There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ cth(th(x))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( cth(th(x))\right)}{dx}\\=&(1 - cth^{2}(th(x)))(1 - th^{2}(x))\\=&th^{2}(x)cth^{2}(th(x)) - cth^{2}(th(x)) - th^{2}(x) + 1\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( th^{2}(x)cth^{2}(th(x)) - cth^{2}(th(x)) - th^{2}(x) + 1\right)}{dx}\\=&2th(x)(1 - th^{2}(x))cth^{2}(th(x)) + th^{2}(x)*2cth(th(x))(1 - cth^{2}(th(x)))(1 - th^{2}(x)) - 2cth(th(x))(1 - cth^{2}(th(x)))(1 - th^{2}(x)) - 2th(x)(1 - th^{2}(x)) + 0\\=&2th(x)cth^{2}(th(x)) - 2th^{3}(x)cth^{2}(th(x)) + 4th^{2}(x)cth(th(x)) - 2th^{4}(x)cth(th(x)) - 4th^{2}(x)cth^{3}(th(x)) + 2th^{4}(x)cth^{3}(th(x)) - 2cth(th(x)) + 2cth^{3}(th(x)) - 2th(x) + 2th^{3}(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 2th(x)cth^{2}(th(x)) - 2th^{3}(x)cth^{2}(th(x)) + 4th^{2}(x)cth(th(x)) - 2th^{4}(x)cth(th(x)) - 4th^{2}(x)cth^{3}(th(x)) + 2th^{4}(x)cth^{3}(th(x)) - 2cth(th(x)) + 2cth^{3}(th(x)) - 2th(x) + 2th^{3}(x)\right)}{dx}\\=&2(1 - th^{2}(x))cth^{2}(th(x)) + 2th(x)*2cth(th(x))(1 - cth^{2}(th(x)))(1 - th^{2}(x)) - 2*3th^{2}(x)(1 - th^{2}(x))cth^{2}(th(x)) - 2th^{3}(x)*2cth(th(x))(1 - cth^{2}(th(x)))(1 - th^{2}(x)) + 4*2th(x)(1 - th^{2}(x))cth(th(x)) + 4th^{2}(x)(1 - cth^{2}(th(x)))(1 - th^{2}(x)) - 2*4th^{3}(x)(1 - th^{2}(x))cth(th(x)) - 2th^{4}(x)(1 - cth^{2}(th(x)))(1 - th^{2}(x)) - 4*2th(x)(1 - th^{2}(x))cth^{3}(th(x)) - 4th^{2}(x)*3cth^{2}(th(x))(1 - cth^{2}(th(x)))(1 - th^{2}(x)) + 2*4th^{3}(x)(1 - th^{2}(x))cth^{3}(th(x)) + 2th^{4}(x)*3cth^{2}(th(x))(1 - cth^{2}(th(x)))(1 - th^{2}(x)) - 2(1 - cth^{2}(th(x)))(1 - th^{2}(x)) + 2*3cth^{2}(th(x))(1 - cth^{2}(th(x)))(1 - th^{2}(x)) - 2(1 - th^{2}(x)) + 2*3th^{2}(x)(1 - th^{2}(x))\\=&10cth^{2}(th(x)) - 32th^{2}(x)cth^{2}(th(x)) + 12th(x)cth(th(x)) - 24th^{3}(x)cth(th(x)) - 12th(x)cth^{3}(th(x)) + 24th^{3}(x)cth^{3}(th(x)) + 30th^{4}(x)cth^{2}(th(x)) + 12th^{5}(x)cth(th(x)) - 12th^{5}(x)cth^{3}(th(x)) + 18th^{2}(x)cth^{4}(th(x)) - 18th^{4}(x)cth^{4}(th(x)) - 8th^{6}(x)cth^{2}(th(x)) + 6th^{6}(x)cth^{4}(th(x)) + 14th^{2}(x) - 12th^{4}(x) + 2th^{6}(x) - 6cth^{4}(th(x)) - 4\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 10cth^{2}(th(x)) - 32th^{2}(x)cth^{2}(th(x)) + 12th(x)cth(th(x)) - 24th^{3}(x)cth(th(x)) - 12th(x)cth^{3}(th(x)) + 24th^{3}(x)cth^{3}(th(x)) + 30th^{4}(x)cth^{2}(th(x)) + 