There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{(x + y + z)}{(sin(x) + sin(y) + sin(z))}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{x}{(sin(x) + sin(y) + sin(z))} + \frac{y}{(sin(x) + sin(y) + sin(z))} + \frac{z}{(sin(x) + sin(y) + sin(z))}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{x}{(sin(x) + sin(y) + sin(z))} + \frac{y}{(sin(x) + sin(y) + sin(z))} + \frac{z}{(sin(x) + sin(y) + sin(z))}\right)}{dx}\\=&(\frac{-(cos(x) + cos(y)*0 + cos(z)*0)}{(sin(x) + sin(y) + sin(z))^{2}})x + \frac{1}{(sin(x) + sin(y) + sin(z))} + (\frac{-(cos(x) + cos(y)*0 + cos(z)*0)}{(sin(x) + sin(y) + sin(z))^{2}})y + 0 + (\frac{-(cos(x) + cos(y)*0 + cos(z)*0)}{(sin(x) + sin(y) + sin(z))^{2}})z + 0\\=&\frac{-xcos(x)}{(sin(x) + sin(y) + sin(z))^{2}} - \frac{ycos(x)}{(sin(x) + sin(y) + sin(z))^{2}} - \frac{zcos(x)}{(sin(x) + sin(y) + sin(z))^{2}} + \frac{1}{(sin(x) + sin(y) + sin(z))}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-xcos(x)}{(sin(x) + sin(y) + sin(z))^{2}} - \frac{ycos(x)}{(sin(x) + sin(y) + sin(z))^{2}} - \frac{zcos(x)}{(sin(x) + sin(y) + sin(z))^{2}} + \frac{1}{(sin(x) + sin(y) + sin(z))}\right)}{dx}\\=&-(\frac{-2(cos(x) + cos(y)*0 + cos(z)*0)}{(sin(x) + sin(y) + sin(z))^{3}})xcos(x) - \frac{cos(x)}{(sin(x) + sin(y) + sin(z))^{2}} - \frac{x*-sin(x)}{(sin(x) + sin(y) + sin(z))^{2}} - (\frac{-2(cos(x) + cos(y)*0 + cos(z)*0)}{(sin(x) + sin(y) + sin(z))^{3}})ycos(x) - \frac{y*-sin(x)}{(sin(x) + sin(y) + sin(z))^{2}} - (\frac{-2(cos(x) + cos(y)*0 + cos(z)*0)}{(sin(x) + sin(y) + sin(z))^{3}})zcos(x) - \frac{z*-sin(x)}{(sin(x) + sin(y) + sin(z))^{2}} + (\frac{-(cos(x) + cos(y)*0 + cos(z)*0)}{(sin(x) + sin(y) + sin(z))^{2}})\\=&\frac{2xcos^{2}(x)}{(sin(x) + sin(y) + sin(z))^{3}} - \frac{2cos(x)}{(sin(x) + sin(y) + sin(z))^{2}} + \frac{xsin(x)}{(sin(x) + sin(y) + sin(z))^{2}} + \frac{2ycos^{2}(x)}{(sin(x) + sin(y) + sin(z))^{3}} + \frac{ysin(x)}{(sin(x) + sin(y) + sin(z))^{2}} + \frac{2zcos^{2}(x)}{(sin(x) + sin(y) + sin(z))^{3}} + \frac{zsin(x)}{(sin(x) + sin(y) + sin(z))^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{2xcos^{2}(x)}{(sin(x) + sin(y) + sin(z))^{3}} - \frac{2cos(x)}{(sin(x) + sin(y) + sin(z))^{2}} + \frac{xsin(x)}{(sin(x) + sin(y) + sin(z))^{2}} + \frac{2ycos^{2}(x)}{(sin(x) + sin(y) + sin(z))^{3}} + \frac{ysin(x)}{(sin(x) + sin(y) + sin(z))^{2}} + \frac{2zcos^{2}(x)}{(sin(x) + sin(y) + sin(z))^{3}} + \frac{zsin(x)}{(sin(x) + sin(y) + sin(z))^{2}}\right)}{dx}\\=&2(\frac{-3(cos(x) + cos(y)*0 + cos(z)*0)}{(sin(x) + sin(y) + sin(z))^{4}})xcos^{2}(x) + \frac{2cos^{2}(x)}{(sin(x) + sin(y) + sin(z))^{3}} + \frac{2x*-2cos(x)sin(x)}{(sin(x) + sin(y) + sin(z))^{3}} - 