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

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