There are 1 questions in this calculation: for each question, the 3 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ third\ derivative\ of\ function\ {e}^{(\frac{x}{2})}cos(2x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = {e}^{(\frac{1}{2}x)}cos(2x)\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( {e}^{(\frac{1}{2}x)}cos(2x)\right)}{dx}\\=&({e}^{(\frac{1}{2}x)}((\frac{1}{2})ln(e) + \frac{(\frac{1}{2}x)(0)}{(e)}))cos(2x) + {e}^{(\frac{1}{2}x)}*-sin(2x)*2\\=&\frac{{e}^{(\frac{1}{2}x)}cos(2x)}{2} - 2{e}^{(\frac{1}{2}x)}sin(2x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{{e}^{(\frac{1}{2}x)}cos(2x)}{2} - 2{e}^{(\frac{1}{2}x)}sin(2x)\right)}{dx}\\=&\frac{({e}^{(\frac{1}{2}x)}((\frac{1}{2})ln(e) + \frac{(\frac{1}{2}x)(0)}{(e)}))cos(2x)}{2} + \frac{{e}^{(\frac{1}{2}x)}*-sin(2x)*2}{2} - 2({e}^{(\frac{1}{2}x)}((\frac{1}{2})ln(e) + \frac{(\frac{1}{2}x)(0)}{(e)}))sin(2x) - 2{e}^{(\frac{1}{2}x)}cos(2x)*2\\=&\frac{-15{e}^{(\frac{1}{2}x)}cos(2x)}{4} - 2{e}^{(\frac{1}{2}x)}sin(2x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-15{e}^{(\frac{1}{2}x)}cos(2x)}{4} - 2{e}^{(\frac{1}{2}x)}sin(2x)\right)}{dx}\\=&\frac{-15({e}^{(\frac{1}{2}x)}((\frac{1}{2})ln(e) + \frac{(\frac{1}{2}x)(0)}{(e)}))cos(2x)}{4} - \frac{15{e}^{(\frac{1}{2}x)}*-sin(2x)*2}{4} - 2({e}^{(\frac{1}{2}x)}((\frac{1}{2})ln(e) + \frac{(\frac{1}{2}x)(0)}{(e)}))sin(2x) - 2{e}^{(\frac{1}{2}x)}cos(2x)*2\\=&\frac{-47{e}^{(\frac{1}{2}x)}cos(2x)}{8} + \frac{13{e}^{(\frac{1}{2}x)}sin(2x)}{2}\\ \end{split}\end{equation} \]

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