There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ a + bx + {e}^{x}((f)cos(2x) + (g)sin(2x))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = a + bx + f{e}^{x}cos(2x) + g{e}^{x}sin(2x)\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( a + bx + f{e}^{x}cos(2x) + g{e}^{x}sin(2x)\right)}{dx}\\=&0 + b + f({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))cos(2x) + f{e}^{x}*-sin(2x)*2 + g({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))sin(2x) + g{e}^{x}cos(2x)*2\\=&b + f{e}^{x}cos(2x) - 2f{e}^{x}sin(2x) + g{e}^{x}sin(2x) + 2g{e}^{x}cos(2x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( b + f{e}^{x}cos(2x) - 2f{e}^{x}sin(2x) + g{e}^{x}sin(2x) + 2g{e}^{x}cos(2x)\right)}{dx}\\=&0 + f({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))cos(2x) + f{e}^{x}*-sin(2x)*2 - 2f({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))sin(2x) - 2f{e}^{x}cos(2x)*2 + g({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))sin(2x) + g{e}^{x}cos(2x)*2 + 2g({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))cos(2x) + 2g{e}^{x}*-sin(2x)*2\\=&-3f{e}^{x}cos(2x) - 4f{e}^{x}sin(2x) - 3g{e}^{x}sin(2x) + 4g{e}^{x}cos(2x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( -3f{e}^{x}cos(2x) - 4f{e}^{x}sin(2x) - 3g{e}^{x}sin(2x) + 4g{e}^{x}cos(2x)\right)}{dx}\\=&-3f({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))cos(2x) - 3f{e}^{x}*-sin(2x)*2 - 4f({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))sin(2x) - 4f{e}^{x}cos(2x)*2 - 3g({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))sin(2x) - 3g{e}^{x}cos(2x)*2 + 4g({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))cos(2x) + 4g{e}^{x}*-sin(2x)*2\\=&-11f{e}^{x}cos(2x) + 2f{e}^{x}sin(2x) - 11g{e}^{x}sin(2x) - 2g{e}^{x}cos(2x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( -11f{e}^{x}cos(2x) + 2f{e}^{x}sin(2x) - 11g{e}^{x}sin(2x) - 2g{e}^{x}cos(2x)\right)}{dx}\\=&-11f({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))cos(2x) - 11f{e}^{x}*-sin(2x)*2 + 2f({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))sin(2x) + 2f{e}^{x}cos(2x)*2 - 11g({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))sin(2x) - 11g{e}^{x}cos(2x)*2 - 2g({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))cos(2x) - 2g{e}^{x}*-sin(2x)*2\\=&-7f{e}^{x}cos(2x) + 24f{e}^{x}sin(2x) - 7g{e}^{x}sin(2x) - 24g{e}^{x}cos(2x)\\ \end{split}\end{equation} \]

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