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

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