There are 1 questions in this calculation: for each question, the 1 derivative of x is calculated.
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\[ \begin{equation}\begin{split}[1/1]Find\ the\ first\ derivative\ of\ function\ {(\frac{({(x + 4)}^{4}({x}^{2} + 1))}{({(x - 1)}^{5})})}^{\frac{1}{3}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = (\frac{x^{6}}{(x - 1)^{5}} + \frac{97x^{4}}{(x - 1)^{5}} + \frac{16x^{5}}{(x - 1)^{5}} + \frac{272x^{3}}{(x - 1)^{5}} + \frac{352x^{2}}{(x - 1)^{5}} + \frac{256x}{(x - 1)^{5}} + \frac{256}{(x - 1)^{5}})^{\frac{1}{3}}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( (\frac{x^{6}}{(x - 1)^{5}} + \frac{97x^{4}}{(x - 1)^{5}} + \frac{16x^{5}}{(x - 1)^{5}} + \frac{272x^{3}}{(x - 1)^{5}} + \frac{352x^{2}}{(x - 1)^{5}} + \frac{256x}{(x - 1)^{5}} + \frac{256}{(x - 1)^{5}})^{\frac{1}{3}}\right)}{dx}\\=&(\frac{\frac{1}{3}((\frac{-5(1 + 0)}{(x - 1)^{6}})x^{6} + \frac{6x^{5}}{(x - 1)^{5}} + 97(\frac{-5(1 + 0)}{(x - 1)^{6}})x^{4} + \frac{97*4x^{3}}{(x - 1)^{5}} + 16(\frac{-5(1 + 0)}{(x - 1)^{6}})x^{5} + \frac{16*5x^{4}}{(x - 1)^{5}} + 272(\frac{-5(1 + 0)}{(x - 1)^{6}})x^{3} + \frac{272*3x^{2}}{(x - 1)^{5}} + 352(\frac{-5(1 + 0)}{(x - 1)^{6}})x^{2} + \frac{352*2x}{(x - 1)^{5}} + 256(\frac{-5(1 + 0)}{(x - 1)^{6}})x + \frac{256}{(x - 1)^{5}} + 256(\frac{-5(1 + 0)}{(x - 1)^{6}}))}{(\frac{x^{6}}{(x - 1)^{5}} + \frac{97x^{4}}{(x - 1)^{5}} + \frac{16x^{5}}{(x - 1)^{5}} + \frac{272x^{3}}{(x - 1)^{5}} + \frac{352x^{2}}{(x - 1)^{5}} + \frac{256x}{(x - 1)^{5}} + \frac{256}{(x - 1)^{5}})^{\frac{2}{3}}})\\=&\frac{-5x^{6}}{3(\frac{x^{6}}{(x - 1)^{5}} + \frac{97x^{4}}{(x - 1)^{5}} + \frac{16x^{5}}{(x - 1)^{5}} + \frac{272x^{3}}{(x - 1)^{5}} + \frac{352x^{2}}{(x - 1)^{5}} + \frac{256x}{(x - 1)^{5}} + \frac{256}{(x - 1)^{5}})^{\frac{2}{3}}(x - 1)^{6}} + \frac{2x^{5}}{(\frac{x^{6}}{(x - 1)^{5}} + \frac{97x^{4}}{(x - 1)^{5}} + \frac{16x^{5}}{(x - 1)^{5}} + \frac{272x^{3}}{(x - 1)^{5}} + \frac{352x^{2}}{(x - 1)^{5}} + \frac{256x}{(x - 1)^{5}} + \frac{256}{(x - 1)^{5}})^{\frac{2}{3}}(x - 1)^{5}} - \frac{485x^{4}}{3(\frac{x^{6}}{(x - 1)^{5}} + \frac{97x^{4}}{(x - 1)^{5}} + \frac{16x^{5}}{(x - 1)^{5}} + \frac{272x^{3}}{(x - 1)^{5}} + \frac{352x^{2}}{(x - 1)^{5}} + \frac{256x}{(x - 1)^{5}} + \frac{256}{(x - 1)^{5}})^{\frac{2}{3}}(x - 1)^{6}} + \frac{388x^{3}}{3(\frac{x^{6}}{(x - 1)^{5}} + \frac{97x^{4}}{(x - 1)^{5}} + \frac{16x^{5}}{(x - 1)^{5}} + \frac{272x^{3}}{(x - 1)^{5}} + \frac{352x^{2}}{(x - 1)^{5}} + \frac{256x}{(x - 1)^{5}} + \frac{256}{(x - 1)^{5}})^{\frac{2}{3}}(x - 1)^{5}} - \frac{80x^{5}}{3(\frac{x^{6}}{(x - 1)^{5}} + \frac{97x^{4}}{(x - 1)^{5}} + \frac{16x^{5}}{(x - 1)^{5}} + \frac{272x^{3}}{(x - 1)^{5}} + \frac{352x^{2}}{(x - 1)^{5}} + \frac{256x}{(x - 1)^{5}} + \frac{256}{(x - 1)^{5}})^{\frac{2}{3}}(x - 1)^{6}} + \frac{80x^{4}}{3(\frac{x^{6}}{(x - 1)^{5}} + \frac{97x^{4}}{(x - 1)^{5}} + \frac{16x^{5}}{(x - 1)^{5}} + \frac{272x^{3}}{(x - 1)^{5}} + \frac{352x^{2}}{(x - 1)^{5}} + \frac{256x}{(x - 1)^{5}} + \frac{256}{(x - 1)^{5}})^{\frac{2}{3}}(x - 1)^{5}} - \frac{1360x^{3}}{3(\frac{x^{6}}{(x - 1)^{5}} + \frac{97x^{4}}{(x - 1)^{5}} + \frac{16x^{5}}{(x - 1)^{5}} + \frac{272x^{3}}{(x - 1)^{5}} + \frac{352x^{2}}{(x - 1)^{5}} + \frac{256x}{(x - 1)^{5}} + \frac{256}{(x - 1)^{5}})^{\frac{2}{3}}(x - 1)^{6}} + \frac{272x^{2}}{(\frac{x^{6}}{(x - 1)^{5}} + \frac{97x^{4}}{(x - 1)^{5}} + \frac{16x^{5}}{(x - 1)^{5}} + \frac{272x^{3}}{(x - 1)^{5}} + \frac{352x^{2}}{(x - 1)^{5}} + \frac{256x}{(x - 1)^{5}} + \frac{256}{(x - 1)^{5}})^{\frac{2}{3}}(x - 1)^{5}} - \frac{1760x^{2}}{3(\frac{x^{6}}{(x - 1)^{5}} + \frac{97x^{4}}{(x - 1)^{5}} + \frac{16x^{5}}{(x - 1)^{5}} + \frac{272x^{3}}{(x - 1)^{5}} + \frac{352x^{2}}{(x - 1)^{5}} + \frac{256x}{(x - 1)^{5}} + \frac{256}{(x - 1)^{5}})^{\frac{2}{3}}(x - 1)^{6}} + \frac{704x}{3(\frac{x^{6}}{(x - 1)^{5}} + \frac{97x^{4}}{(x - 1)^{5}} + \frac{16x^{5}}{(x - 1)^{5}} + \frac{272x^{3}}{(x - 1)^{5}} + \frac{352x^{2}}{(x - 1)^{5}} + \frac{256x}{(x - 1)^{5}} + \frac{256}{(x - 1)^{5}})^{\frac{2}{3}}(x - 1)^{5}} - \frac{1280x}{3(\frac{x^{6}}{(x - 1)^{5}} + \frac{97x^{4}}{(x - 1)^{5}} + \frac{16x^{5}}{(x - 1)^{5}} + \frac{272x^{3}}{(x - 1)^{5}} + \frac{352x^{2}}{(x - 1)^{5}} + \frac{256x}{(x - 1)^{5}} + \frac{256}{(x - 1)^{5}})^{\frac{2}{3}}(x - 1)^{6}} + \frac{256}{3(\frac{x^{6}}{(x - 1)^{5}} + \frac{97x^{4}}{(x - 1)^{5}} + \frac{16x^{5}}{(x - 1)^{5}} + \frac{272x^{3}}{(x - 1)^{5}} + \frac{352x^{2}}{(x - 1)^{5}} + \frac{256x}{(x - 1)^{5}} + \frac{256}{(x - 1)^{5}})^{\frac{2}{3}}(x - 1)^{5}} - \frac{1280}{3(\frac{x^{6}}{(x - 1)^{5}} + \frac{97x^{4}}{(x - 1)^{5}} + \frac{16x^{5}}{(x - 1)^{5}} + \frac{272x^{3}}{(x - 1)^{5}} + \frac{352x^{2}}{(x - 1)^{5}} + \frac{256x}{(x - 1)^{5}} + \frac{256}{(x - 1)^{5}})^{\frac{2}{3}}(x - 1)^{6}}\\ \end{split}\end{equation} \]

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