There are 1 questions in this calculation: for each question, the 5 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 5th\ derivative\ of\ function\ \frac{(xsqrt({x}^{2} + 1))}{2} + (\frac{1}{2})ln(x + sqrt({x}^{2} + 1))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{1}{2}xsqrt(x^{2} + 1) + \frac{1}{2}ln(x + sqrt(x^{2} + 1))\\\\ &\color{blue}{The\ 5th\ derivative\ of\ function:} \\=&\frac{105x^{6}}{2(x^{2} + 1)^{\frac{9}{2}}} - \frac{225x^{4}}{2(x^{2} + 1)^{\frac{7}{2}}} + \frac{135x^{2}}{2(x^{2} + 1)^{\frac{5}{2}}} + \frac{45x^{5}}{(x + sqrt(x^{2} + 1))^{3}(x^{2} + 1)^{\frac{7}{2}}} + \frac{30x^{5}}{(x + sqrt(x^{2} + 1))^{4}(x^{2} + 1)^{3}} + \frac{75x^{4}}{(x + sqrt(x^{2} + 1))^{3}(x^{2} + 1)^{3}} - \frac{46x^{3}}{(x^{2} + 1)^{\frac{5}{2}}(x + sqrt(x^{2} + 1))^{3}} + \frac{16x^{3}}{(x + sqrt(x^{2} + 1))^{3}(x^{2} + 1)^{\frac{5}{2}}} + \frac{72x^{3}}{(x + sqrt(x^{2} + 1))^{4}(x^{2} + 1)^{2}} - \frac{29x^{2}}{(x + sqrt(x^{2} + 1))^{3}(x^{2} + 1)^{2}} - \frac{6x}{(x^{2} + 1)^{\frac{3}{2}}(x + sqrt(x^{2} + 1))^{3}} + \frac{105x^{5}}{2(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)^{4}} - \frac{56x^{3}}{(x^{2} + 1)^{3}(x + sqrt(x^{2} + 1))^{2}} + \frac{31x}{2(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)^{2}} + \frac{12x^{5}}{(x + sqrt(x^{2} + 1))^{5}(x^{2} + 1)^{\frac{5}{2}}} + \frac{90x^{4}}{(x + sqrt(x^{2} + 1))^{4}(x^{2} + 1)^{\frac{5}{2}}} - \frac{12x^{3}}{(x^{2} + 1)^{2}(x + sqrt(x^{2} + 1))^{4}} + \frac{60x^{4}}{(x + sqrt(x^{2} + 1))^{5}(x^{2} + 1)^{2}} - \frac{61x^{2}}{(x^{2} + 1)^{2}(x + sqrt(x^{2} + 1))^{3}} + \frac{75x^{4}}{2(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)^{\frac{7}{2}}} - \frac{24x^{2}}{(x + sqrt(x^{2} + 1))^{4}(x^{2} + 1)^{\frac{3}{2}}} - \frac{27x^{2}}{(x^{2} + 1)^{\frac{5}{2}}(x + sqrt(x^{2} + 1))^{2}} + \frac{120x^{3}}{(x + sqrt(x^{2} + 1))^{5}(x^{2} + 1)^{\frac{3}{2}}} - \frac{36x^{2}}{(x^{2} + 1)^{\frac{3}{2}}(x + sqrt(x^{2} + 1))^{4}} - \frac{19x^{3}}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)^{3}} + \frac{7x}{(x^{2} + 1)^{2}(x + sqrt(x^{2} + 1))^{2}} - \frac{36x}{(x^{2} + 1)(x + sqrt(x^{2} + 1))^{4}} + \frac{120x^{2}}{(x + sqrt(x^{2} + 1))^{5}(x^{2} + 1)} - \frac{54x}{(x + sqrt(x^{2} + 1))^{4}(x^{2} + 1)} + \frac{60x}{(x + sqrt(x^{2} + 1))^{5}(x^{2} + 1)^{\frac{1}{2}}} - \frac{18x^{2}}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)^{\frac{5}{2}}} + \frac{105x^{5}}{2(x + sqrt(x^{2} + 1))(x^{2} + 1)^{\frac{9}{2}}} - \frac{135x^{3}}{2(x^{2} + 1)^{\frac{7}{2}}(x + sqrt(x^{2} + 1))} - \frac{9x}{(x + sqrt(x^{2} + 1))^{3}(x^{2} + 1)^{\frac{3}{2}}} + \frac{33x}{2(x + sqrt(x^{2} + 1))(x^{2} + 1)^{\frac{5}{2}}} - \frac{15x^{3}}{2(x + sqrt(x^{2} + 1))(x^{2} + 1)^{\frac{7}{2}}} + \frac{6x}{(x^{2} + 1)^{\frac{5}{2}}(x + sqrt(x^{2} + 1))} + \frac{6}{(x^{2} + 1)(x + sqrt(x^{2} + 1))^{3}} + \frac{9}{2(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)^{\frac{3}{2}}} - \frac{30}{(x + sqrt(x^{2} + 1))^{4}(x^{2} + 1)^{\frac{1}{2}}} + \frac{9}{(x + sqrt(x^{2} + 1))^{3}(x^{2} + 1)} + \frac{3}{(x^{2} + 1)^{\frac{3}{2}}(x + sqrt(x^{2} + 1))^{2}} - \frac{15}{2(x^{2} + 1)^{\frac{3}{2}}} + \frac{12}{(x + sqrt(x^{2} + 1))^{5}}\\ \end{split}\end{equation} \]

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