There are 1 questions in this calculation: for each question, the 1 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ first\ derivative\ of\ function\ \frac{{(x + 2)}^{3}}{sqrt({(x + 1)}^{5}(x + 3))}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{x^{3}}{sqrt(x^{6} + 8x^{5} + 25x^{4} + 40x^{3} + 35x^{2} + 16x + 3)} + \frac{6x^{2}}{sqrt(x^{6} + 8x^{5} + 25x^{4} + 40x^{3} + 35x^{2} + 16x + 3)} + \frac{12x}{sqrt(x^{6} + 8x^{5} + 25x^{4} + 40x^{3} + 35x^{2} + 16x + 3)} + \frac{8}{sqrt(x^{6} + 8x^{5} + 25x^{4} + 40x^{3} + 35x^{2} + 16x + 3)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{x^{3}}{sqrt(x^{6} + 8x^{5} + 25x^{4} + 40x^{3} + 35x^{2} + 16x + 3)} + \frac{6x^{2}}{sqrt(x^{6} + 8x^{5} + 25x^{4} + 40x^{3} + 35x^{2} + 16x + 3)} + \frac{12x}{sqrt(x^{6} + 8x^{5} + 25x^{4} + 40x^{3} + 35x^{2} + 16x + 3)} + \frac{8}{sqrt(x^{6} + 8x^{5} + 25x^{4} + 40x^{3} + 35x^{2} + 16x + 3)}\right)}{dx}\\=&\frac{3x^{2}}{sqrt(x^{6} + 8x^{5} + 25x^{4} + 40x^{3} + 35x^{2} + 16x + 3)} + \frac{x^{3}*-(6x^{5} + 8*5x^{4} + 25*4x^{3} + 40*3x^{2} + 35*2x + 16 + 0)*\frac{1}{2}}{(x^{6} + 8x^{5} + 25x^{4} + 40x^{3} + 35x^{2} + 16x + 3)(x^{6} + 8x^{5} + 25x^{4} + 40x^{3} + 35x^{2} + 16x + 3)^{\frac{1}{2}}} + \frac{6*2x}{sqrt(x^{6} + 8x^{5} + 25x^{4} + 40x^{3} + 35x^{2} + 16x + 3)} + \frac{6x^{2}*-(6x^{5} + 8*5x^{4} + 25*4x^{3} + 40*3x^{2} + 35*2x + 16 + 0)*\frac{1}{2}}{(x^{6} + 8x^{5} + 25x^{4} + 40x^{3} + 35x^{2} + 16x + 3)(x^{6} + 8x^{5} + 25x^{4} + 40x^{3} + 35x^{2} + 16x + 3)^{\frac{1}{2}}} + \frac{12}{sqrt(x^{6} + 8x^{5} + 25x^{4} + 40x^{3} + 35x^{2} + 16x + 3)} + \frac{12x*-(6x^{5} + 8*5x^{4} + 25*4x^{3} + 40*3x^{2} + 35*2x + 16 + 0)*\frac{1}{2}}{(x^{6} + 8x^{5} + 25x^{4} + 40x^{3} + 35x^{2} + 16x + 3)(x^{6} + 8x^{5} + 25x^{4} + 40x^{3} + 35x^{2} + 16x + 3)^{\frac{1}{2}}} + \frac{8*-(6x^{5} + 8*5x^{4} + 25*4x^{3} + 40*3x^{2} + 35*2x + 16 + 0)*\frac{1}{2}}{(x^{6} + 8x^{5} + 25x^{4} + 40x^{3} + 35x^{2} + 16x + 3)(x^{6} + 8x^{5} + 25x^{4} + 40x^{3} + 35x^{2} + 16x + 3)^{\frac{1}{2}}}\\=&\frac{3x^{2}}{sqrt(x^{6} + 8x^{5} + 25x^{4} + 40x^{3} + 35x^{2} + 16x + 3)} - \frac{3x^{8}}{(x^{6} + 8x^{5} + 25x^{4} + 40x^{3} + 35x^{2} + 16x + 3)^{\frac{3}{2}}} - \frac{38x^{7}}{(x^{6} + 8x^{5} + 25x^{4} + 40x^{3} + 35x^{2} + 16x + 3)^{\frac{3}{2}}} - \frac{206x^{6}}{(x^{6} + 8x^{5} + 25x^{4} + 40x^{3} + 35x^{2} + 16x + 3)^{\frac{3}{2}}} - \frac{624x^{5}}{(x^{6} + 8x^{5} + 25x^{4} + 40x^{3} + 35x^{2} + 16x + 3)^{\frac{3}{2}}} - \frac{1155x^{4}}{(x^{6} + 8x^{5} + 25x^{4} + 40x^{3} + 35x^{2} + 16x + 3)^{\frac{3}{2}}} - \frac{1338x^{3}}{(x^{6} + 8x^{5} + 25x^{4} + 40x^{3} + 35x^{2} + 16x + 3)^{\frac{3}{2}}} + \frac{12x}{sqrt(x^{6} + 8x^{5} + 25x^{4} + 40x^{3} + 35x^{2} + 16x + 3)} - \frac{948x^{2}}{(x^{6} + 8x^{5} + 25x^{4} + 40x^{3} + 35x^{2} + 16x + 3)^{\frac{3}{2}}} + \frac{12}{sqrt(x^{6} + 8x^{5} + 25x^{4} + 40x^{3} + 35x^{2} + 16x + 3)} - \frac{376x}{(x^{6} + 8x^{5} + 25x^{4} + 40x^{3} + 35x^{2} + 16x + 3)^{\frac{3}{2}}} - \frac{64}{(x^{6} + 8x^{5} + 25x^{4} + 40x^{3} + 35x^{2} + 16x + 3)^{\frac{3}{2}}}\\ \end{split}\end{equation} \]

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