There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ {(1 + {(cos(x))}^{2})}^{\frac{1}{2}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = (cos^{2}(x) + 1)^{\frac{1}{2}}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( (cos^{2}(x) + 1)^{\frac{1}{2}}\right)}{dx}\\=&(\frac{\frac{1}{2}(-2cos(x)sin(x) + 0)}{(cos^{2}(x) + 1)^{\frac{1}{2}}})\\=&\frac{-sin(x)cos(x)}{(cos^{2}(x) + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-sin(x)cos(x)}{(cos^{2}(x) + 1)^{\frac{1}{2}}}\right)}{dx}\\=&-(\frac{\frac{-1}{2}(-2cos(x)sin(x) + 0)}{(cos^{2}(x) + 1)^{\frac{3}{2}}})sin(x)cos(x) - \frac{cos(x)cos(x)}{(cos^{2}(x) + 1)^{\frac{1}{2}}} - \frac{sin(x)*-sin(x)}{(cos^{2}(x) + 1)^{\frac{1}{2}}}\\=&\frac{-sin^{2}(x)cos^{2}(x)}{(cos^{2}(x) + 1)^{\frac{3}{2}}} - \frac{cos^{2}(x)}{(cos^{2}(x) + 1)^{\frac{1}{2}}} + \frac{sin^{2}(x)}{(cos^{2}(x) + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-sin^{2}(x)cos^{2}(x)}{(cos^{2}(x) + 1)^{\frac{3}{2}}} - \frac{cos^{2}(x)}{(cos^{2}(x) + 1)^{\frac{1}{2}}} + \frac{sin^{2}(x)}{(cos^{2}(x) + 1)^{\frac{1}{2}}}\right)}{dx}\\=&-(\frac{\frac{-3}{2}(-2cos(x)sin(x) + 0)}{(cos^{2}(x) + 1)^{\frac{5}{2}}})sin^{2}(x)cos^{2}(x) - \frac{2sin(x)cos(x)cos^{2}(x)}{(cos^{2}(x) + 1)^{\frac{3}{2}}} - \frac{sin^{2}(x)*-2cos(x)sin(x)}{(cos^{2}(x) + 1)^{\frac{3}{2}}} - (\frac{\frac{-1}{2}(-2cos(x)sin(x) + 0)}{(cos^{2}(x) + 1)^{\frac{3}{2}}})cos^{2}(x) - \frac{-2cos(x)sin(x)}{(cos^{2}(x) + 1)^{\frac{1}{2}}} + (\frac{\frac{-1}{2}(-2cos(x)sin(x) + 0)}{(cos^{2}(x) + 1)^{\frac{3}{2}}})sin^{2}(x) + \frac{2sin(x)cos(x)}{(cos^{2}(x) + 1)^{\frac{1}{2}}}\\=&\frac{-3sin^{3}(x)cos^{3}(x)}{(cos^{2}(x) + 1)^{\frac{5}{2}}} - \frac{3sin(x)cos^{3}(x)}{(cos^{2}(x) + 1)^{\frac{3}{2}}} + \frac{3sin^{3}(x)cos(x)}{(cos^{2}(x) + 1)^{\frac{3}{2}}} + \frac{4sin(x)cos(x)}{(cos^{2}(x) + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-3sin^{3}(x)cos^{3}(x)}{(cos^{2}(x) + 1)^{\frac{5}{2}}} - \frac{3sin(x)cos^{3}(x)}{(cos^{2}(x) + 1)^{\frac{3}{2}}} + \frac{3sin^{3}(x)cos(x)}{(cos^{2}(x) + 1)^{\frac{3}{2}}} + \frac{4sin(x)cos(x)}{(cos^{2}(x) + 1)^{\frac{1}{2}}}\right)}{dx}\\=&-3(\frac{\frac{-5}{2}(-2cos(x)sin(x) + 0)}{(cos^{2}(x) + 1)^{\frac{7}{2}}})sin^{3}(x)cos^{3}(x) - \frac{3*3sin^{2}(x)cos(x)cos^{3}(x)}{(cos^{2}(x) + 1)^{\frac{5}{2}}} - \frac{3sin^{3}(x)*-3cos^{2}(x)sin(x)}{(cos^{2}(x) + 1)^{\frac{5}{2}}} - 3(\frac{\frac{-3}{2}(-2cos(x)sin(x) + 0)}{(cos^{2}(x) + 1)^{\frac{5}{2}}})sin(x)cos^{3}(x) - \frac{3cos(x)cos^{3}(x)}{(cos^{2}(x) + 1)^{\frac{3}{2}}} - \frac{3sin(x)*-3cos^{2}(x)sin(x)}{(cos^{2}(x) + 1)^{\frac{3}{2}}} + 3(\frac{\frac{-3}{2}(-2cos(x)sin(x) + 0)}{(cos^{2}(x) + 1)^{\frac{5}{2}}})sin^{3}(x)cos(x) + \frac{3*3sin^{2}(x)cos(x)cos(x)}{(cos^{2}(x) + 1)^{\frac{3}{2}}} + \frac{3sin^{3}(x)*-sin(x)}{(cos^{2}(x) + 1)^{\frac{3}{2}}} + 4(\frac{\frac{-1}{2}(-2cos(x)sin(x) + 0)}{(cos^{2}(x) + 1)^{\frac{3}{2}}})sin(x)cos(x) + \frac{4cos(x)cos(x)}{(cos^{2}(x) + 1)^{\frac{1}{2}}} + \frac{4sin(x)*-sin(x)}{(cos^{2}(x) + 1)^{\frac{1}{2}}}\\=&\frac{-15sin^{4}(x)cos^{4}(x)}{(cos^{2}(x) + 1)^{\frac{7}{2}}} - \frac{18sin^{2}(x)cos^{4}(x)}{(cos^{2}(x) + 1)^{\frac{5}{2}}} + \frac{18sin^{4}(x)cos^{2}(x)}{(cos^{2}(x) + 1)^{\frac{5}{2}}} - \frac{3cos^{4}(x)}{(cos^{2}(x) + 1)^{\frac{3}{2}}} + \frac{22sin^{2}(x)cos^{2}(x)}{(cos^{2}(x) + 1)^{\frac{3}{2}}} - \frac{3sin^{4}(x)}{(cos^{2}(x) + 1)^{\frac{3}{2}}} + \frac{4cos^{2}(x)}{(cos^{2}(x) + 1)^{\frac{1}{2}}} - \frac{4sin^{2}(x)}{(cos^{2}(x) + 1)^{\frac{1}{2}}}\\ \end{split}\end{equation} \]

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