There are 1 questions in this calculation: for each question, the 3 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ third\ derivative\ of\ function\ \frac{{({e}^{x})}^{\frac{1}{2}}}{cos(2x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{{e}^{(\frac{1}{2}x)}}{cos(2x)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{{e}^{(\frac{1}{2}x)}}{cos(2x)}\right)}{dx}\\=&\frac{({e}^{(\frac{1}{2}x)}((\frac{1}{2})ln(e) + \frac{(\frac{1}{2}x)(0)}{(e)}))}{cos(2x)} + \frac{{e}^{(\frac{1}{2}x)}sin(2x)*2}{cos^{2}(2x)}\\=&\frac{{e}^{(\frac{1}{2}x)}}{2cos(2x)} + \frac{2{e}^{(\frac{1}{2}x)}sin(2x)}{cos^{2}(2x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{{e}^{(\frac{1}{2}x)}}{2cos(2x)} + \frac{2{e}^{(\frac{1}{2}x)}sin(2x)}{cos^{2}(2x)}\right)}{dx}\\=&\frac{({e}^{(\frac{1}{2}x)}((\frac{1}{2})ln(e) + \frac{(\frac{1}{2}x)(0)}{(e)}))}{2cos(2x)} + \frac{{e}^{(\frac{1}{2}x)}sin(2x)*2}{2cos^{2}(2x)} + \frac{2({e}^{(\frac{1}{2}x)}((\frac{1}{2})ln(e) + \frac{(\frac{1}{2}x)(0)}{(e)}))sin(2x)}{cos^{2}(2x)} + \frac{2{e}^{(\frac{1}{2}x)}cos(2x)*2}{cos^{2}(2x)} + \frac{2{e}^{(\frac{1}{2}x)}sin(2x)*2sin(2x)*2}{cos^{3}(2x)}\\=&\frac{17{e}^{(\frac{1}{2}x)}}{4cos(2x)} + \frac{2{e}^{(\frac{1}{2}x)}sin(2x)}{cos^{2}(2x)} + \frac{8{e}^{(\frac{1}{2}x)}sin^{2}(2x)}{cos^{3}(2x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{17{e}^{(\frac{1}{2}x)}}{4cos(2x)} + \frac{2{e}^{(\frac{1}{2}x)}sin(2x)}{cos^{2}(2x)} + \frac{8{e}^{(\frac{1}{2}x)}sin^{2}(2x)}{cos^{3}(2x)}\right)}{dx}\\=&\frac{17({e}^{(\frac{1}{2}x)}((\frac{1}{2})ln(e) + \frac{(\frac{1}{2}x)(0)}{(e)}))}{4cos(2x)} + \frac{17{e}^{(\frac{1}{2}x)}sin(2x)*2}{4cos^{2}(2x)} + \frac{2({e}^{(\frac{1}{2}x)}((\frac{1}{2})ln(e) + \frac{(\frac{1}{2}x)(0)}{(e)}))sin(2x)}{cos^{2}(2x)} + \frac{2{e}^{(\frac{1}{2}x)}cos(2x)*2}{cos^{2}(2x)} + \frac{2{e}^{(\frac{1}{2}x)}sin(2x)*2sin(2x)*2}{cos^{3}(2x)} + \frac{8({e}^{(\frac{1}{2}x)}((\frac{1}{2})ln(e) + \frac{(\frac{1}{2}x)(0)}{(e)}))sin^{2}(2x)}{cos^{3}(2x)} + \frac{8{e}^{(\frac{1}{2}x)}*2sin(2x)cos(2x)*2}{cos^{3}(2x)} + \frac{8{e}^{(\frac{1}{2}x)}sin^{2}(2x)*3sin(2x)*2}{cos^{4}(2x)}\\=&\frac{49{e}^{(\frac{1}{2}x)}}{8cos(2x)} + \frac{83{e}^{(\frac{1}{2}x)}sin(2x)}{2cos^{2}(2x)} + \frac{12{e}^{(\frac{1}{2}x)}sin^{2}(2x)}{cos^{3}(2x)} + \frac{48{e}^{(\frac{1}{2}x)}sin^{3}(2x)}{cos^{4}(2x)}\\ \end{split}\end{equation} \]

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