There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ {(1 - {a}^{2}cos(x)cos(x))}^{\frac{1}{2}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = (-a^{2}cos^{2}(x) + 1)^{\frac{1}{2}}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( (-a^{2}cos^{2}(x) + 1)^{\frac{1}{2}}\right)}{dx}\\=&(\frac{\frac{1}{2}(-a^{2}*-2cos(x)sin(x) + 0)}{(-a^{2}cos^{2}(x) + 1)^{\frac{1}{2}}})\\=&\frac{a^{2}sin(x)cos(x)}{(-a^{2}cos^{2}(x) + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{a^{2}sin(x)cos(x)}{(-a^{2}cos^{2}(x) + 1)^{\frac{1}{2}}}\right)}{dx}\\=&(\frac{\frac{-1}{2}(-a^{2}*-2cos(x)sin(x) + 0)}{(-a^{2}cos^{2}(x) + 1)^{\frac{3}{2}}})a^{2}sin(x)cos(x) + \frac{a^{2}cos(x)cos(x)}{(-a^{2}cos^{2}(x) + 1)^{\frac{1}{2}}} + \frac{a^{2}sin(x)*-sin(x)}{(-a^{2}cos^{2}(x) + 1)^{\frac{1}{2}}}\\=&\frac{-a^{4}sin^{2}(x)cos^{2}(x)}{(-a^{2}cos^{2}(x) + 1)^{\frac{3}{2}}} + \frac{a^{2}cos^{2}(x)}{(-a^{2}cos^{2}(x) + 1)^{\frac{1}{2}}} - \frac{a^{2}sin^{2}(x)}{(-a^{2}cos^{2}(x) + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-a^{4}sin^{2}(x)cos^{2}(x)}{(-a^{2}cos^{2}(x) + 1)^{\frac{3}{2}}} + \frac{a^{2}cos^{2}(x)}{(-a^{2}cos^{2}(x) + 1)^{\frac{1}{2}}} - \frac{a^{2}sin^{2}(x)}{(-a^{2}cos^{2}(x) + 1)^{\frac{1}{2}}}\right)}{dx}\\=&-(\frac{\frac{-3}{2}(-a^{2}*-2cos(x)sin(x) + 0)}{(-a^{2}cos^{2}(x) + 1)^{\frac{5}{2}}})a^{4}sin^{2}(x)cos^{2}(x) - \frac{a^{4}*2sin(x)cos(x)cos^{2}(x)}{(-a^{2}cos^{2}(x) + 1)^{\frac{3}{2}}} - \frac{a^{4}sin^{2}(x)*-2cos(x)sin(x)}{(-a^{2}cos^{2}(x) + 1)^{\frac{3}{2}}} + (\frac{\frac{-1}{2}(-a^{2}*-2cos(x)sin(x) + 0)}{(-a^{2}cos^{2}(x) + 1)^{\frac{3}{2}}})a^{2}cos^{2}(x) + \frac{a^{2}*-2cos(x)sin(x)}{(-a^{2}cos^{2}(x) + 1)^{\frac{1}{2}}} - (\frac{\frac{-1}{2}(-a^{2}*-2cos(x)sin(x) + 0)}{(-a^{2}cos^{2}(x) + 1)^{\frac{3}{2}}})a^{2}sin^{2}(x) - \frac{a^{2}*2sin(x)cos(x)}{(-a^{2}cos^{2}(x) + 1)^{\frac{1}{2}}}\\=&\frac{3a^{6}sin^{3}(x)cos^{3}(x)}{(-a^{2}cos^{2}(x) + 1)^{\frac{5}{2}}} - \frac{3a^{4}sin(x)cos^{3}(x)}{(-a^{2}cos^{2}(x) + 1)^{\frac{3}{2}}} + \frac{3a^{4}sin^{3}(x)cos(x)}{(-a^{2}cos^{2}(x) + 