There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
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\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ 10000(\frac{sin(2x)sin(3x)sin(4x)}{2*3*4})\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{1250}{3}sin(2x)sin(3x)sin(4x)\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{1250}{3}sin(2x)sin(3x)sin(4x)\right)}{dx}\\=&\frac{1250}{3}cos(2x)*2sin(3x)sin(4x) + \frac{1250}{3}sin(2x)cos(3x)*3sin(4x) + \frac{1250}{3}sin(2x)sin(3x)cos(4x)*4\\=&\frac{2500sin(3x)sin(4x)cos(2x)}{3} + 1250sin(2x)sin(4x)cos(3x) + \frac{5000sin(3x)sin(2x)cos(4x)}{3}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{2500sin(3x)sin(4x)cos(2x)}{3} + 1250sin(2x)sin(4x)cos(3x) + \frac{5000sin(3x)sin(2x)cos(4x)}{3}\right)}{dx}\\=&\frac{2500cos(3x)*3sin(4x)cos(2x)}{3} + \frac{2500sin(3x)cos(4x)*4cos(2x)}{3} + \frac{2500sin(3x)sin(4x)*-sin(2x)*2}{3} + 1250cos(2x)*2sin(4x)cos(3x) + 1250sin(2x)cos(4x)*4cos(3x) + 1250sin(2x)sin(4x)*-sin(3x)*3 + \frac{5000cos(3x)*3sin(2x)cos(4x)}{3} + \frac{5000sin(3x)cos(2x)*2cos(4x)}{3} + \frac{5000sin(3x)sin(2x)*-sin(4x)*4}{3}\\=&2500sin(4x)cos(3x)cos(2x) + \frac{10000sin(3x)cos(4x)cos(2x)}{3} - \frac{5000sin(3x)sin(2x)sin(4x)}{3} + 2500sin(4x)cos(2x)cos(3x) + 5000sin(2x)cos(4x)cos(3x) - 3750sin(2x)sin(3x)sin(4x) + 5000sin(2x)cos(3x)cos(4x) + \frac{10000sin(3x)cos(2x)cos(4x)}{3} - \frac{20000sin(3x)sin(4x)sin(2x)}{3}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 2500sin(4x)cos(3x)cos(2x) + \frac{10000sin(3x)cos(4x)cos(2x)}{3} - \frac{5000sin(3x)sin(2x)sin(4x)}{3} + 2500sin(4x)cos(2x)cos(3x) + 5000sin(2x)cos(4x)cos(3x) - 3750sin(2x)sin(3x)sin(4x) + 5000sin(2x)cos(3x)cos(4x) + \frac{10000sin(3x)cos(2x)cos(4x)}{3} - \frac{20000sin(3x)sin(4x)sin(2x)}{3}\right)}{dx}\\=&2500cos(4x)*4cos(3x)cos(2x) + 2500sin(4x)*-sin(3x)*3cos(2x) + 2500sin(4x)cos(3x)*-sin(2x)*2 + \frac{10000cos(3x)*3cos(4x)cos(2x)}{3} + \frac{10000sin(3x)*-sin(4x)*4cos(2x)}{3} + \frac{10000sin(3x)cos(4x)*-sin(2x)*2}{3} - \frac{5000cos(3x)*3sin(2x)sin(4x)}{3} - \frac{5000sin(3x)cos(2x)*2sin(4x)}{3} - \frac{5000sin(3x)sin(2x)cos(4x)*4}{3} + 2500cos(4x)*4cos(2x)cos(3x) + 2500sin(4x)*-sin(2x)*2cos(3x) + 2500sin(4x)cos(2x)*-sin(3x)*3 + 5000cos(2x)*2cos(4x)cos(3x) + 5000sin(2x)*-sin(4x)*4cos(3x) + 5000sin(2x)cos(4x)*-sin(3x)*3 - 3750cos(2x)*2sin(3x)sin(4x) - 3750sin(2x)cos(3x)*3sin(4x) - 3750sin(2x)sin(3x)cos(4x)*4 + 5000cos(2x)*2cos(3x)cos(4x) + 5000sin(2x)*-sin(3x)*3cos(4x) + 5000sin(2x)cos(3x)*-sin(4x)*4 + \frac{10000cos(3x)*3cos(2x)cos(4x)}{3} + \frac{10000sin(3x)*-sin(2x)*2cos(4x)}{3} + \frac{10000sin(3x)cos(2x)*-sin(4x)*4}{3} - \frac{20000cos(3x)*3sin(4x)sin(2x)}{3} - \frac{20000sin(3x)cos(4x)*4sin(2x)}{3} - \frac{20000sin(3x)sin(4x)cos(2x)*2}{3}\\=&10000cos(4x)cos(3x)cos(2x) - \frac{77500sin(3x)sin(4x)cos(2x)}{3} - 26250sin(2x)sin(4x)cos(3x) + 10000cos(3x)cos(4x)cos(2x) - 40000sin(4x)sin(3x)cos(2x) - 20000sin(2x)sin(3x)cos(4x) + 10000cos(4x)cos(2x)cos(3x) + 10000cos(2x)cos(4x)cos(3x) - 60000sin(4x)sin(2x)cos(3x) - \frac{215000sin(3x)sin(2x)cos(4x)}{3} + 10000cos(2x)cos(3x)cos(4x) + 10000cos(3x)cos(2x)cos(4x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 10000cos(4x)cos(3x)cos(2x) - \frac{77500sin(3x)sin(4x)cos(2x)}{3} - 26250sin(2x)sin(4x)cos(3x) + 10000cos(3x)cos(4x)cos(2x) - 40000sin(4x)sin(3x)cos(2x) - 20000sin(2x)sin(3x)cos(4x) + 10000cos(4x)cos(2x)cos(3x) + 10000cos(2x)cos(4x)cos(3x) - 60000sin(4x)sin(2x)cos(3x) - \frac{215000sin(3x)sin(2x)cos(4x)}{3} + 10000cos(2x)cos(3x)cos(4x) + 10000cos(3x)cos(2x)cos(4x)\right)}{dx}\\=&10000*-sin(4x)*4cos(3x)cos(2x) + 10000cos(4x)*-sin(3x)*3cos(2x) + 10000cos(4x)cos(3x)*-sin(2x)*2 - \frac{77500cos(3x)*3sin(4x)cos(2x)}{3} - \frac{77500sin(3x)cos(4x)*4cos(2x)}{3} - \frac{77500sin(3x)sin(4x)*-sin(2x)*2}{3} - 26250cos(2x)*2sin(4x)cos(3x) - 26250sin(2x)cos(4x)*4cos(3x) - 26250sin(2x)sin(4x)*-sin(3x)*3 + 10000*-sin(3x)*3cos(4x)cos(2x) + 10000cos(3x)*-sin(4x)*4cos(2x) + 10000cos(3x)cos(4x)*-sin(2x)*2 - 40000cos(4x)*4sin(3x)cos(2x) - 40000sin(4x)cos(3x)*3cos(2x) - 40000sin(4x)sin(3x)*-sin(2x)*2 - 20000cos(2x)*2sin(3x)cos(4x) - 20000sin(2x)cos(3x)*3cos(4x) - 20000sin(2x)sin(3x)*-sin(4x)*4 + 10000*-sin(4x)*4cos(2x)cos(3x) + 10000cos(4x)*-sin(2x)*2cos(3x) + 10000cos(4x)cos(2x)*-sin(3x)*3 + 10000*-sin(2x)*2cos(4x)cos(3x) + 10000cos(2x)*-sin(4x)*4cos(3x) + 10000cos(2x)cos(4x)*-sin(3x)*3 - 60000cos(4x)*4sin(2x)cos(3x) - 60000sin(4x)cos(2x)*2cos(3x) - 60000sin(4x)sin(2x)*-sin(3x)*3 - \frac{215000cos(3x)*3sin(2x)cos(4x)}{3} - \frac{215000sin(3x)cos(2x)*2cos(4x)}{3} - \frac{215000sin(3x)sin(2x)*-sin(4x)*4}{3} + 10000*-sin(2x)*2cos(3x)cos(4x) + 10000cos(2x)*-sin(3x)*3cos(4x) + 10000cos(2x)cos(3x)*-sin(4x)*4 + 10000*-sin(3x)*3cos(2x)cos(4x) + 10000cos(3x)*-sin(2x)*2cos(4x) + 10000cos(3x)cos(2x)*-sin(4x)*4\\=&-317500sin(4x)cos(3x)cos(2x) - \frac{1060000sin(3x)cos(4x)cos(2x)}{3} - 335000sin(2x)cos(3x)cos(4x) + \frac{155000sin(3x)sin(2x)sin(4x)}{3} - 292500sin(4x)cos(2x)cos(3x) - 405000sin(2x)cos(4x)cos(3x) + 78750sin(2x)sin(3x)sin(4x) + 80000sin(4x)sin(2x)sin(3x) - \frac{820000sin(3x)cos(2x)cos(4x)}{3} + 80000sin(2x)sin(4x)sin(3x) + 180000sin(4x)sin(3x)sin(2x) + \frac{860000sin(3x)sin(4x)sin(2x)}{3}\\ \end{split}\end{equation} \]

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