There are 1 questions in this calculation: for each question, the 2 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ second\ derivative\ of\ function\ {(sqrt(\frac{x}{(1 + {x}^{2})}))}^{x}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = {sqrt(\frac{x}{(x^{2} + 1)})}^{x}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( {sqrt(\frac{x}{(x^{2} + 1)})}^{x}\right)}{dx}\\=&({sqrt(\frac{x}{(x^{2} + 1)})}^{x}((1)ln(sqrt(\frac{x}{(x^{2} + 1)})) + \frac{(x)(\frac{((\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x + \frac{1}{(x^{2} + 1)})*\frac{1}{2}}{(\frac{x}{(x^{2} + 1)})^{\frac{1}{2}}})}{(sqrt(\frac{x}{(x^{2} + 1)}))}))\\=&{sqrt(\frac{x}{(x^{2} + 1)})}^{x}ln(sqrt(\frac{x}{(x^{2} + 1)})) - \frac{x^{\frac{5}{2}}{sqrt(\frac{x}{(x^{2} + 1)})}^{x}}{(x^{2} + 1)^{\frac{3}{2}}sqrt(\frac{x}{(x^{2} + 1)})} + \frac{x^{\frac{1}{2}}{sqrt(\frac{x}{(x^{2} + 1)})}^{x}}{2(x^{2} + 1)^{\frac{1}{2}}sqrt(\frac{x}{(x^{2} + 1)})}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( {sqrt(\frac{x}{(x^{2} + 1)})}^{x}ln(sqrt(\frac{x}{(x^{2} + 1)})) - \frac{x^{\frac{5}{2}}{sqrt(\frac{x}{(x^{2} + 1)})}^{x}}{(x^{2} + 1)^{\frac{3}{2}}sqrt(\frac{x}{(x^{2} + 1)})} + \frac{x^{\frac{1}{2}}{sqrt(\frac{x}{(x^{2} + 1)})}^{x}}{2(x^{2} + 1)^{\frac{1}{2}}sqrt(\frac{x}{(x^{2} + 1)})}\right)}{dx}\\=&({sqrt(\frac{x}{(x^{2} + 1)})}^{x}((1)ln(sqrt(\frac{x}{(x^{2} + 1)})) + \frac{(x)(\frac{((\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x + \frac{1}{(x^{2} + 1)})*\frac{1}{2}}{(\frac{x}{(x^{2} + 1)})^{\frac{1}{2}}})}{(sqrt(\frac{x}{(x^{2} + 1)}))}))ln(sqrt(\frac{x}{(x^{2} + 1)})) + \frac{{sqrt(\frac{x}{(x^{2} + 1)})}^{x}((\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x + \frac{1}{(x^{2} + 1)})*\frac{1}{2}}{(sqrt(\frac{x}{(x^{2} + 1)}))(\frac{x}{(x^{2} + 1)})^{\frac{1}{2}}} - \frac{(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} + 1)^{\frac{5}{2}}})x^{\frac{5}{2}}{sqrt(\frac{x}{(x^{2} + 1)})}^{x}}{sqrt(\frac{x}{(x^{2} + 1)})} - \frac{\frac{5}{2}x^{\frac{3}{2}}{sqrt(\frac{x}{(x^{2} + 1)})}^{x}}{(x^{2} + 1)^{\frac{3}{2}}sqrt(\frac{x}{(x^{2} + 1)})} - \frac{x^{\frac{5}{2}}({sqrt(\frac{x}{(x^{2} + 1)})}^{x}((1)ln(sqrt(\frac{x}{(x^{2} + 1)})) + \frac{(x)(\frac{((\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x + \frac{1}{(x^{2} + 1)})*\frac{1}{2}}{(\frac{x}{(x^{2} + 1)})^{\frac{1}{2}}})}{(sqrt(\frac{x}{(x^{2} + 1)}))}))}{(x^{2} + 