There are 1 questions in this calculation: for each question, the 3 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ third\ derivative\ of\ function\ ln({cos(x)}^{2} + sin(x))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = ln(cos^{2}(x) + sin(x))\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( ln(cos^{2}(x) + sin(x))\right)}{dx}\\=&\frac{(-2cos(x)sin(x) + cos(x))}{(cos^{2}(x) + sin(x))}\\=&\frac{-2sin(x)cos(x)}{(cos^{2}(x) + sin(x))} + \frac{cos(x)}{(cos^{2}(x) + sin(x))}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2sin(x)cos(x)}{(cos^{2}(x) + sin(x))} + \frac{cos(x)}{(cos^{2}(x) + sin(x))}\right)}{dx}\\=&-2(\frac{-(-2cos(x)sin(x) + cos(x))}{(cos^{2}(x) + sin(x))^{2}})sin(x)cos(x) - \frac{2cos(x)cos(x)}{(cos^{2}(x) + sin(x))} - \frac{2sin(x)*-sin(x)}{(cos^{2}(x) + sin(x))} + (\frac{-(-2cos(x)sin(x) + cos(x))}{(cos^{2}(x) + sin(x))^{2}})cos(x) + \frac{-sin(x)}{(cos^{2}(x) + sin(x))}\\=&\frac{-4sin^{2}(x)cos^{2}(x)}{(cos^{2}(x) + sin(x))^{2}} + \frac{4sin(x)cos^{2}(x)}{(cos^{2}(x) + sin(x))^{2}} - \frac{2cos^{2}(x)}{(cos^{2}(x) + sin(x))} + \frac{2sin^{2}(x)}{(cos^{2}(x) + sin(x))} - \frac{cos^{2}(x)}{(cos^{2}(x) + sin(x))^{2}} - \frac{sin(x)}{(cos^{2}(x) + sin(x))}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-4sin^{2}(x)cos^{2}(x)}{(cos^{2}(x) + sin(x))^{2}} + \frac{4sin(x)cos^{2}(x)}{(cos^{2}(x) + sin(x))^{2}} - \frac{2cos^{2}(x)}{(cos^{2}(x) + sin(x))} + \frac{2sin^{2}(x)}{(cos^{2}(x) + sin(x))} - \frac{cos^{2}(x)}{(cos^{2}(x) + sin(x))^{2}} - \frac{sin(x)}{(cos^{2}(x) + sin(x))}\right)}{dx}\\=&-4(\frac{-2(-2cos(x)sin(x) + cos(x))}{(cos^{2}(x) + sin(x))^{3}})sin^{2}(x)cos^{2}(x) - \frac{4*2sin(x)cos(x)cos^{2}(x)}{(cos^{2}(x) + sin(x))^{2}} - \frac{4sin^{2}(x)*-2cos(x)sin(x)}{(cos^{2}(x) + sin(x))^{2}} + 4(\frac{-2(-2cos(x)sin(x) + cos(x))}{(cos^{2}(x) + sin(x))^{3}})sin(x)cos^{2}(x) + \frac{4cos(x)cos^{2}(x)}{(cos^{2}(x) + sin(x))^{2}} + \frac{4sin(x)*-2cos(x)sin(x)}{(cos^{2}(x) + sin(x))^{2}} - 2(\frac{-(-2cos(x)sin(x) + cos(x))}{(cos^{2}(x) + sin(x))^{2}})cos^{2}(x) - \frac{2*-2cos(x)sin(x)}{(cos^{2}(x) + sin(x))} + 2(\frac{-(-2cos(x)sin(x) + cos(x))}{(cos^{2}(x) + sin(x))^{2}})sin^{2}(x) + \frac{2*2sin(x)cos(x)}{(cos^{2}(x) + sin(x))} - (\frac{-2(-2cos(x)sin(x) + cos(x))}{(cos^{2}(x) + sin(x))^{3}})cos^{2}(x) - \frac{-2cos(x)sin(x)}{(cos^{2}(x) + sin(x))^{2}} - (\frac{-(-2cos(x)sin(x) + cos(x))}{(cos^{2}(x) + sin(x))^{2}})sin(x) - \frac{cos(x)}{(cos^{2}(x) + sin(x))}\\=&\frac{-16sin^{3}(x)cos^{3}(x)}{(cos^{2}(x) + sin(x))^{3}} + \frac{24sin^{2}(x)cos^{3}(x)}{(cos^{2}(x) + sin(x))^{3}} - \frac{12sin(x)cos^{3}(x)}{(cos^{2}(x) + sin(x))^{2}} + \frac{12sin^{3}(x)cos(x)}{(cos^{2}(x) + sin(x))^{2}} - \frac{12sin(x)cos^{3}(x)}{(cos^{2}(x) + sin(x))^{3}} + \frac{6cos^{3}(x)}{(cos^{2}(x) + sin(x))^{2}} - \frac{12sin^{2}(x)cos(x)}{(cos^{2}(x) + sin(x))^{2}} + \frac{8sin(x)cos(x)}{(cos^{2}(x) + sin(x))} + \frac{2cos^{3}(x)}{(cos^{2}(x) + sin(x))^{3}} + \frac{3sin(x)cos(x)}{(cos^{2}(x) + sin(x))^{2}} - \frac{cos(x)}{(cos^{2}(x) + sin(x))}\\ \end{split}\end{equation} \]

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