There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{2x}{({e}^{x} - 1)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{2x}{({e}^{x} - 1)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{2x}{({e}^{x} - 1)}\right)}{dx}\\=&2(\frac{-(({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})) + 0)}{({e}^{x} - 1)^{2}})x + \frac{2}{({e}^{x} - 1)}\\=&\frac{-2x{e}^{x}}{({e}^{x} - 1)^{2}} + \frac{2}{({e}^{x} - 1)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2x{e}^{x}}{({e}^{x} - 1)^{2}} + \frac{2}{({e}^{x} - 1)}\right)}{dx}\\=&-2(\frac{-2(({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})) + 0)}{({e}^{x} - 1)^{3}})x{e}^{x} - \frac{2{e}^{x}}{({e}^{x} - 1)^{2}} - \frac{2x({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))}{({e}^{x} - 1)^{2}} + 2(\frac{-(({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})) + 0)}{({e}^{x} - 1)^{2}})\\=&\frac{4x{e}^{(2x)}}{({e}^{x} - 1)^{3}} - \frac{4{e}^{x}}{({e}^{x} - 1)^{2}} - \frac{2x{e}^{x}}{({e}^{x} - 1)^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{4x{e}^{(2x)}}{({e}^{x} - 1)^{3}} - \frac{4{e}^{x}}{({e}^{x} - 1)^{2}} - \frac{2x{e}^{x}}{({e}^{x} - 1)^{2}}\right)}{dx}\\=&4(\frac{-3(({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})) + 0)}{({e}^{x} - 1)^{4}})x{e}^{(2x)} + \frac{4{e}^{(2x)}}{({e}^{x} - 1)^{3}} + \frac{4x({e}^{(2x)}((2)ln(e) + \frac{(2x)(0)}{(e)}))}{({e}^{x} - 1)^{3}} - 4(\frac{-2(({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})) + 0)}{({e}^{x} - 1)^{3}}){e}^{x} - \frac{4({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))}{({e}^{x} - 1)^{2}} - 2(\frac{-2(({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})) + 0)}{({e}^{x} - 1)^{3}})x{e}^{x} - \frac{2{e}^{x}}{({e}^{x} - 1)^{2}} - \frac{2x({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))}{({e}^{x} - 1)^{2}}\\=&\frac{-12x{e}^{(3x)}}{({e}^{x} - 1)^{4}} + \frac{12{e}^{(2x)}}{({e}^{x} - 1)^{3}} + \frac{12x{e}^{(2x)}}{({e}^{x} - 1)^{3}} - \frac{6{e}^{x}}{({e}^{x} - 1)^{2}} - \frac{2x{e}^{x}}{({e}^{x} - 1)^{2}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-12x{e}^{(3x)}}{({e}^{x} - 1)^{4}} + \frac{12{e}^{(2x)}}{({e}^{x} - 1)^{3}} + \frac{12x{e}^{(2x)}}{({e}^{x} - 1)^{3}} - \frac{6{e}^{x}}{({e}^{x} - 1)^{2}} - \frac{2x{e}^{x}}{({e}^{x} - 1)^{2}}\right)}{dx}\\=&-12(\frac{-4(({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})) + 0)}{({e}^{x} - 1)^{5}})x{e}^{(3x)} - \frac{12{e}^{(3x)}}{({e}^{x} - 1)^{4}} - \frac{12x({e}^{(3x)}((3)ln(e) + \frac{(3x)(0)}{(e)}))}{({e}^{x} - 1)^{4}} + 12(\frac{-3(({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})) + 0)}{({e}^{x} - 1)^{4}}){e}^{(2x)} + \frac{12({e}^{(2x)}((2)ln(e) + \frac{(2x)(0)}{(e)}))}{({e}^{x} - 1)^{3}} + 12(\frac{-3(({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})) + 0)}{({e}^{x} - 1)^{4}})x{e}^{(2x)} + \frac{12{e}^{(2x)}}{({e}^{x} - 1)^{3}} + \frac{12x({e}^{(2x)}((2)ln(e) + \frac{(2x)(0)}{(e)}))}{({e}^{x} - 1)^{3}} - 6(\frac{-2(({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})) + 0)}{({e}^{x} - 1)^{3}}){e}^{x} - \frac{6({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))}{({e}^{x} - 1)^{2}} - 2(\frac{-2(({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})) + 0)}{({e}^{x} - 1)^{3}})x{e}^{x} - \frac{2{e}^{x}}{({e}^{x} - 1)^{2}} - \frac{2x({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))}{({e}^{x} - 1)^{2}}\\=&\frac{48x{e}^{(4x)}}{({e}^{x} - 1)^{5}} - \frac{48{e}^{(3x)}}{({e}^{x} - 1)^{4}} - \frac{72x{e}^{(3x)}}{({e}^{x} - 1)^{4}} + \frac{48{e}^{(2x)}}{({e}^{x} - 1)^{3}} + \frac{28x{e}^{(2x)}}{({e}^{x} - 1)^{3}} - \frac{8{e}^{x}}{({e}^{x} - 1)^{2}} - \frac{2x{e}^{x}}{({e}^{x} - 1)^{2}}\\ \end{split}\end{equation} \]

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