There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ {sin(x)}^{e}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( {sin(x)}^{e}\right)}{dx}\\=&({sin(x)}^{e}((0)ln(sin(x)) + \frac{(e)(cos(x))}{(sin(x))}))\\=&\frac{{sin(x)}^{e}ecos(x)}{sin(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{{sin(x)}^{e}ecos(x)}{sin(x)}\right)}{dx}\\=&\frac{({sin(x)}^{e}((0)ln(sin(x)) + \frac{(e)(cos(x))}{(sin(x))}))ecos(x)}{sin(x)} + \frac{{sin(x)}^{e}*0cos(x)}{sin(x)} + \frac{{sin(x)}^{e}e*-cos(x)cos(x)}{sin^{2}(x)} + \frac{{sin(x)}^{e}e*-sin(x)}{sin(x)}\\=&\frac{{sin(x)}^{e}e^{2}cos^{2}(x)}{sin^{2}(x)} - \frac{{sin(x)}^{e}ecos^{2}(x)}{sin^{2}(x)} - {sin(x)}^{e}e\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{{sin(x)}^{e}e^{2}cos^{2}(x)}{sin^{2}(x)} - \frac{{sin(x)}^{e}ecos^{2}(x)}{sin^{2}(x)} - {sin(x)}^{e}e\right)}{dx}\\=&\frac{({sin(x)}^{e}((0)ln(sin(x)) + \frac{(e)(cos(x))}{(sin(x))}))e^{2}cos^{2}(x)}{sin^{2}(x)} + \frac{{sin(x)}^{e}*2e*0cos^{2}(x)}{sin^{2}(x)} + \frac{{sin(x)}^{e}e^{2}*-2cos(x)cos^{2}(x)}{sin^{3}(x)} + \frac{{sin(x)}^{e}e^{2}*-2cos(x)sin(x)}{sin^{2}(x)} - \frac{({sin(x)}^{e}((0)ln(sin(x)) + \frac{(e)(cos(x))}{(sin(x))}))ecos^{2}(x)}{sin^{2}(x)} - \frac{{sin(x)}^{e}*0cos^{2}(x)}{sin^{2}(x)} - \frac{{sin(x)}^{e}e*-2cos(x)cos^{2}(x)}{sin^{3}(x)} - \frac{{sin(x)}^{e}e*-2cos(x)sin(x)}{sin^{2}(x)} - ({sin(x)}^{e}((0)ln(sin(x)) + \frac{(e)(cos(x))}{(sin(x))}))e - {sin(x)}^{e}*0\\=&\frac{{sin(x)}^{e}e^{3}cos^{3}(x)}{sin^{3}(x)} - \frac{3{sin(x)}^{e}e^{2}cos^{3}(x)}{sin^{3}(x)} - \frac{3{sin(x)}^{e}e^{2}cos(x)}{sin(x)} + \frac{2{sin(x)}^{e}ecos^{3}(x)}{sin^{3}(x)} + \frac{2{sin(x)}^{e}ecos(x)}{sin(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{{sin(x)}^{e}e^{3}cos^{3}(x)}{sin^{3}(x)} - \frac{3{sin(x)}^{e}e^{2}cos^{3}(x)}{sin^{3}(x)} - \frac{3{sin(x)}^{e}e^{2}cos(x)}{sin(x)} + \frac{2{sin(x)}^{e}ecos^{3}(x)}{sin^{3}(x)} + \frac{2{sin(x)}^{e}ecos(x)}{sin(x)}\right)}{dx}\\=&\frac{({sin(x)}^{e}((0)ln(sin(x)) + \frac{(e)(cos(x))}{(sin(x))}))e^{3}cos^{3}(x)}{sin^{3}(x)} + \frac{{sin(x)}^{e}*3e^{2}*0cos^{3}(x)}{sin^{3}(x)} + \frac{{sin(x)}^{e}e^{3}*-3cos(x)cos^{3}(x)}{sin^{4}(x)} + \frac{{sin(x)}^{e}e^{3}*-3cos^{2}(x)sin(x)}{sin^{3}(x)} - \frac{3({sin(x)}^{e}((0)ln(sin(x)) + \frac{(e)(cos(x))}{(sin(x))}))e^{2}cos^{3}(x)}{sin^{3}(x)} - \frac{3{sin(x)}^{e}*2e*0cos^{3}(x)}{sin^{3}(x)} - \frac{3{sin(x)}^{e}e^{2}*-3cos(x)cos^{3}(x)}{sin^{4}(x)} - \frac{3{sin(x)}^{e}e^{2}*-3cos^{2}(x)sin(x)}{sin^{3}(x)} - \frac{3({sin(x)}^{e}((0)ln(sin(x)) + \frac{(e)(cos(x))}{(sin(x))}))e^{2}cos(x)}{sin(x)} - \frac{3{sin(x)}^{e}*2e*0cos(x)}{sin(x)} - \frac{3{sin(x)}^{e}e^{2}*-cos(x)cos(x)}{sin^{2}(x)} - \frac{3{sin(x)}^{e}e^{2}*-sin(x)}{sin(x)} + \frac{2({sin(x)}^{e}((0)ln(sin(x)) + \frac{(e)(cos(x))}{(sin(x))}))ecos^{3}(x)}{sin^{3}(x)} + \frac{2{sin(x)}^{e}*0cos^{3}(x)}{sin^{3}(x)} + \frac{2{sin(x)}^{e}e*-3cos(x)cos^{3}(x)}{sin^{4}(x)} + \frac{2{sin(x)}^{e}e*-3cos^{2}(x)sin(x)}{sin^{3}(x)} + \frac{2({sin(x)}^{e}((0)ln(sin(x)) + \frac{(e)(cos(x))}{(sin(x))}))ecos(x)}{sin(x)} + \frac{2{sin(x)}^{e}*0cos(x)}{sin(x)} + \frac{2{sin(x)}^{e}e*-cos(x)cos(x)}{sin^{2}(x)} + \frac{2{sin(x)}^{e}e*-sin(x)}{sin(x)}\\=&\frac{{sin(x)}^{e}e^{4}cos^{4}(x)}{sin^{4}(x)} - \frac{6{sin(x)}^{e}e^{3}cos^{4}(x)}{sin^{4}(x)} - \frac{6{sin(x)}^{e}e^{3}cos^{2}(x)}{sin^{2}(x)} + \frac{11{sin(x)}^{e}e^{2}cos^{4}(x)}{sin^{4}(x)} + \frac{14{sin(x)}^{e}e^{2}cos^{2}(x)}{sin^{2}(x)} - \frac{6{sin(x)}^{e}ecos^{4}(x)}{sin^{4}(x)} - \frac{8{sin(x)}^{e}ecos^{2}(x)}{sin^{2}(x)} + 3{sin(x)}^{e}e^{2} - 2{sin(x)}^{e}e\\ \end{split}\end{equation} \]

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