There are 1 questions in this calculation: for each question, the 1 derivative of x is calculated.
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\[ \begin{equation}\begin{split}[1/1]Find\ the\ first\ derivative\ of\ function\ \frac{(\frac{{(x + 1)}^{2}{({x}^{2} + 1)}^{1}}{3})sqrt(x)}{({x}^{2} + 2)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{\frac{1}{3}x^{4}sqrt(x)}{(x^{2} + 2)} + \frac{\frac{2}{3}x^{3}sqrt(x)}{(x^{2} + 2)} + \frac{\frac{2}{3}x^{2}sqrt(x)}{(x^{2} + 2)} + \frac{\frac{2}{3}xsqrt(x)}{(x^{2} + 2)} + \frac{\frac{1}{3}sqrt(x)}{(x^{2} + 2)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{\frac{1}{3}x^{4}sqrt(x)}{(x^{2} + 2)} + \frac{\frac{2}{3}x^{3}sqrt(x)}{(x^{2} + 2)} + \frac{\frac{2}{3}x^{2}sqrt(x)}{(x^{2} + 2)} + \frac{\frac{2}{3}xsqrt(x)}{(x^{2} + 2)} + \frac{\frac{1}{3}sqrt(x)}{(x^{2} + 2)}\right)}{dx}\\=&\frac{1}{3}(\frac{-(2x + 0)}{(x^{2} + 2)^{2}})x^{4}sqrt(x) + \frac{\frac{1}{3}*4x^{3}sqrt(x)}{(x^{2} + 2)} + \frac{\frac{1}{3}x^{4}*\frac{1}{2}}{(x^{2} + 2)(x)^{\frac{1}{2}}} + \frac{2}{3}(\frac{-(2x + 0)}{(x^{2} + 2)^{2}})x^{3}sqrt(x) + \frac{\frac{2}{3}*3x^{2}sqrt(x)}{(x^{2} + 2)} + \frac{\frac{2}{3}x^{3}*\frac{1}{2}}{(x^{2} + 2)(x)^{\frac{1}{2}}} + \frac{2}{3}(\frac{-(2x + 0)}{(x^{2} + 2)^{2}})x^{2}sqrt(x) + \frac{\frac{2}{3}*2xsqrt(x)}{(x^{2} + 2)} + \frac{\frac{2}{3}x^{2}*\frac{1}{2}}{(x^{2} + 2)(x)^{\frac{1}{2}}} + \frac{2}{3}(\frac{-(2x + 0)}{(x^{2} + 2)^{2}})xsqrt(x) + \frac{\frac{2}{3}sqrt(x)}{(x^{2} + 2)} + \frac{\frac{2}{3}x*\frac{1}{2}}{(x^{2} + 2)(x)^{\frac{1}{2}}} + \frac{1}{3}(\frac{-(2x + 0)}{(x^{2} + 2)^{2}})sqrt(x) + \frac{\frac{1}{3}*\frac{1}{2}}{(x^{2} + 2)(x)^{\frac{1}{2}}}\\=&\frac{-2x^{5}sqrt(x)}{3(x^{2} + 2)^{2}} + \frac{4x^{3}sqrt(x)}{3(x^{2} + 2)} - \frac{4x^{4}sqrt(x)}{3(x^{2} + 2)^{2}} - \frac{4x^{3}sqrt(x)}{3(x^{2} + 2)^{2}} + \frac{2x^{2}sqrt(x)}{(x^{2} + 2)} - \frac{4x^{2}sqrt(x)}{3(x^{2} + 2)^{2}} + \frac{4xsqrt(x)}{3(x^{2} + 2)} - \frac{2xsqrt(x)}{3(x^{2} + 2)^{2}} + \frac{x^{\frac{3}{2}}}{3(x^{2} + 2)} + \frac{x^{\frac{5}{2}}}{3(x^{2} + 2)} + \frac{2sqrt(x)}{3(x^{2} + 2)} + \frac{x^{\frac{1}{2}}}{3(x^{2} + 2)} + \frac{x^{\frac{7}{2}}}{6(x^{2} + 2)} + \frac{1}{6(x^{2} + 2)x^{\frac{1}{2}}}\\ \end{split}\end{equation} \]

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