There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ {e}^{sin({x}^{2})}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = {e}^{sin(x^{2})}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( {e}^{sin(x^{2})}\right)}{dx}\\=&({e}^{sin(x^{2})}((cos(x^{2})*2x)ln(e) + \frac{(sin(x^{2}))(0)}{(e)}))\\=&2x{e}^{sin(x^{2})}cos(x^{2})\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( 2x{e}^{sin(x^{2})}cos(x^{2})\right)}{dx}\\=&2{e}^{sin(x^{2})}cos(x^{2}) + 2x({e}^{sin(x^{2})}((cos(x^{2})*2x)ln(e) + \frac{(sin(x^{2}))(0)}{(e)}))cos(x^{2}) + 2x{e}^{sin(x^{2})}*-sin(x^{2})*2x\\=&2{e}^{sin(x^{2})}cos(x^{2}) + 4x^{2}{e}^{sin(x^{2})}cos^{2}(x^{2}) - 4x^{2}{e}^{sin(x^{2})}sin(x^{2})\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 2{e}^{sin(x^{2})}cos(x^{2}) + 4x^{2}{e}^{sin(x^{2})}cos^{2}(x^{2}) - 4x^{2}{e}^{sin(x^{2})}sin(x^{2})\right)}{dx}\\=&2({e}^{sin(x^{2})}((cos(x^{2})*2x)ln(e) + \frac{(sin(x^{2}))(0)}{(e)}))cos(x^{2}) + 2{e}^{sin(x^{2})}*-sin(x^{2})*2x + 4*2x{e}^{sin(x^{2})}cos^{2}(x^{2}) + 4x^{2}({e}^{sin(x^{2})}((cos(x^{2})*2x)ln(e) + \frac{(sin(x^{2}))(0)}{(e)}))cos^{2}(x^{2}) + 4x^{2}{e}^{sin(x^{2})}*-2cos(x^{2})sin(x^{2})*2x - 4*2x{e}^{sin(x^{2})}sin(x^{2}) - 4x^{2}({e}^{sin(x^{2})}((cos(x^{2})*2x)ln(e) + \frac{(sin(x^{2}))(0)}{(e)}))sin(x^{2}) - 4x^{2}{e}^{sin(x^{2})}cos(x^{2})*2x\\=&12x{e}^{sin(x^{2})}cos^{2}(x^{2}) - 24x^{3}{e}^{sin(x^{2})}sin(x^{2})cos(x^{2}) + 8x^{3}{e}^{sin(x^{2})}cos^{3}(x^{2}) - 12x{e}^{sin(x^{2})}sin(x^{2}) - 8x^{3}{e}^{sin(x^{2})}cos(x^{2})\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 12x{e}^{sin(x^{2})}cos^{2}(x^{2}) - 24x^{3}{e}^{sin(x^{2})}sin(x^{2})cos(x^{2}) + 8x^{3}{e}^{sin(x^{2})}cos^{3}(x^{2}) - 12x{e}^{sin(x^{2})}sin(x^{2}) - 8x^{3}{e}^{sin(x^{2})}cos(x^{2})\right)}{dx}\\=&12{e}^{sin(x^{2})}cos^{2}(x^{2}) + 12x({e}^{sin(x^{2})}((cos(x^{2})*2x)ln(e) + \frac{(sin(x^{2}))(0)}{(e)}))cos^{2}(x^{2}) + 12x{e}^{sin(x^{2})}*-2cos(x^{2})sin(x^{2})*2x - 24*3x^{2}{e}^{sin(x^{2})}sin(x^{2})cos(x^{2}) - 24x^{3}({e}^{sin(x^{2})}((cos(x^{2})*2x)ln(e) + \frac{(sin(x^{2}))(0)}{(e)}))sin(x^{2})cos(x^{2}) - 24x^{3}{e}^{sin(x^{2})}cos(x^{2})*2xcos(x^{2}) - 24x^{3}{e}^{sin(x^{2})}sin(x^{2})*-sin(x^{2})*2x + 8*3x^{2}{e}^{sin(x^{2})}cos^{3}(x^{2}) + 8x^{3}({e}^{sin(x^{2})}((cos(x^{2})*2x)ln(e) + \frac{(sin(x^{2}))(0)}{(e)}))cos^{3}(x^{2}) + 8x^{3}{e}^{sin(x^{2})}*-3cos^{2}(x^{2})sin(x^{2})*2x - 12{e}^{sin(x^{2})}sin(x^{2}) - 12x({e}^{sin(x^{2})}((cos(x^{2})*2x)ln(e) + \frac{(sin(x^{2}))(0)}{(e)}))sin(x^{2}) - 12x{e}^{sin(x^{2})}cos(x^{2})*2x - 8*3x^{2}{e}^{sin(x^{2})}cos(x^{2}) - 8x^{3}({e}^{sin(x^{2})}((cos(x^{2})*2x)ln(e) + \frac{(sin(x^{2}))(0)}{(e)}))cos(x^{2}) - 8x^{3}{e}^{sin(x^{2})}*-sin(x^{2})*2x\\=&12{e}^{sin(x^{2})}cos^{2}(x^{2}) + 48x^{2}{e}^{sin(x^{2})}cos^{3}(x^{2}) - 144x^{2}{e}^{sin(x^{2})}sin(x^{2})cos(x^{2}) - 96x^{4}{e}^{sin(x^{2})}sin(x^{2})cos^{2}(x^{2}) - 64x^{4}{e}^{sin(x^{2})}cos^{2}(x^{2}) + 16x^{4}{e}^{sin(x^{2})}cos^{4}(x^{2}) + 48x^{4}{e}^{sin(x^{2})}sin^{2}(x^{2}) - 12{e}^{sin(x^{2})}sin(x^{2}) - 48x^{2}{e}^{sin(x^{2})}cos(x^{2}) + 16x^{4}{e}^{sin(x^{2})}sin(x^{2})\\ \end{split}\end{equation} \]

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