There are 1 questions in this calculation: for each question, the 2 derivative of x is calculated.
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\[ \begin{equation}\begin{split}[1/1]Find\ the\ second\ derivative\ of\ function\ \frac{(r - lrsin(x))}{({(l - sin(x))}^{2} + rrcos(x)cos(x))}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = - \frac{rlsin(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))} + \frac{r}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( - \frac{rlsin(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))} + \frac{r}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))}\right)}{dx}\\=& - (\frac{-(-2lcos(x) + 0 + 2sin(x)cos(x) + r^{2}*-2cos(x)sin(x))}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{2}})rlsin(x) - \frac{rlcos(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))} + (\frac{-(-2lcos(x) + 0 + 2sin(x)cos(x) + r^{2}*-2cos(x)sin(x))}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{2}})r + 0\\=&\frac{-2rl^{2}sin(x)cos(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{2}} + \frac{2rlsin^{2}(x)cos(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{2}} - \frac{2r^{3}lsin^{2}(x)cos(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{2}} - \frac{rlcos(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))} + \frac{2rlcos(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{2}} - \frac{2rsin(x)cos(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{2}} + \frac{2r^{3}sin(x)cos(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{2}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2rl^{2}sin(x)cos(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{2}} + \frac{2rlsin^{2}(x)cos(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{2}} - \frac{2r^{3}lsin^{2}(x)cos(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{2}} - \frac{rlcos(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))} + \frac{2rlcos(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{2}} - \frac{2rsin(x)cos(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{2}} + \frac{2r^{3}sin(x)cos(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{2}}\right)}{dx}\\=&-2(\frac{-2(-2lcos(x) + 0 + 2sin(x)cos(x) + r^{2}*-2cos(x)sin(x))}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{3}})rl^{2}sin(x)cos(x) - \frac{2rl^{2}cos(x)cos(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{2}} - \frac{2rl^{2}sin(x)*-sin(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{2}} + 2(\frac{-2(-2lcos(x) + 0 + 2sin(x)cos(x) + r^{2}*-2cos(x)sin(x))}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{3}})rlsin^{2}(x)cos(x) + \frac{2rl*2sin(x)cos(x)cos(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{2}} + \frac{2rlsin^{2}(x)*-sin(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{2}} - 2(\frac{-2(-2lcos(x) + 0 + 2sin(x)cos(x) + r^{2}*-2cos(x)sin(x))}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{3}})r^{3}lsin^{2}(x)cos(x) - \frac{2r^{3}l*2sin(x)cos(x)cos(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{2}} - \frac{2r^{3}lsin^{2}(x)*-sin(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{2}} - (\frac{-(-2lcos(x) + 0 + 2sin(x)cos(x) + r^{2}*-2cos(x)sin(x))}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{2}})rlcos(x) - \frac{rl*-sin(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))} + 2(\frac{-2(-2lcos(x) + 0 + 2sin(x)cos(x) + r^{2}*-2cos(x)sin(x))}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{3}})rlcos(x) + \frac{2rl*-sin(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{2}} - 2(\frac{-2(-2lcos(x) + 0 + 2sin(x)cos(x) + r^{2}*-2cos(x)sin(x))}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{3}})rsin(x)cos(x) - \frac{2rcos(x)cos(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{2}} - \frac{2rsin(x)*-sin(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{2}} + 2(\frac{-2(-2lcos(x) + 0 + 2sin(x)cos(x) + r^{2}*-2cos(x)sin(x))}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{3}})r^{3}sin(x)cos(x) + \frac{2r^{3}cos(x)cos(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{2}} + \frac{2r^{3}sin(x)*-sin(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{2}}\\=&\frac{-8rl^{3}sin(x)cos^{2}(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{3}} + \frac{16rl^{2}sin^{2}(x)cos^{2}(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{3}} - \frac{16r^{3}l^{2}sin^{2}(x)cos^{2}(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{3}} - \frac{4rl^{2}cos^{2}(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{2}} + \frac{6rlsin(x)cos^{2}(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{2}} - \frac{8rlsin^{3}(x)cos^{2}(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{3}} + \frac{16r^{3}lsin^{3}(x)cos^{2}(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{3}} - \frac{6r^{3}lsin(x)cos^{2}(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{2}} - \frac{8r^{5}lsin^{3}(x)cos^{2}(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{3}} - \frac{16rlsin(x)cos^{2}(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{3}} + \frac{16r^{3}lsin(x)cos^{2}(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{3}} + \frac{2r^{3}lsin^{3}(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{2}} + \frac{rlsin(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))} + \frac{8rl^{2}cos^{2}(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{3}} - \frac{2rlsin^{3}(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{2}} + \frac{2rl^{2}sin^{2}(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{2}} - \frac{2rlsin(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{2}} + \frac{8rsin^{2}(x)cos^{2}(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{3}} - \frac{16r^{3}sin^{2}(x)cos^{2}(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{3}} - \frac{2rcos^{2}(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{2}} + \frac{8r^{5}sin^{2}(x)cos^{2}(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{3}} + \frac{2rsin^{2}(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{2}} + \frac{2r^{3}cos^{2}(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{2}} - \frac{2r^{3}sin^{2}(x)}{(-2lsin(x) + l^{2} + sin^{2}(x) + r^{2}cos^{2}(x))^{2}}\\ \end{split}\end{equation} \]

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