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

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