12th^{5}(x)cth(th(x)) - 12th^{5}(x)cth^{3}(th(x)) + 18th^{2}(x)cth^{4}(th(x)) - 18th^{4}(x)cth^{4}(th(x)) - 8th^{6}(x)cth^{2}(th(x)) + 6th^{6}(x)cth^{4}(th(x)) + 14th^{2}(x) - 12th^{4}(x) + 2th^{6}(x) - 6cth^{4}(th(x)) - 4\right)}{dx}\\=&10*2cth(th(x))(1 - cth^{2}(th(x)))(1 - th^{2}(x)) - 32*2th(x)(1 - th^{2}(x))cth^{2}(th(x)) - 32th^{2}(x)*2cth(th(x))(1 - cth^{2}(th(x)))(1 - th^{2}(x)) + 12(1 - th^{2}(x))cth(th(x)) + 12th(x)(1 - cth^{2}(th(x)))(1 - th^{2}(x)) - 24*3th^{2}(x)(1 - th^{2}(x))cth(th(x)) - 24th^{3}(x)(1 - cth^{2}(th(x)))(1 - th^{2}(x)) - 12(1 - th^{2}(x))cth^{3}(th(x)) - 12th(x)*3cth^{2}(th(x))(1 - cth^{2}(th(x)))(1 - th^{2}(x)) + 24*3th^{2}(x)(1 - th^{2}(x))cth^{3}(th(x)) + 24th^{3}(x)*3cth^{2}(th(x))(1 - cth^{2}(th(x)))(1 - th^{2}(x)) + 30*4th^{3}(x)(1 - th^{2}(x))cth^{2}(th(x)) + 30th^{4}(x)*2cth(th(x))(1 - cth^{2}(th(x)))(1 - th^{2}(x)) + 12*5th^{4}(x)(1 - th^{2}(x))cth(th(x)) + 12th^{5}(x)(1 - cth^{2}(th(x)))(1 - th^{2}(x)) - 12*5th^{4}(x)(1 - th^{2}(x))cth^{3}(th(x)) - 12th^{5}(x)*3cth^{2}(th(x))(1 - cth^{2}(th(x)))(1 - th^{2}(x)) + 18*2th(x)(1 - th^{2}(x))cth^{4}(th(x)) + 18th^{2}(x)*4cth^{3}(th(x))(1 - cth^{2}(th(x)))(1 - th^{2}(x)) - 18*4th^{3}(x)(1 - th^{2}(x))cth^{4}(th(x)) - 18th^{4}(x)*4cth^{3}(th(x))(1 - cth^{2}(th(x)))(1 - th^{2}(x)) - 8*6th^{5}(x)(1 - th^{2}(x))cth^{2}(th(x)) - 8th^{6}(x)*2cth(th(x))(1 - cth^{2}(th(x)))(1 - th^{2}(x)) + 6*6th^{5}(x)(1 - th^{2}(x))cth^{4}(th(x)) + 6th^{6}(x)*4cth^{3}(th(x))(1 - cth^{2}(th(x)))(1 - th^{2}(x)) + 14*2th(x)(1 - th^{2}(x)) - 12*4th^{3}(x)(1 - th^{2}(x)) + 2*6th^{5}(x)(1 - th^{2}(x)) - 6*4cth^{3}(th(x))(1 - cth^{2}(th(x)))(1 - th^{2}(x)) + 0\\=&32cth(th(x)) + 264th^{2}(x)cth^{3}(th(x)) - 56cth^{3}(th(x)) - 112th(x)cth^{2}(th(x)) + 328th^{3}(x)cth^{2}(th(x)) - 168th^{2}(x)cth(th(x)) + 256th^{4}(x)cth(th(x)) + 72th(x)cth^{4}(th(x)) - 400th^{4}(x)cth^{3}(th(x)) - 216th^{3}(x)cth^{4}(th(x)) - 312th^{5}(x)cth^{2}(th(x)) + 216th^{5}(x)cth^{4}(th(x)) - 136th^{6}(x)cth(th(x)) + 232th^{6}(x)cth^{3}(th(x)) + 96th^{7}(x)cth^{2}(th(x)) - 96th^{2}(x)cth^{5}(th(x)) - 72th^{7}(x)cth^{4}(th(x)) + 144th^{4}(x)cth^{5}(th(x)) - 96th^{6}(x)cth^{5}(th(x)) + 16th^{8}(x)cth(th(x)) - 40th^{8}(x)cth^{3}(th(x)) + 24th^{8}(x)cth^{5}(th(x)) + 96th^{5}(x) - 24th^{7}(x) - 112th^{3}(x) + 40th(x) + 24cth^{5}(th(x))\\ \end{split}\end{equation} \]

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