1)^{\frac{3}{2}}} - \frac{4a^{2}sin(x)cos(x)}{(-a^{2}cos^{2}(x) + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{3a^{6}sin^{3}(x)cos^{3}(x)}{(-a^{2}cos^{2}(x) + 1)^{\frac{5}{2}}} - \frac{3a^{4}sin(x)cos^{3}(x)}{(-a^{2}cos^{2}(x) + 1)^{\frac{3}{2}}} + \frac{3a^{4}sin^{3}(x)cos(x)}{(-a^{2}cos^{2}(x) + 1)^{\frac{3}{2}}} - \frac{4a^{2}sin(x)cos(x)}{(-a^{2}cos^{2}(x) + 1)^{\frac{1}{2}}}\right)}{dx}\\=&3(\frac{\frac{-5}{2}(-a^{2}*-2cos(x)sin(x) + 0)}{(-a^{2}cos^{2}(x) + 1)^{\frac{7}{2}}})a^{6}sin^{3}(x)cos^{3}(x) + \frac{3a^{6}*3sin^{2}(x)cos(x)cos^{3}(x)}{(-a^{2}cos^{2}(x) + 1)^{\frac{5}{2}}} + \frac{3a^{6}sin^{3}(x)*-3cos^{2}(x)sin(x)}{(-a^{2}cos^{2}(x) + 1)^{\frac{5}{2}}} - 3(\frac{\frac{-3}{2}(-a^{2}*-2cos(x)sin(x) + 0)}{(-a^{2}cos^{2}(x) + 1)^{\frac{5}{2}}})a^{4}sin(x)cos^{3}(x) - \frac{3a^{4}cos(x)cos^{3}(x)}{(-a^{2}cos^{2}(x) + 1)^{\frac{3}{2}}} - \frac{3a^{4}sin(x)*-3cos^{2}(x)sin(x)}{(-a^{2}cos^{2}(x) + 1)^{\frac{3}{2}}} + 3(\frac{\frac{-3}{2}(-a^{2}*-2cos(x)sin(x) + 0)}{(-a^{2}cos^{2}(x) + 1)^{\frac{5}{2}}})a^{4}sin^{3}(x)cos(x) + \frac{3a^{4}*3sin^{2}(x)cos(x)cos(x)}{(-a^{2}cos^{2}(x) + 1)^{\frac{3}{2}}} + \frac{3a^{4}sin^{3}(x)*-sin(x)}{(-a^{2}cos^{2}(x) + 1)^{\frac{3}{2}}} - 4(\frac{\frac{-1}{2}(-a^{2}*-2cos(x)sin(x) + 0)}{(-a^{2}cos^{2}(x) + 1)^{\frac{3}{2}}})a^{2}sin(x)cos(x) - \frac{4a^{2}cos(x)cos(x)}{(-a^{2}cos^{2}(x) + 1)^{\frac{1}{2}}} - \frac{4a^{2}sin(x)*-sin(x)}{(-a^{2}cos^{2}(x) + 1)^{\frac{1}{2}}}\\=&\frac{-15a^{8}sin^{4}(x)cos^{4}(x)}{(-a^{2}cos^{2}(x) + 1)^{\frac{7}{2}}} + \frac{18a^{6}sin^{2}(x)cos^{4}(x)}{(-a^{2}cos^{2}(x) + 1)^{\frac{5}{2}}} - \frac{18a^{6}sin^{4}(x)cos^{2}(x)}{(-a^{2}cos^{2}(x) + 1)^{\frac{5}{2}}} - \frac{3a^{4}cos^{4}(x)}{(-a^{2}cos^{2}(x) + 1)^{\frac{3}{2}}} + \frac{22a^{4}sin^{2}(x)cos^{2}(x)}{(-a^{2}cos^{2}(x) + 1)^{\frac{3}{2}}} - \frac{3a^{4}sin^{4}(x)}{(-a^{2}cos^{2}(x) + 1)^{\frac{3}{2}}} - \frac{4a^{2}cos^{2}(x)}{(-a^{2}cos^{2}(x) + 1)^{\frac{1}{2}}} + \frac{4a^{2}sin^{2}(x)}{(-a^{2}cos^{2}(x) + 1)^{\frac{1}{2}}}\\ \end{split}\end{equation} \]

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