2(\frac{-2(cos(x) + cos(y)*0 + cos(z)*0)}{(sin(x) + sin(y) + sin(z))^{3}})cos(x) - \frac{2*-sin(x)}{(sin(x) + sin(y) + sin(z))^{2}} + (\frac{-2(cos(x) + cos(y)*0 + cos(z)*0)}{(sin(x) + sin(y) + sin(z))^{3}})xsin(x) + \frac{sin(x)}{(sin(x) + sin(y) + sin(z))^{2}} + \frac{xcos(x)}{(sin(x) + sin(y) + sin(z))^{2}} + 2(\frac{-3(cos(x) + cos(y)*0 + cos(z)*0)}{(sin(x) + sin(y) + sin(z))^{4}})ycos^{2}(x) + \frac{2y*-2cos(x)sin(x)}{(sin(x) + sin(y) + sin(z))^{3}} + (\frac{-2(cos(x) + cos(y)*0 + cos(z)*0)}{(sin(x) + sin(y) + sin(z))^{3}})ysin(x) + \frac{ycos(x)}{(sin(x) + sin(y) + sin(z))^{2}} + 2(\frac{-3(cos(x) + cos(y)*0 + cos(z)*0)}{(sin(x) + sin(y) + sin(z))^{4}})zcos^{2}(x) + \frac{2z*-2cos(x)sin(x)}{(sin(x) + sin(y) + sin(z))^{3}} + (\frac{-2(cos(x) + cos(y)*0 + cos(z)*0)}{(sin(x) + sin(y) + sin(z))^{3}})zsin(x) + \frac{zcos(x)}{(sin(x) + sin(y) + sin(z))^{2}}\\=&\frac{-6xcos^{3}(x)}{(sin(x) + sin(y) + sin(z))^{4}} + \frac{6cos^{2}(x)}{(sin(x) + sin(y) + sin(z))^{3}} - \frac{6xsin(x)cos(x)}{(sin(x) + sin(y) + sin(z))^{3}} + \frac{3sin(x)}{(sin(x) + sin(y) + sin(z))^{2}} + \frac{xcos(x)}{(sin(x) + sin(y) + sin(z))^{2}} - \frac{6ycos^{3}(x)}{(sin(x) + sin(y) + sin(z))^{4}} - \frac{6ysin(x)cos(x)}{(sin(x) + sin(y) + sin(z))^{3}} + \frac{ycos(x)}{(sin(x) + sin(y) + sin(z))^{2}} - \frac{6zcos^{3}(x)}{(sin(x) + sin(y) + sin(z))^{4}} - \frac{6zsin(x)cos(x)}{(sin(x) + sin(y) + sin(z))^{3}} + \frac{zcos(x)}{(sin(x) + sin(y) + sin(z))^{2}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-6xcos^{3}(x)}{(sin(x) + sin(y) + sin(z))^{4}} + \frac{6cos^{2}(x)}{(sin(x) + sin(y) + sin(z))^{3}} - \frac{6xsin(x)cos(x)}{(sin(x) + sin(y) + sin(z))^{3}} + \frac{3sin(x)}{(sin(x) + sin(y) + sin(z))^{2}} + \frac{xcos(x)}{(sin(x) + sin(y) + sin(z))^{2}} - \frac{6ycos^{3}(x)}{(sin(x) + sin(y) + sin(z))^{4}} - \frac{6ysin(x)cos(x)}{(sin(x) + sin(y) + sin(z))^{3}} + \frac{ycos(x)}{(sin(x) + sin(y) + sin(z))^{2}} - \frac{6zcos^{3}(x)}{(sin(x) + sin(y) + sin(z))^{4}} - \frac{6zsin(x)cos(x)}{(sin(x) + sin(y) + sin(z))^{3}} + \frac{zcos(x)}{(sin(x) + sin(y) + sin(z))^{2}}\right)}{dx}\\=&-6(\frac{-4(cos(x) + cos(y)*0 + cos(z)*0)}{(sin(x) + sin(y) + sin(z))^{5}})xcos^{3}(x) - \frac{6cos^{3}(x)}{(sin(x) + sin(y) + sin(z))^{4}} - \frac{6x*-3cos^{2}(x)sin(x)}{(sin(x) + sin(y) + sin(z))^{4}} + 6(\frac{-3(cos(x) + cos(y)*0 + cos(z)*0)}{(sin(x) + sin(y) + sin(z))^{4}})cos^{2}(x) + \frac{6*-2cos(x)sin(x)}{(sin(x) + sin(y) + sin(z))^{3}} - 6(\frac{-3(cos(x) + cos(y)*0 + cos(z)*0)}{(sin(x) + sin(y) + sin(z))^{4}})xsin(x)cos(x) - \frac{6sin(x)cos(x)}{(sin(x) + sin(y) + sin(z))^{3}} - \frac{6xcos(x)cos(x)}{(sin(x) + sin(y) + sin(z))^{3}} - \frac{6xsin(x)*-sin(x)}{(sin(x) + sin(y) + sin(z))^{3}} + 3(\frac{-2(cos(x) + cos(y)*0 + cos(z)*0)}{(sin(x) + sin(y) + sin(z))^{3}})sin(x) + \frac{3cos(x)}{(sin(x) + sin(y) + sin(z))^{2}} + (\frac{-2(cos(x) + cos(y)*0 + cos(z)*0)}{(sin(x) + sin(y) + sin(z))^{3}})xcos(x) + \frac{cos(x)}{(sin(x) + sin(y) + sin(z))^{2}} + \frac{x*-sin(x)}{(sin(x) + sin(y) + sin(z))^{2}} - 6(\frac{-4(cos(x) + cos(y)*0 + cos(z)*0)}{(sin(x) + sin(y) + sin(z))^{5}})ycos^{3}(x) - \frac{6y*-3cos^{2}(x)sin(x)}{(sin(x) + sin(y) + sin(z))^{4}} - 6(\frac{-3(cos(x) + cos(y)*0 + cos(z)*0)}{(sin(x) + sin(y) + sin(z))^{4}})ysin(x)cos(x) - \frac{6ycos(x)cos(x)}{(sin(x) + sin(y) + sin(z))^{3}} - \frac{6ysin(x)*-sin(x)}{(sin(x) + sin(y) + sin(z))^{3}} + (\frac{-2(cos(x) + cos(y)*0 + cos(z)*0)}{(sin(x) + sin(y) + sin(z))^{3}})ycos(x) + \frac{y*-sin(x)}{(sin(x) + sin(y) + sin(z))^{2}} - 6(\frac{-4(cos(x) + cos(y)*0 + cos(z)*0)}{(sin(x) + sin(y) + sin(z))^{5}})zcos^{3}(x) - \frac{6z*-3cos^{2}(x)sin(x)}{(sin(x) + sin(y) + sin(z))^{4}} - 6(\frac{-3(cos(x) + cos(y)*0 + cos(z)*0)}{(sin(x) + sin(y) + sin(z))^{4}})zsin(x)cos(x) - \frac{6zcos(x)cos(x)}{(sin(x) + sin(y) + sin(z))^{3}} - \frac{6zsin(x)*-sin(x)}{(sin(x) + sin(y) + sin(z))^{3}} + (\frac{-2(cos(x) + cos(y)*0 + cos(z)*0)}{(sin(x) + sin(y) + sin(z))^{3}})zcos(x) + \frac{z*-sin(x)}{(sin(x) + sin(y) + sin(z))^{2}}\\=&\frac{24xcos^{4}(x)}{(sin(x) + sin(y) + sin(z))^{5}} - \frac{24cos^{3}(x)}{(sin(x) + sin(y) + sin(z))^{4}} + \frac{36xsin(x)cos^{2}(x)}{(sin(x) + sin(y) + sin(z))^{4}} - \frac{24sin(x)cos(x)}{(sin(x) + sin(y) + sin(z))^{3}} - \frac{8xcos^{2}(x)}{(sin(x) + sin(y) + sin(z))^{3}} + \frac{6xsin^{2}(x)}{(sin(x) + sin(y) + sin(z))^{3}} + \frac{4cos(x)}{(sin(x) + sin(y) + sin(z))^{2}} - \frac{xsin(x)}{(sin(x) + sin(y) + sin(z))^{2}} + \frac{24ycos^{4}(x)}{(sin(x) + sin(y) + sin(z))^{5}} + \frac{36ysin(x)cos^{2}(x)}{(sin(x) + sin(y) + sin(z))^{4}} - \frac{8ycos^{2}(x)}{(sin(x) + sin(y) + sin(z))^{3}} + \frac{6ysin^{2}(x)}{(sin(x) + sin(y) + sin(z))^{3}} - \frac{ysin(x)}{(sin(x) + sin(y) + sin(z))^{2}} + \frac{24zcos^{4}(x)}{(sin(x) + sin(y) + sin(z))^{5}} + \frac{36zsin(x)cos^{2}(x)}{(sin(x) + sin(y) + sin(z))^{4}} - \frac{8zcos^{2}(x)}{(sin(x) + sin(y) + sin(z))^{3}} + \frac{6zsin^{2}(x)}{(sin(x) + sin(y) + sin(z))^{3}} - \frac{zsin(x)}{(sin(x) + sin(y) + sin(z))^{2}}\\ \end{split}\end{equation} \]

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