1)^{\frac{3}{2}}sqrt(\frac{x}{(x^{2} + 1)})} - \frac{x^{\frac{5}{2}}{sqrt(\frac{x}{(x^{2} + 1)})}^{x}*-((\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x + \frac{1}{(x^{2} + 1)})*\frac{1}{2}}{(x^{2} + 1)^{\frac{3}{2}}(\frac{x}{(x^{2} + 1)})(\frac{x}{(x^{2} + 1)})^{\frac{1}{2}}} + \frac{(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{3}{2}}})x^{\frac{1}{2}}{sqrt(\frac{x}{(x^{2} + 1)})}^{x}}{2sqrt(\frac{x}{(x^{2} + 1)})} + \frac{\frac{1}{2}{sqrt(\frac{x}{(x^{2} + 1)})}^{x}}{2(x^{2} + 1)^{\frac{1}{2}}x^{\frac{1}{2}}sqrt(\frac{x}{(x^{2} + 1)})} + \frac{x^{\frac{1}{2}}({sqrt(\frac{x}{(x^{2} + 1)})}^{x}((1)ln(sqrt(\frac{x}{(x^{2} + 1)})) + \frac{(x)(\frac{((\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x + \frac{1}{(x^{2} + 1)})*\frac{1}{2}}{(\frac{x}{(x^{2} + 1)})^{\frac{1}{2}}})}{(sqrt(\frac{x}{(x^{2} + 1)}))}))}{2(x^{2} + 1)^{\frac{1}{2}}sqrt(\frac{x}{(x^{2} + 1)})} + \frac{x^{\frac{1}{2}}{sqrt(\frac{x}{(x^{2} + 1)})}^{x}*-((\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x + \frac{1}{(x^{2} + 1)})*\frac{1}{2}}{2(x^{2} + 1)^{\frac{1}{2}}(\frac{x}{(x^{2} + 1)})(\frac{x}{(x^{2} + 1)})^{\frac{1}{2}}}\\=&{sqrt(\frac{x}{(x^{2} + 1)})}^{x}ln^{2}(sqrt(\frac{x}{(x^{2} + 1)})) - \frac{2x^{\frac{5}{2}}{sqrt(\frac{x}{(x^{2} + 1)})}^{x}ln(sqrt(\frac{x}{(x^{2} + 1)}))}{(x^{2} + 1)^{\frac{3}{2}}sqrt(\frac{x}{(x^{2} + 1)})} + \frac{x^{\frac{1}{2}}{sqrt(\frac{x}{(x^{2} + 1)})}^{x}ln(sqrt(\frac{x}{(x^{2} + 1)}))}{(x^{2} + 1)^{\frac{1}{2}}sqrt(\frac{x}{(x^{2} + 1)})} - \frac{4x^{\frac{3}{2}}{sqrt(\frac{x}{(x^{2} + 1)})}^{x}}{(x^{2} + 1)^{\frac{3}{2}}sqrt(\frac{x}{(x^{2} + 1)})} + \frac{3{sqrt(\frac{x}{(x^{2} + 1)})}^{x}}{4(x^{2} + 1)^{\frac{1}{2}}x^{\frac{1}{2}}sqrt(\frac{x}{(x^{2} + 1)})} + \frac{3x^{\frac{7}{2}}{sqrt(\frac{x}{(x^{2} + 1)})}^{x}}{(x^{2} + 1)^{\frac{5}{2}}sqrt(\frac{x}{(x^{2} + 1)})} + \frac{x^{5}{sqrt(\frac{x}{(x^{2} + 1)})}^{x}}{(x^{2} + 1)^{3}sqrt(\frac{x}{(x^{2} + 1)})^{2}} - \frac{x^{3}{sqrt(\frac{x}{(x^{2} + 1)})}^{x}}{(x^{2} + 1)^{2}sqrt(\frac{x}{(x^{2} + 1)})^{2}} + \frac{x{sqrt(\frac{x}{(x^{2} + 1)})}^{x}}{4(x^{2} + 1)sqrt(\frac{x}{(x^{2} + 1)})^{2}} + \frac{x{sqrt(\frac{x}{(x^{2} + 1)})}^{x}}{(x^{2} + 1)} - \frac{x^{3}{sqrt(\frac{x}{(x^{2} + 1)})}^{x}}{(x^{2} + 1)^{2}} - \frac{{sqrt(\frac{x}{(x^{2} + 1)})}^{x}}{4x}\\ \end{split}\end{equation} \]

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