There are 1 questions in this calculation: for each question, the 3 derivative of t is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ third\ derivative\ of\ function\ \frac{{t}^{\frac{5}{2}}}{({e}^{t} - 1)}\ with\ respect\ to\ t：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{t^{\frac{5}{2}}}{({e}^{t} - 1)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{t^{\frac{5}{2}}}{({e}^{t} - 1)}\right)}{dt}\\=&(\frac{-(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{2}})t^{\frac{5}{2}} + \frac{\frac{5}{2}t^{\frac{3}{2}}}{({e}^{t} - 1)}\\=&\frac{-t^{\frac{5}{2}}{e}^{t}}{({e}^{t} - 1)^{2}} + \frac{5t^{\frac{3}{2}}}{2({e}^{t} - 1)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-t^{\frac{5}{2}}{e}^{t}}{({e}^{t} - 1)^{2}} + \frac{5t^{\frac{3}{2}}}{2({e}^{t} - 1)}\right)}{dt}\\=&-(\frac{-2(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{3}})t^{\frac{5}{2}}{e}^{t} - \frac{\frac{5}{2}t^{\frac{3}{2}}{e}^{t}}{({e}^{t} - 1)^{2}} - \frac{t^{\frac{5}{2}}({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)}))}{({e}^{t} - 1)^{2}} + \frac{5(\frac{-(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{2}})t^{\frac{3}{2}}}{2} + \frac{5*\frac{3}{2}t^{\frac{1}{2}}}{2({e}^{t} - 1)}\\=&\frac{2t^{\frac{5}{2}}{e}^{(2t)}}{({e}^{t} - 1)^{3}} - \frac{5t^{\frac{3}{2}}{e}^{t}}{({e}^{t} - 1)^{2}} - \frac{t^{\frac{5}{2}}{e}^{t}}{({e}^{t} - 1)^{2}} + \frac{15t^{\frac{1}{2}}}{4({e}^{t} - 1)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{2t^{\frac{5}{2}}{e}^{(2t)}}{({e}^{t} - 1)^{3}} - \frac{5t^{\frac{3}{2}}{e}^{t}}{({e}^{t} - 1)^{2}} - \frac{t^{\frac{5}{2}}{e}^{t}}{({e}^{t} - 1)^{2}} + \frac{15t^{\frac{1}{2}}}{4({e}^{t} - 1)}\right)}{dt}\\=&2(\frac{-3(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{4}})t^{\frac{5}{2}}{e}^{(2t)} + \frac{2*\frac{5}{2}t^{\frac{3}{2}}{e}^{(2t)}}{({e}^{t} - 1)^{3}} + \frac{2t^{\frac{5}{2}}({e}^{(2t)}((2)ln(e) + \frac{(2t)(0)}{(e)}))}{({e}^{t} - 1)^{3}} - 5(\frac{-2(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{3}})t^{\frac{3}{2}}{e}^{t} - \frac{5*\frac{3}{2}t^{\frac{1}{2}}{e}^{t}}{({e}^{t} - 1)^{2}} - \frac{5t^{\frac{3}{2}}({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)}))}{({e}^{t} - 1)^{2}} - (\frac{-2(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{3}})t^{\frac{5}{2}}{e}^{t} - \frac{\frac{5}{2}t^{\frac{3}{2}}{e}^{t}}{({e}^{t} - 1)^{2}} - \frac{t^{\frac{5}{2}}({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)}))}{({e}^{t} - 1)^{2}} + \frac{15(\frac{-(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{2}})t^{\frac{1}{2}}}{4} + \frac{15*\frac{1}{2}}{4({e}^{t} - 1)t^{\frac{1}{2}}}\\=&\frac{-6t^{\frac{5}{2}}{e}^{(3t)}}{({e}^{t} - 1)^{4}} + \frac{15t^{\frac{3}{2}}{e}^{(2t)}}{({e}^{t} - 1)^{3}} + \frac{6t^{\frac{5}{2}}{e}^{(2t)}}{({e}^{t} - 1)^{3}} - \frac{45t^{\frac{1}{2}}{e}^{t}}{4({e}^{t} - 1)^{2}} - \frac{15t^{\frac{3}{2}}{e}^{t}}{2({e}^{t} - 1)^{2}} - \frac{t^{\frac{5}{2}}{e}^{t}}{({e}^{t} - 1)^{2}} + \frac{15}{8({e}^{t} - 1)t^{\frac{1}{2}}}\\ \end{split}\end{equation} \]

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