There are 1 questions in this calculation: for each question, the 1 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ first\ derivative\ of\ function\ \frac{(sin(x + 120) - sin(x))}{(sin(x + 240) - sin(x))}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{sin(x + 120)}{(sin(x + 240) - sin(x))} - \frac{sin(x)}{(sin(x + 240) - sin(x))}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{sin(x + 120)}{(sin(x + 240) - sin(x))} - \frac{sin(x)}{(sin(x + 240) - sin(x))}\right)}{dx}\\=&(\frac{-(cos(x + 240)(1 + 0) - cos(x))}{(sin(x + 240) - sin(x))^{2}})sin(x + 120) + \frac{cos(x + 120)(1 + 0)}{(sin(x + 240) - sin(x))} - (\frac{-(cos(x + 240)(1 + 0) - cos(x))}{(sin(x + 240) - sin(x))^{2}})sin(x) - \frac{cos(x)}{(sin(x + 240) - sin(x))}\\=&\frac{-sin(x + 120)cos(x + 240)}{(sin(x + 240) - sin(x))^{2}} + \frac{sin(x + 120)cos(x)}{(sin(x + 240) - sin(x))^{2}} + \frac{cos(x + 120)}{(sin(x + 240) - sin(x))} + \frac{sin(x)cos(x + 240)}{(sin(x + 240) - sin(x))^{2}} - \frac{sin(x)cos(x)}{(sin(x + 240) - sin(x))^{2}} - \frac{cos(x)}{(sin(x + 240) - sin(x))}\\ \end{split}\end{equation} \]

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