There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ ln(cos(x) + sin(x))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( ln(cos(x) + sin(x))\right)}{dx}\\=&\frac{(-sin(x) + cos(x))}{(cos(x) + sin(x))}\\=&\frac{-sin(x)}{(cos(x) + sin(x))} + \frac{cos(x)}{(cos(x) + sin(x))}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-sin(x)}{(cos(x) + sin(x))} + \frac{cos(x)}{(cos(x) + sin(x))}\right)}{dx}\\=&-(\frac{-(-sin(x) + cos(x))}{(cos(x) + sin(x))^{2}})sin(x) - \frac{cos(x)}{(cos(x) + sin(x))} + (\frac{-(-sin(x) + cos(x))}{(cos(x) + sin(x))^{2}})cos(x) + \frac{-sin(x)}{(cos(x) + sin(x))}\\=&\frac{2sin(x)cos(x)}{(cos(x) + sin(x))^{2}} - \frac{sin^{2}(x)}{(cos(x) + sin(x))^{2}} - \frac{cos(x)}{(cos(x) + sin(x))} - \frac{cos^{2}(x)}{(cos(x) + sin(x))^{2}} - \frac{sin(x)}{(cos(x) + sin(x))}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{2sin(x)cos(x)}{(cos(x) + sin(x))^{2}} - \frac{sin^{2}(x)}{(cos(x) + sin(x))^{2}} - \frac{cos(x)}{(cos(x) + sin(x))} - \frac{cos^{2}(x)}{(cos(x) + sin(x))^{2}} - \frac{sin(x)}{(cos(x) + sin(x))}\right)}{dx}\\=&2(\frac{-2(-sin(x) + cos(x))}{(cos(x) + sin(x))^{3}})sin(x)cos(x) + \frac{2cos(x)cos(x)}{(cos(x) + sin(x))^{2}} + \frac{2sin(x)*-sin(x)}{(cos(x) + sin(x))^{2}} - (\frac{-2(-sin(x) + cos(x))}{(cos(x) + sin(x))^{3}})sin^{2}(x) - \frac{2sin(x)cos(x)}{(cos(x) + sin(x))^{2}} - (\frac{-(-sin(x) + cos(x))}{(cos(x) + sin(x))^{2}})cos(x) - \frac{-sin(x)}{(cos(x) + sin(x))} - (\frac{-2(-sin(x) + cos(x))}{(cos(x) + sin(x))^{3}})cos^{2}(x) - \frac{-2cos(x)sin(x)}{(cos(x) + sin(x))^{2}} - (\frac{-(-sin(x) + cos(x))}{(cos(x) + sin(x))^{2}})sin(x) - \frac{cos(x)}{(cos(x) + sin(x))}\\=&\frac{6sin^{2}(x)cos(x)}{(cos(x) + sin(x))^{3}} - \frac{6sin(x)cos^{2}(x)}{(cos(x) + sin(x))^{3}} + \frac{3cos^{2}(x)}{(cos(x) + sin(x))^{2}} - \frac{3sin^{2}(x)}{(cos(x) + sin(x))^{2}} - \frac{2sin^{3}(x)}{(cos(x) + sin(x))^{3}} + \frac{2cos^{3}(x)}{(cos(x) + sin(x))^{3}} + \frac{sin(x)}{(cos(x) + sin(x))} - \frac{cos(x)}{(cos(x) + sin(x))}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{6sin^{2}(x)cos(x)}{(cos(x) + sin(x))^{3}} - \frac{6sin(x)cos^{2}(x)}{(cos(x) + sin(x))^{3}} + \frac{3cos^{2}(x)}{(cos(x) + sin(x))^{2}} - \frac{3sin^{2}(x)}{(cos(x) + sin(x))^{2}} - \frac{2sin^{3}(x)}{(cos(x) + sin(x))^{3}} + \frac{2cos^{3}(x)}{(cos(x) + sin(x))^{3}} + \frac{sin(x)}{(cos(x) + sin(x))} - \frac{cos(x)}{(cos(x) + sin(x))}\right)}{dx}\\=&6(\frac{-3(-sin(x) + cos(x))}{(cos(x) + sin(x))^{4}})sin^{2}(x)cos(x) + \frac{6*2sin(x)cos(x)cos(x)}{(cos(x) + sin(x))^{3}} + \frac{6sin^{2}(x)*-sin(x)}{(cos(x) + sin(x))^{3}} - 6(\frac{-3(-sin(x) + cos(x))}{(cos(x) + sin(x))^{4}})sin(x)cos^{2}(x) - \frac{6cos(x)cos^{2}(x)}{(cos(x) + sin(x))^{3}} - \frac{6sin(x)*-2cos(x)sin(x)}{(cos(x) + sin(x))^{3}} + 3(\frac{-2(-sin(x) + cos(x))}{(cos(x) + sin(x))^{3}})cos^{2}(x) + \frac{3*-2cos(x)sin(x)}{(cos(x) + sin(x))^{2}} - 3(\frac{-2(-sin(x) + cos(x))}{(cos(x) + sin(x))^{3}})sin^{2}(x) - \frac{3*2sin(x)cos(x)}{(cos(x) + sin(x))^{2}} - 2(\frac{-3(-sin(x) + cos(x))}{(cos(x) + sin(x))^{4}})sin^{3}(x) - \frac{2*3sin^{2}(x)cos(x)}{(cos(x) + sin(x))^{3}} + 2(\frac{-3(-sin(x) + cos(x))}{(cos(x) + sin(x))^{4}})cos^{3}(x) + \frac{2*-3cos^{2}(x)sin(x)}{(cos(x) + sin(x))^{3}} + (\frac{-(-sin(x) + cos(x))}{(cos(x) + sin(x))^{2}})sin(x) + \frac{cos(x)}{(cos(x) + sin(x))} - (\frac{-(-sin(x) + cos(x))}{(cos(x) + sin(x))^{2}})cos(x) - \frac{-sin(x)}{(cos(x) + sin(x))}\\=&\frac{24sin^{3}(x)cos(x)}{(cos(x) + sin(x))^{4}} - \frac{36sin^{2}(x)cos^{2}(x)}{(cos(x) + sin(x))^{4}} + \frac{12sin(x)cos^{2}(x)}{(cos(x) + sin(x))^{3}} + \frac{12sin^{2}(x)cos(x)}{(cos(x) + sin(x))^{3}} + \frac{24sin(x)cos^{3}(x)}{(cos(x) + sin(x))^{4}} - \frac{12cos^{3}(x)}{(cos(x) + sin(x))^{3}} - \frac{14sin(x)cos(x)}{(cos(x) + sin(x))^{2}} - \frac{12sin^{3}(x)}{(cos(x) + sin(x))^{3}} - \frac{6sin^{4}(x)}{(cos(x) + sin(x))^{4}} - \frac{6cos^{4}(x)}{(cos(x) + sin(x))^{4}} + \frac{sin^{2}(x)}{(cos(x) + sin(x))^{2}} + \frac{cos(x)}{(cos(x) + sin(x))} + \frac{cos^{2}(x)}{(cos(x) + sin(x))^{2}} + \frac{sin(x)}{(cos(x) + sin(x))}\\ \end{split}\end{equation} \]

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