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

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