There are 1 questions in this calculation: for each question, the 1 derivative of x is calculated.
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\[ \begin{equation}\begin{split}[1/1]Find\ the\ first\ derivative\ of\ function\ sin(\frac{(1 + {x}^{4})}{({x}^{2} + cos({x}^{3}) - 51{x}^{5})})\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = sin(\frac{x^{4}}{(x^{2} + cos(x^{3}) - 51x^{5})} + \frac{1}{(x^{2} + cos(x^{3}) - 51x^{5})})\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( sin(\frac{x^{4}}{(x^{2} + cos(x^{3}) - 51x^{5})} + \frac{1}{(x^{2} + cos(x^{3}) - 51x^{5})})\right)}{dx}\\=&cos(\frac{x^{4}}{(x^{2} + cos(x^{3}) - 51x^{5})} + \frac{1}{(x^{2} + cos(x^{3}) - 51x^{5})})((\frac{-(2x + -sin(x^{3})*3x^{2} - 51*5x^{4})}{(x^{2} + cos(x^{3}) - 51x^{5})^{2}})x^{4} + \frac{4x^{3}}{(x^{2} + cos(x^{3}) - 51x^{5})} + (\frac{-(2x + -sin(x^{3})*3x^{2} - 51*5x^{4})}{(x^{2} + cos(x^{3}) - 51x^{5})^{2}}))\\=&\frac{-2x^{5}cos(\frac{x^{4}}{(x^{2} + cos(x^{3}) - 51x^{5})} + \frac{1}{(x^{2} + cos(x^{3}) - 51x^{5})})}{(x^{2} + cos(x^{3}) - 51x^{5})^{2}} + \frac{3x^{6}sin(x^{3})cos(\frac{x^{4}}{(x^{2} + cos(x^{3}) - 51x^{5})} + \frac{1}{(x^{2} + cos(x^{3}) - 51x^{5})})}{(x^{2} + cos(x^{3}) - 51x^{5})^{2}} + \frac{255x^{8}cos(\frac{x^{4}}{(x^{2} + cos(x^{3}) - 51x^{5})} + \frac{1}{(x^{2} + cos(x^{3}) - 51x^{5})})}{(x^{2} + cos(x^{3}) - 51x^{5})^{2}} + \frac{4x^{3}cos(\frac{x^{4}}{(x^{2} + cos(x^{3}) - 51x^{5})} + \frac{1}{(x^{2} + cos(x^{3}) - 51x^{5})})}{(x^{2} + cos(x^{3}) - 51x^{5})} - \frac{2xcos(\frac{x^{4}}{(x^{2} + cos(x^{3}) - 51x^{5})} + \frac{1}{(x^{2} + cos(x^{3}) - 51x^{5})})}{(x^{2} + cos(x^{3}) - 51x^{5})^{2}} + \frac{3x^{2}sin(x^{3})cos(\frac{x^{4}}{(x^{2} + cos(x^{3}) - 51x^{5})} + \frac{1}{(x^{2} + cos(x^{3}) - 51x^{5})})}{(x^{2} + cos(x^{3}) - 51x^{5})^{2}} + \frac{255x^{4}cos(\frac{x^{4}}{(x^{2} + cos(x^{3}) - 51x^{5})} + \frac{1}{(x^{2} + cos(x^{3}) - 51x^{5})})}{(x^{2} + cos(x^{3}) - 51x^{5})^{2}}\\ \end{split}\end{equation} \]

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