There are 1 questions in this calculation: for each question, the 1 derivative of x is calculated.
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\[ \begin{equation}\begin{split}[1/1]Find\ the\ first\ derivative\ of\ function\ \frac{({(2 - x)}^{5}sqrt(x + 1))}{({(x + 3)}^{7})}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{-x^{5}sqrt(x + 1)}{(x + 3)^{7}} + \frac{10x^{4}sqrt(x + 1)}{(x + 3)^{7}} - \frac{40x^{3}sqrt(x + 1)}{(x + 3)^{7}} + \frac{80x^{2}sqrt(x + 1)}{(x + 3)^{7}} - \frac{80xsqrt(x + 1)}{(x + 3)^{7}} + \frac{32sqrt(x + 1)}{(x + 3)^{7}}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{-x^{5}sqrt(x + 1)}{(x + 3)^{7}} + \frac{10x^{4}sqrt(x + 1)}{(x + 3)^{7}} - \frac{40x^{3}sqrt(x + 1)}{(x + 3)^{7}} + \frac{80x^{2}sqrt(x + 1)}{(x + 3)^{7}} - \frac{80xsqrt(x + 1)}{(x + 3)^{7}} + \frac{32sqrt(x + 1)}{(x + 3)^{7}}\right)}{dx}\\=&-(\frac{-7(1 + 0)}{(x + 3)^{8}})x^{5}sqrt(x + 1) - \frac{5x^{4}sqrt(x + 1)}{(x + 3)^{7}} - \frac{x^{5}(1 + 0)*\frac{1}{2}}{(x + 3)^{7}(x + 1)^{\frac{1}{2}}} + 10(\frac{-7(1 + 0)}{(x + 3)^{8}})x^{4}sqrt(x + 1) + \frac{10*4x^{3}sqrt(x + 1)}{(x + 3)^{7}} + \frac{10x^{4}(1 + 0)*\frac{1}{2}}{(x + 3)^{7}(x + 1)^{\frac{1}{2}}} - 40(\frac{-7(1 + 0)}{(x + 3)^{8}})x^{3}sqrt(x + 1) - \frac{40*3x^{2}sqrt(x + 1)}{(x + 3)^{7}} - \frac{40x^{3}(1 + 0)*\frac{1}{2}}{(x + 3)^{7}(x + 1)^{\frac{1}{2}}} + 80(\frac{-7(1 + 0)}{(x + 3)^{8}})x^{2}sqrt(x + 1) + \frac{80*2xsqrt(x + 1)}{(x + 3)^{7}} + \frac{80x^{2}(1 + 0)*\frac{1}{2}}{(x + 3)^{7}(x + 1)^{\frac{1}{2}}} - 80(\frac{-7(1 + 0)}{(x + 3)^{8}})xsqrt(x + 1) - \frac{80sqrt(x + 1)}{(x + 3)^{7}} - \frac{80x(1 + 0)*\frac{1}{2}}{(x + 3)^{7}(x + 1)^{\frac{1}{2}}} + 32(\frac{-7(1 + 0)}{(x + 3)^{8}})sqrt(x + 1) + \frac{32(1 + 0)*\frac{1}{2}}{(x + 3)^{7}(x + 1)^{\frac{1}{2}}}\\=&\frac{7x^{5}sqrt(x + 1)}{(x + 3)^{8}} - \frac{5x^{4}sqrt(x + 1)}{(x + 3)^{7}} - \frac{x^{5}}{2(x + 3)^{7}(x + 1)^{\frac{1}{2}}} - \frac{70x^{4}sqrt(x + 1)}{(x + 3)^{8}} + \frac{40x^{3}sqrt(x + 1)}{(x + 3)^{7}} + \frac{5x^{4}}{(x + 3)^{7}(x + 1)^{\frac{1}{2}}} + \frac{280x^{3}sqrt(x + 1)}{(x + 3)^{8}} - \frac{120x^{2}sqrt(x + 1)}{(x + 3)^{7}} - \frac{20x^{3}}{(x + 3)^{7}(x + 1)^{\frac{1}{2}}} - \frac{560x^{2}sqrt(x + 1)}{(x + 3)^{8}} + \frac{160xsqrt(x + 1)}{(x + 3)^{7}} + \frac{40x^{2}}{(x + 3)^{7}(x + 1)^{\frac{1}{2}}} + \frac{560xsqrt(x + 1)}{(x + 3)^{8}} - \frac{80sqrt(x + 1)}{(x + 3)^{7}} - \frac{40x}{(x + 3)^{7}(x + 1)^{\frac{1}{2}}} - \frac{224sqrt(x + 1)}{(x + 3)^{8}} + \frac{16}{(x + 3)^{7}(x + 1)^{\frac{1}{2}}}\\ \end{split}\end{equation} \]

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