There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ {sin(x)}^{\frac{5}{2}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = sin^{\frac{5}{2}}(x)\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( sin^{\frac{5}{2}}(x)\right)}{dx}\\=&\frac{5}{2}sin^{\frac{3}{2}}(x)cos(x)\\=&\frac{5sin^{\frac{3}{2}}(x)cos(x)}{2}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{5sin^{\frac{3}{2}}(x)cos(x)}{2}\right)}{dx}\\=&\frac{5*\frac{3}{2}sin^{\frac{1}{2}}(x)cos(x)cos(x)}{2} + \frac{5sin^{\frac{3}{2}}(x)*-sin(x)}{2}\\=&\frac{15sin^{\frac{1}{2}}(x)cos^{2}(x)}{4} - \frac{5sin^{\frac{5}{2}}(x)}{2}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{15sin^{\frac{1}{2}}(x)cos^{2}(x)}{4} - \frac{5sin^{\frac{5}{2}}(x)}{2}\right)}{dx}\\=&\frac{15*\frac{1}{2}cos(x)cos^{2}(x)}{4sin^{\frac{1}{2}}(x)} + \frac{15sin^{\frac{1}{2}}(x)*-2cos(x)sin(x)}{4} - \frac{5*\frac{5}{2}sin^{\frac{3}{2}}(x)cos(x)}{2}\\=&\frac{15cos^{3}(x)}{8sin^{\frac{1}{2}}(x)} - \frac{55sin^{\frac{3}{2}}(x)cos(x)}{4}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{15cos^{3}(x)}{8sin^{\frac{1}{2}}(x)} - \frac{55sin^{\frac{3}{2}}(x)cos(x)}{4}\right)}{dx}\\=&\frac{15*\frac{-1}{2}cos(x)cos^{3}(x)}{8sin^{\frac{3}{2}}(x)} + \frac{15*-3cos^{2}(x)sin(x)}{8sin^{\frac{1}{2}}(x)} - \frac{55*\frac{3}{2}sin^{\frac{1}{2}}(x)cos(x)cos(x)}{4} - \frac{55sin^{\frac{3}{2}}(x)*-sin(x)}{4}\\=&\frac{-15cos^{4}(x)}{16sin^{\frac{3}{2}}(x)} - \frac{105sin^{\frac{1}{2}}(x)cos^{2}(x)}{4} + \frac{55sin^{\frac{5}{2}}(x)}{4}\\ \end{split}\end{equation} \]

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