{4} - \frac{cos^{3}(x)cos(\frac{1}{2}sin(x))}{8}\right)}{dx}\\=&\frac{--sin(x)cos(\frac{1}{2}sin(x))}{2} - \frac{cos(x)*-sin(\frac{1}{2}sin(x))*\frac{1}{2}cos(x)}{2} + \frac{3cos(x)sin(\frac{1}{2}sin(x))cos(x)}{4} + \frac{3sin(x)cos(\frac{1}{2}sin(x))*\frac{1}{2}cos(x)cos(x)}{4} + \frac{3sin(x)sin(\frac{1}{2}sin(x))*-sin(x)}{4} - \frac{-3cos^{2}(x)sin(x)cos(\frac{1}{2}sin(x))}{8} - \frac{cos^{3}(x)*-sin(\frac{1}{2}sin(x))*\frac{1}{2}cos(x)}{8}\\=&\frac{3sin(x)cos^{2}(x)cos(\frac{1}{2}sin(x))}{4} + sin(\frac{1}{2}sin(x))cos^{2}(x) - \frac{3sin^{2}(x)sin(\frac{1}{2}sin(x))}{4} + \frac{sin(x)cos(\frac{1}{2}sin(x))}{2} + \frac{sin(\frac{1}{2}sin(x))cos^{4}(x)}{16}\\ \end{split}\end{equation} \]

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