There are 1 questions in this calculation: for each question, the 4 derivative of e is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ sin({e}^{x})\ with\ respect\ to\ e：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( sin({e}^{x})\right)}{de}\\=&cos({e}^{x})({e}^{x}((0)ln(e) + \frac{(x)(1)}{(e)}))\\=&\frac{x{e}^{x}cos({e}^{x})}{e}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{x{e}^{x}cos({e}^{x})}{e}\right)}{de}\\=&\frac{x*-{e}^{x}cos({e}^{x})}{e^{2}} + \frac{x({e}^{x}((0)ln(e) + \frac{(x)(1)}{(e)}))cos({e}^{x})}{e} + \frac{x{e}^{x}*-sin({e}^{x})({e}^{x}((0)ln(e) + \frac{(x)(1)}{(e)}))}{e}\\=&\frac{-x{e}^{x}cos({e}^{x})}{e^{2}} + \frac{x^{2}{e}^{x}cos({e}^{x})}{e^{2}} - \frac{x^{2}{e}^{(2x)}sin({e}^{x})}{e^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-x{e}^{x}cos({e}^{x})}{e^{2}} + \frac{x^{2}{e}^{x}cos({e}^{x})}{e^{2}} - \frac{x^{2}{e}^{(2x)}sin({e}^{x})}{e^{2}}\right)}{de}\\=&\frac{-x*-2{e}^{x}cos({e}^{x})}{e^{3}} - \frac{x({e}^{x}((0)ln(e) + \frac{(x)(1)}{(e)}))cos({e}^{x})}{e^{2}} - \frac{x{e}^{x}*-sin({e}^{x})({e}^{x}((0)ln(e) + \frac{(x)(1)}{(e)}))}{e^{2}} + \frac{x^{2}*-2{e}^{x}cos({e}^{x})}{e^{3}} + \frac{x^{2}({e}^{x}((0)ln(e) + \frac{(x)(1)}{(e)}))cos({e}^{x})}{e^{2}} + \frac{x^{2}{e}^{x}*-sin({e}^{x})({e}^{x}((0)ln(e) + \frac{(x)(1)}{(e)}))}{e^{2}} - \frac{x^{2}*-2{e}^{(2x)}sin({e}^{x})}{e^{3}} - \frac{x^{2}({e}^{(2x)}((0)ln(e) + \frac{(2x)(1)}{(e)}))sin({e}^{x})}{e^{2}} - \frac{x^{2}{e}^{(2x)}cos({e}^{x})({e}^{x}((0)ln(e) + \frac{(x)(1)}{(e)}))}{e^{2}}\\=&\frac{2x{e}^{x}cos({e}^{x})}{e^{3}} - \frac{3x^{2}{e}^{x}cos({e}^{x})}{e^{3}} + \frac{3x^{2}{e}^{(2x)}sin({e}^{x})}{e^{3}} + \frac{x^{3}{e}^{x}cos({e}^{x})}{e^{3}} - \frac{3x^{3}{e}^{(2x)}sin({e}^{x})}{e^{3}} - \frac{x^{3}{e}^{(3x)}cos({e}^{x})}{e^{3}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{2x{e}^{x}cos({e}^{x})}{e^{3}} - \frac{3x^{2}{e}^{x}cos({e}^{x})}{e^{3}} + \frac{3x^{2}{e}^{(2x)}sin({e}^{x})}{e^{3}} + \frac{x^{3}{e}^{x}cos({e}^{x})}{e^{3}} - \frac{3x^{3}{e}^{(2x)}sin({e}^{x})}{e^{3}} - \frac{x^{3}{e}^{(3x)}cos({e}^{x})}{e^{3}}\right)}{de}\\=&\frac{2x*-3{e}^{x}cos({e}^{x})}{e^{4}} + \frac{2x({e}^{x}((0)ln(e) + \frac{(x)(1)}{(e)}))cos({e}^{x})}{e^{3}} + \frac{2x{e}^{x}*-sin({e}^{x})({e}^{x}((0)ln(e) + \frac{(x)(1)}{(e)}))}{e^{3}} - \frac{3x^{2}*-3{e}^{x}cos({e}^{x})}{e^{4}} - \frac{3x^{2}({e}^{x}((0)ln(e) + \frac{(x)(1)}{(e)}))cos({e}^{x})}{e^{3}} - \frac{3x^{2}{e}^{x}*-sin({e}^{x})({e}^{x}((0)ln(e) + \frac{(x)(1)}{(e)}))}{e^{3}} + \frac{3x^{2}*-3{e}^{(2x)}sin({e}^{x})}{e^{4}} + \frac{3x^{2}({e}^{(2x)}((0)ln(e) + \frac{(2x)(1)}{(e)}))sin({e}^{x})}{e^{3}} + \frac{3x^{2}{e}^{(2x)}cos({e}^{x})({e}^{x}((0)ln(e) + \frac{(x)(1)}{(e)}))}{e^{3}} + \frac{x^{3}*-3{e}^{x}cos({e}^{x})}{e^{4}} + \frac{x^{3}({e}^{x}((0)ln(e) + \frac{(x)(1)}{(e)}))cos({e}^{x})}{e^{3}} + \frac{x^{3}{e}^{x}*-sin({e}^{x})({e}^{x}((0)ln(e) + \frac{(x)(1)}{(e)}))}{e^{3}} - \frac{3x^{3}*-3{e}^{(2x)}sin({e}^{x})}{e^{4}} - \frac{3x^{3}({e}^{(2x)}((0)ln(e) + \frac{(2x)(1)}{(e)}))sin({e}^{x})}{e^{3}} - \frac{3x^{3}{e}^{(2x)}cos({e}^{x})({e}^{x}((0)ln(e) + \frac{(x)(1)}{(e)}))}{e^{3}} - \frac{x^{3}*-3{e}^{(3x)}cos({e}^{x})}{e^{4}} - \frac{x^{3}({e}^{(3x)}((0)ln(e) + \frac{(3x)(1)}{(e)}))cos({e}^{x})}{e^{3}} - \frac{x^{3}{e}^{(3x)}*-sin({e}^{x})({e}^{x}((0)ln(e) + \frac{(x)(1)}{(e)}))}{e^{3}}\\=&\frac{-6x{e}^{x}cos({e}^{x})}{e^{4}} + \frac{11x^{2}{e}^{x}cos({e}^{x})}{e^{4}} - \frac{11x^{2}{e}^{(2x)}sin({e}^{x})}{e^{4}} - \frac{6x^{3}{e}^{x}cos({e}^{x})}{e^{4}} + \frac{18x^{3}{e}^{(2x)}sin({e}^{x})}{e^{4}} + \frac{6x^{3}{e}^{(3x)}cos({e}^{x})}{e^{4}} + \frac{x^{4}{e}^{x}cos({e}^{x})}{e^{4}} - \frac{7x^{4}{e}^{(2x)}sin({e}^{x})}{e^{4}} - \frac{6x^{4}{e}^{(3x)}cos({e}^{x})}{e^{4}} + \frac{x^{4}{e}^{(4x)}sin({e}^{x})}{e^{4}}\\ \end{split}\end{equation} \]

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