There are 1 questions in this calculation: for each question, the 2 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ second\ derivative\ of\ function\ \frac{(1 + 2sin(x))}{(5 + 4sin(x))}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{2sin(x)}{(4sin(x) + 5)} + \frac{1}{(4sin(x) + 5)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{2sin(x)}{(4sin(x) + 5)} + \frac{1}{(4sin(x) + 5)}\right)}{dx}\\=&2(\frac{-(4cos(x) + 0)}{(4sin(x) + 5)^{2}})sin(x) + \frac{2cos(x)}{(4sin(x) + 5)} + (\frac{-(4cos(x) + 0)}{(4sin(x) + 5)^{2}})\\=& - \frac{8sin(x)cos(x)}{(4sin(x) + 5)^{2}} + \frac{2cos(x)}{(4sin(x) + 5)} - \frac{4cos(x)}{(4sin(x) + 5)^{2}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( - \frac{8sin(x)cos(x)}{(4sin(x) + 5)^{2}} + \frac{2cos(x)}{(4sin(x) + 5)} - \frac{4cos(x)}{(4sin(x) + 5)^{2}}\right)}{dx}\\=& - 8(\frac{-2(4cos(x) + 0)}{(4sin(x) + 5)^{3}})sin(x)cos(x) - \frac{8cos(x)cos(x)}{(4sin(x) + 5)^{2}} - \frac{8sin(x)*-sin(x)}{(4sin(x) + 5)^{2}} + 2(\frac{-(4cos(x) + 0)}{(4sin(x) + 5)^{2}})cos(x) + \frac{2*-sin(x)}{(4sin(x) + 5)} - 4(\frac{-2(4cos(x) + 0)}{(4sin(x) + 5)^{3}})cos(x) - \frac{4*-sin(x)}{(4sin(x) + 5)^{2}}\\=&\frac{64sin(x)cos^{2}(x)}{(4sin(x) + 5)^{3}} - \frac{16cos^{2}(x)}{(4sin(x) + 5)^{2}} + \frac{8sin^{2}(x)}{(4sin(x) + 5)^{2}} - \frac{2sin(x)}{(4sin(x) + 5)} + \frac{32cos^{2}(x)}{(4sin(x) + 5)^{3}} + \frac{4sin(x)}{(4sin(x) + 5)^{2}}\\ \end{split}\end{equation} \]

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