There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ sin(\frac{lg(x)}{2})\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = sin(\frac{1}{2}lg(x))\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( sin(\frac{1}{2}lg(x))\right)}{dx}\\=&\frac{cos(\frac{1}{2}lg(x))*\frac{1}{2}}{ln{10}(x)}\\=&\frac{cos(\frac{1}{2}lg(x))}{2xln{10}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{cos(\frac{1}{2}lg(x))}{2xln{10}}\right)}{dx}\\=&\frac{-cos(\frac{1}{2}lg(x))}{2x^{2}ln{10}} + \frac{-0cos(\frac{1}{2}lg(x))}{2xln^{2}{10}} + \frac{-sin(\frac{1}{2}lg(x))*\frac{1}{2}}{2xln{10}ln{10}(x)}\\=&\frac{-cos(\frac{1}{2}lg(x))}{2x^{2}ln{10}} - \frac{sin(\frac{1}{2}lg(x))}{4x^{2}ln^{2}{10}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-cos(\frac{1}{2}lg(x))}{2x^{2}ln{10}} - \frac{sin(\frac{1}{2}lg(x))}{4x^{2}ln^{2}{10}}\right)}{dx}\\=&\frac{--2cos(\frac{1}{2}lg(x))}{2x^{3}ln{10}} - \frac{-0cos(\frac{1}{2}lg(x))}{2x^{2}ln^{2}{10}} - \frac{-sin(\frac{1}{2}lg(x))*\frac{1}{2}}{2x^{2}ln{10}ln{10}(x)} - \frac{-2sin(\frac{1}{2}lg(x))}{4x^{3}ln^{2}{10}} - \frac{-2*0sin(\frac{1}{2}lg(x))}{4x^{2}ln^{3}{10}} - \frac{cos(\frac{1}{2}lg(x))*\frac{1}{2}}{4x^{2}ln^{2}{10}ln{10}(x)}\\=&\frac{cos(\frac{1}{2}lg(x))}{x^{3}ln{10}} + \frac{3sin(\frac{1}{2}lg(x))}{4x^{3}ln^{2}{10}} - \frac{cos(\frac{1}{2}lg(x))}{8x^{3}ln^{3}{10}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{cos(\frac{1}{2}lg(x))}{x^{3}ln{10}} + \frac{3sin(\frac{1}{2}lg(x))}{4x^{3}ln^{2}{10}} - \frac{cos(\frac{1}{2}lg(x))}{8x^{3}ln^{3}{10}}\right)}{dx}\\=&\frac{-3cos(\frac{1}{2}lg(x))}{x^{4}ln{10}} + \frac{-0cos(\frac{1}{2}lg(x))}{x^{3}ln^{2}{10}} + \frac{-sin(\frac{1}{2}lg(x))*\frac{1}{2}}{x^{3}ln{10}ln{10}(x)} + \frac{3*-3sin(\frac{1}{2}lg(x))}{4x^{4}ln^{2}{10}} + \frac{3*-2*0sin(\frac{1}{2}lg(x))}{4x^{3}ln^{3}{10}} + \frac{3cos(\frac{1}{2}lg(x))*\frac{1}{2}}{4x^{3}ln^{2}{10}ln{10}(x)} - \frac{-3cos(\frac{1}{2}lg(x))}{8x^{4}ln^{3}{10}} - \frac{-3*0cos(\frac{1}{2}lg(x))}{8x^{3}ln^{4}{10}} - \frac{-sin(\frac{1}{2}lg(x))*\frac{1}{2}}{8x^{3}ln^{3}{10}ln{10}(x)}\\=&\frac{-3cos(\frac{1}{2}lg(x))}{x^{4}ln{10}} - \frac{11sin(\frac{1}{2}lg(x))}{4x^{4}ln^{2}{10}} + \frac{3cos(\frac{1}{2}lg(x))}{4x^{4}ln^{3}{10}} + \frac{sin(\frac{1}{2}lg(x))}{16x^{4}ln^{4}{10}}\\ \end{split}\end{equation} \]

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