There are 1 questions in this calculation: for each question, the 3 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ third\ derivative\ of\ function\ {(x + 1)}^{\frac{1}{2}}{e}^{{x}^{2}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = (x + 1)^{\frac{1}{2}}{e}^{x^{2}}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( (x + 1)^{\frac{1}{2}}{e}^{x^{2}}\right)}{dx}\\=&(\frac{\frac{1}{2}(1 + 0)}{(x + 1)^{\frac{1}{2}}}){e}^{x^{2}} + (x + 1)^{\frac{1}{2}}({e}^{x^{2}}((2x)ln(e) + \frac{(x^{2})(0)}{(e)}))\\=&\frac{{e}^{x^{2}}}{2(x + 1)^{\frac{1}{2}}} + 2(x + 1)^{\frac{1}{2}}x{e}^{x^{2}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{{e}^{x^{2}}}{2(x + 1)^{\frac{1}{2}}} + 2(x + 1)^{\frac{1}{2}}x{e}^{x^{2}}\right)}{dx}\\=&\frac{(\frac{\frac{-1}{2}(1 + 0)}{(x + 1)^{\frac{3}{2}}}){e}^{x^{2}}}{2} + \frac{({e}^{x^{2}}((2x)ln(e) + \frac{(x^{2})(0)}{(e)}))}{2(x + 1)^{\frac{1}{2}}} + 2(\frac{\frac{1}{2}(1 + 0)}{(x + 1)^{\frac{1}{2}}})x{e}^{x^{2}} + 2(x + 1)^{\frac{1}{2}}{e}^{x^{2}} + 2(x + 1)^{\frac{1}{2}}x({e}^{x^{2}}((2x)ln(e) + \frac{(x^{2})(0)}{(e)}))\\=&\frac{-{e}^{x^{2}}}{4(x + 1)^{\frac{3}{2}}} + \frac{2x{e}^{x^{2}}}{(x + 1)^{\frac{1}{2}}} + 2(x + 1)^{\frac{1}{2}}{e}^{x^{2}} + 4(x + 1)^{\frac{1}{2}}x^{2}{e}^{x^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-{e}^{x^{2}}}{4(x + 1)^{\frac{3}{2}}} + \frac{2x{e}^{x^{2}}}{(x + 1)^{\frac{1}{2}}} + 2(x + 1)^{\frac{1}{2}}{e}^{x^{2}} + 4(x + 1)^{\frac{1}{2}}x^{2}{e}^{x^{2}}\right)}{dx}\\=&\frac{-(\frac{\frac{-3}{2}(1 + 0)}{(x + 1)^{\frac{5}{2}}}){e}^{x^{2}}}{4} - \frac{({e}^{x^{2}}((2x)ln(e) + \frac{(x^{2})(0)}{(e)}))}{4(x + 1)^{\frac{3}{2}}} + 2(\frac{\frac{-1}{2}(1 + 0)}{(x + 1)^{\frac{3}{2}}})x{e}^{x^{2}} + \frac{2{e}^{x^{2}}}{(x + 1)^{\frac{1}{2}}} + \frac{2x({e}^{x^{2}}((2x)ln(e) + \frac{(x^{2})(0)}{(e)}))}{(x + 1)^{\frac{1}{2}}} + 2(\frac{\frac{1}{2}(1 + 0)}{(x + 1)^{\frac{1}{2}}}){e}^{x^{2}} + 2(x + 1)^{\frac{1}{2}}({e}^{x^{2}}((2x)ln(e) + \frac{(x^{2})(0)}{(e)})) + 4(\frac{\frac{1}{2}(1 + 0)}{(x + 1)^{\frac{1}{2}}})x^{2}{e}^{x^{2}} + 4(x + 1)^{\frac{1}{2}}*2x{e}^{x^{2}} + 4(x + 1)^{\frac{1}{2}}x^{2}({e}^{x^{2}}((2x)ln(e) + \frac{(x^{2})(0)}{(e)}))\\=&\frac{3{e}^{x^{2}}}{8(x + 1)^{\frac{5}{2}}} - \frac{3x{e}^{x^{2}}}{2(x + 1)^{\frac{3}{2}}} + \frac{3{e}^{x^{2}}}{(x + 1)^{\frac{1}{2}}} + 12(x + 1)^{\frac{1}{2}}x{e}^{x^{2}} + \frac{6x^{2}{e}^{x^{2}}}{(x + 1)^{\frac{1}{2}}} + 8(x + 1)^{\frac{1}{2}}x^{3}{e}^{x^{2}}\\ \end{split}\end{equation} \]

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