There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ {sin(x)}^{(\frac{2}{5})}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( {sin(x)}^{\frac{2}{5}}\right)}{dx}\\=&({sin(x)}^{\frac{2}{5}}((0)ln(sin(x)) + \frac{(\frac{2}{5})(cos(x))}{(sin(x))}))\\=&\frac{2cos(x)}{5sin^{\frac{3}{5}}(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{2cos(x)}{5sin^{\frac{3}{5}}(x)}\right)}{dx}\\=&\frac{2*\frac{-3}{5}cos(x)cos(x)}{5sin^{\frac{8}{5}}(x)} + \frac{2*-sin(x)}{5sin^{\frac{3}{5}}(x)}\\=&\frac{-6cos^{2}(x)}{25sin^{\frac{8}{5}}(x)} - \frac{2sin^{\frac{2}{5}}(x)}{5}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-6cos^{2}(x)}{25sin^{\frac{8}{5}}(x)} - \frac{2sin^{\frac{2}{5}}(x)}{5}\right)}{dx}\\=&\frac{-6*\frac{-8}{5}cos(x)cos^{2}(x)}{25sin^{\frac{13}{5}}(x)} - \frac{6*-2cos(x)sin(x)}{25sin^{\frac{8}{5}}(x)} - \frac{2*\frac{2}{5}cos(x)}{5sin^{\frac{3}{5}}(x)}\\=&\frac{48cos^{3}(x)}{125sin^{\frac{13}{5}}(x)} + \frac{8cos(x)}{25sin^{\frac{3}{5}}(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{48cos^{3}(x)}{125sin^{\frac{13}{5}}(x)} + \frac{8cos(x)}{25sin^{\frac{3}{5}}(x)}\right)}{dx}\\=&\frac{48*\frac{-13}{5}cos(x)cos^{3}(x)}{125sin^{\frac{18}{5}}(x)} + \frac{48*-3cos^{2}(x)sin(x)}{125sin^{\frac{13}{5}}(x)} + \frac{8*\frac{-3}{5}cos(x)cos(x)}{25sin^{\frac{8}{5}}(x)} + \frac{8*-sin(x)}{25sin^{\frac{3}{5}}(x)}\\=&\frac{-624cos^{4}(x)}{625sin^{\frac{18}{5}}(x)} - \frac{168cos^{2}(x)}{125sin^{\frac{8}{5}}(x)} - \frac{8sin^{\frac{2}{5}}(x)}{25}\\ \end{split}\end{equation} \]

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