There are 1 questions in this calculation: for each question, the 1 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ first\ derivative\ of\ function\ \frac{(({(x + 2)}^{\frac{1}{2}}{(3 - x)}^{4}))}{({(x + 1)}^{5})}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{(x + 2)^{\frac{1}{2}}x^{4}}{(x + 1)^{5}} - \frac{12(x + 2)^{\frac{1}{2}}x^{3}}{(x + 1)^{5}} + \frac{54(x + 2)^{\frac{1}{2}}x^{2}}{(x + 1)^{5}} - \frac{108(x + 2)^{\frac{1}{2}}x}{(x + 1)^{5}} + \frac{81(x + 2)^{\frac{1}{2}}}{(x + 1)^{5}}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{(x + 2)^{\frac{1}{2}}x^{4}}{(x + 1)^{5}} - \frac{12(x + 2)^{\frac{1}{2}}x^{3}}{(x + 1)^{5}} + \frac{54(x + 2)^{\frac{1}{2}}x^{2}}{(x + 1)^{5}} - \frac{108(x + 2)^{\frac{1}{2}}x}{(x + 1)^{5}} + \frac{81(x + 2)^{\frac{1}{2}}}{(x + 1)^{5}}\right)}{dx}\\=&(\frac{-5(1 + 0)}{(x + 1)^{6}})(x + 2)^{\frac{1}{2}}x^{4} + \frac{(\frac{\frac{1}{2}(1 + 0)}{(x + 2)^{\frac{1}{2}}})x^{4}}{(x + 1)^{5}} + \frac{(x + 2)^{\frac{1}{2}}*4x^{3}}{(x + 1)^{5}} - 12(\frac{-5(1 + 0)}{(x + 1)^{6}})(x + 2)^{\frac{1}{2}}x^{3} - \frac{12(\frac{\frac{1}{2}(1 + 0)}{(x + 2)^{\frac{1}{2}}})x^{3}}{(x + 1)^{5}} - \frac{12(x + 2)^{\frac{1}{2}}*3x^{2}}{(x + 1)^{5}} + 54(\frac{-5(1 + 0)}{(x + 1)^{6}})(x + 2)^{\frac{1}{2}}x^{2} + \frac{54(\frac{\frac{1}{2}(1 + 0)}{(x + 2)^{\frac{1}{2}}})x^{2}}{(x + 1)^{5}} + \frac{54(x + 2)^{\frac{1}{2}}*2x}{(x + 1)^{5}} - 108(\frac{-5(1 + 0)}{(x + 1)^{6}})(x + 2)^{\frac{1}{2}}x - \frac{108(\frac{\frac{1}{2}(1 + 0)}{(x + 2)^{\frac{1}{2}}})x}{(x + 1)^{5}} - \frac{108(x + 2)^{\frac{1}{2}}}{(x + 1)^{5}} + 81(\frac{-5(1 + 0)}{(x + 1)^{6}})(x + 2)^{\frac{1}{2}} + \frac{81(\frac{\frac{1}{2}(1 + 0)}{(x + 2)^{\frac{1}{2}}})}{(x + 1)^{5}}\\=&\frac{-5(x + 2)^{\frac{1}{2}}x^{4}}{(x + 1)^{6}} + \frac{x^{4}}{2(x + 2)^{\frac{1}{2}}(x + 1)^{5}} + \frac{4(x + 2)^{\frac{1}{2}}x^{3}}{(x + 1)^{5}} + \frac{60(x + 2)^{\frac{1}{2}}x^{3}}{(x + 1)^{6}} - \frac{6x^{3}}{(x + 2)^{\frac{1}{2}}(x + 1)^{5}} - \frac{36(x + 2)^{\frac{1}{2}}x^{2}}{(x + 1)^{5}} - \frac{270(x + 2)^{\frac{1}{2}}x^{2}}{(x + 1)^{6}} + \frac{27x^{2}}{(x + 2)^{\frac{1}{2}}(x + 1)^{5}} + \frac{108(x + 2)^{\frac{1}{2}}x}{(x + 1)^{5}} + \frac{540(x + 2)^{\frac{1}{2}}x}{(x + 1)^{6}} - \frac{54x}{(x + 2)^{\frac{1}{2}}(x + 1)^{5}} - \frac{108(x + 2)^{\frac{1}{2}}}{(x + 1)^{5}} + \frac{81}{2(x + 2)^{\frac{1}{2}}(x + 1)^{5}} - \frac{405(x + 2)^{\frac{1}{2}}}{(x + 1)^{6}}\\ \end{split}\end{equation} \]

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