There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ {cos(x)}^{(\frac{2}{7})}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( {cos(x)}^{\frac{2}{7}}\right)}{dx}\\=&({cos(x)}^{\frac{2}{7}}((0)ln(cos(x)) + \frac{(\frac{2}{7})(-sin(x))}{(cos(x))}))\\=&\frac{-2sin(x)}{7cos^{\frac{5}{7}}(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2sin(x)}{7cos^{\frac{5}{7}}(x)}\right)}{dx}\\=&\frac{-2cos(x)}{7cos^{\frac{5}{7}}(x)} - \frac{2sin(x)*\frac{5}{7}sin(x)}{7cos^{\frac{12}{7}}(x)}\\=&\frac{-2cos^{\frac{2}{7}}(x)}{7} - \frac{10sin^{2}(x)}{49cos^{\frac{12}{7}}(x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2cos^{\frac{2}{7}}(x)}{7} - \frac{10sin^{2}(x)}{49cos^{\frac{12}{7}}(x)}\right)}{dx}\\=&\frac{-2*\frac{-2}{7}sin(x)}{7cos^{\frac{5}{7}}(x)} - \frac{10*2sin(x)cos(x)}{49cos^{\frac{12}{7}}(x)} - \frac{10sin^{2}(x)*\frac{12}{7}sin(x)}{49cos^{\frac{19}{7}}(x)}\\=& - \frac{16sin(x)}{49cos^{\frac{5}{7}}(x)} - \frac{120sin^{3}(x)}{343cos^{\frac{19}{7}}(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( - \frac{16sin(x)}{49cos^{\frac{5}{7}}(x)} - \frac{120sin^{3}(x)}{343cos^{\frac{19}{7}}(x)}\right)}{dx}\\=& - \frac{16cos(x)}{49cos^{\frac{5}{7}}(x)} - \frac{16sin(x)*\frac{5}{7}sin(x)}{49cos^{\frac{12}{7}}(x)} - \frac{120*3sin^{2}(x)cos(x)}{343cos^{\frac{19}{7}}(x)} - \frac{120sin^{3}(x)*\frac{19}{7}sin(x)}{343cos^{\frac{26}{7}}(x)}\\=& - \frac{16cos^{\frac{2}{7}}(x)}{49} - \frac{440sin^{2}(x)}{343cos^{\frac{12}{7}}(x)} - \frac{2280sin^{4}(x)}{2401cos^{\frac{26}{7}}(x)}\\ \end{split}\end{equation} \]

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