Derivative function：
Enter an original function (that is, the function to be derived), then set the variable to be derived and the order of the derivative, and click the "Next" button to obtain the derivative function of the corresponding order of the function.
Note that the input function supports mathematical functions and other constants.
Enter an original function (that is, the function to be derived), then set the variable to be derived and the order of the derivative, and click the "Next" button to obtain the derivative function of the corresponding order of the function.
Note that the input function supports mathematical functions and other constants.
Current location：Derivative function > Derivative function calculation history > Answer
There are 1 questions in this calculation: for each question, the 4 derivative of t is calculated.
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{{t}^{(x - 1)}}{({e}^{t} - 1)}\ with\ respect\ to\ t：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{{t}^{(x - 1)}}{({e}^{t} - 1)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{{t}^{(x - 1)}}{({e}^{t} - 1)}\right)}{dt}\\=&(\frac{-(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{2}}){t}^{(x - 1)} + \frac{({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)}\\=&\frac{-{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}} + \frac{x{t}^{(x - 1)}}{({e}^{t} - 1)t} - \frac{{t}^{(x - 1)}}{({e}^{t} - 1)t}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}} + \frac{x{t}^{(x - 1)}}{({e}^{t} - 1)t} - \frac{{t}^{(x - 1)}}{({e}^{t} - 1)t}\right)}{dt}\\=&-(\frac{-2(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{3}}){e}^{t}{t}^{(x - 1)} - \frac{({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})){t}^{(x - 1)}}{({e}^{t} - 1)^{2}} - \frac{{e}^{t}({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)^{2}} + \frac{(\frac{-(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{2}})x{t}^{(x - 1)}}{t} + \frac{x*-{t}^{(x - 1)}}{({e}^{t} - 1)t^{2}} + \frac{x({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)t} - \frac{(\frac{-(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{2}}){t}^{(x - 1)}}{t} - \frac{-{t}^{(x - 1)}}{({e}^{t} - 1)t^{2}} - \frac{({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)t}\\=&\frac{2{e}^{(2t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{3}} - \frac{{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}} - \frac{x{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t} + \frac{{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t} - \frac{x{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t} - \frac{3x{t}^{(x - 1)}}{({e}^{t} - 1)t^{2}} + \frac{x^{2}{t}^{(x - 1)}}{({e}^{t} - 1)t^{2}} + \frac{{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t} + \frac{2{t}^{(x - 1)}}{({e}^{t} - 1)t^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{2{e}^{(2t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{3}} - \frac{{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}} - \frac{x{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t} + \frac{{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t} - \frac{x{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t} - \frac{3x{t}^{(x - 1)}}{({e}^{t} - 1)t^{2}} + \frac{x^{2}{t}^{(x - 1)}}{({e}^{t} - 1)t^{2}} + \frac{{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t} + \frac{2{t}^{(x - 1)}}{({e}^{t} - 1)t^{2}}\right)}{dt}\\=&2(\frac{-3(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{4}}){e}^{(2t)}{t}^{(x - 1)} + \frac{2({e}^{(2t)}((2)ln(e) + \frac{(2t)(0)}{(e)})){t}^{(x - 1)}}{({e}^{t} - 1)^{3}} + \frac{2{e}^{(2t)}({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)^{3}} - (\frac{-2(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{3}}){e}^{t}{t}^{(x - 1)} - \frac{({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})){t}^{(x - 1)}}{({e}^{t} - 1)^{2}} - \frac{{e}^{t}({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)^{2}} - \frac{(\frac{-2(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{3}})x{t}^{(x - 1)}{e}^{t}}{t} - \frac{x*-{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t^{2}} - \frac{x({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)})){e}^{t}}{({e}^{t} - 1)^{2}t} - \frac{x{t}^{(x - 1)}({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)}))}{({e}^{t} - 1)^{2}t} + \frac{(\frac{-2(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{3}}){t}^{(x - 1)}{e}^{t}}{t} + \frac{-{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t^{2}} + \frac{({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)})){e}^{t}}{({e}^{t} - 1)^{2}t} + \frac{{t}^{(x - 1)}({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)}))}{({e}^{t} - 1)^{2}t} - \frac{(\frac{-2(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{3}})x{e}^{t}{t}^{(x - 1)}}{t} - \frac{x*-{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{2}} - \frac{x({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})){t}^{(x - 1)}}{({e}^{t} - 1)^{2}t} - \frac{x{e}^{t}({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)^{2}t} - \frac{3(\frac{-(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{2}})x{t}^{(x - 1)}}{t^{2}} - \frac{3x*-2{t}^{(x - 1)}}{({e}^{t} - 1)t^{3}} - \frac{3x({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)t^{2}} + \frac{(\frac{-(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{2}})x^{2}{t}^{(x - 1)}}{t^{2}} + \frac{x^{2}*-2{t}^{(x - 1)}}{({e}^{t} - 1)t^{3}} + \frac{x^{2}({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)t^{2}} + \frac{(\frac{-2(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{3}}){e}^{t}{t}^{(x - 1)}}{t} + \frac{-{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{2}} + \frac{({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})){t}^{(x - 1)}}{({e}^{t} - 1)^{2}t} + \frac{{e}^{t}({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)^{2}t} + \frac{2(\frac{-(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{2}}){t}^{(x - 1)}}{t^{2}} + \frac{2*-2{t}^{(x - 1)}}{({e}^{t} - 1)t^{3}} + \frac{2({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)t^{2}}\\=&\frac{-6{e}^{(3t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{4}} + \frac{6{e}^{(2t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{3}} + \frac{2x{t}^{(x - 1)}{e}^{(2t)}}{({e}^{t} - 1)^{3}t} - \frac{2{t}^{(x - 1)}{e}^{(2t)}}{({e}^{t} - 1)^{3}t} - \frac{{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}} - \frac{x{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t} + \frac{{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t} + \frac{4x{e}^{(2t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{3}t} + \frac{5x{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t^{2}} - \frac{2x^{2}{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t^{2}} - \frac{2x{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t} - \frac{4{e}^{(2t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{3}t} - \frac{3{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t^{2}} + \frac{2{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t} + \frac{4x{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{2}} - \frac{x^{2}{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{2}} - \frac{6x^{2}{t}^{(x - 1)}}{({e}^{t} - 1)t^{3}} + \frac{11x{t}^{(x - 1)}}{({e}^{t} - 1)t^{3}} + \frac{x^{3}{t}^{(x - 1)}}{({e}^{t} - 1)t^{3}} - \frac{3{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{2}} - \frac{6{t}^{(x - 1)}}{({e}^{t} - 1)t^{3}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-6{e}^{(3t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{4}} + \frac{6{e}^{(2t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{3}} + \frac{2x{t}^{(x - 1)}{e}^{(2t)}}{({e}^{t} - 1)^{3}t} - \frac{2{t}^{(x - 1)}{e}^{(2t)}}{({e}^{t} - 1)^{3}t} - \frac{{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}} - \frac{x{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t} + \frac{{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t} + \frac{4x{e}^{(2t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{3}t} + \frac{5x{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t^{2}} - \frac{2x^{2}{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t^{2}} - \frac{2x{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t} - \frac{4{e}^{(2t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{3}t} - \frac{3{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t^{2}} + \frac{2{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t} + \frac{4x{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{2}} - \frac{x^{2}{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{2}} - \frac{6x^{2}{t}^{(x - 1)}}{({e}^{t} - 1)t^{3}} + \frac{11x{t}^{(x - 1)}}{({e}^{t} - 1)t^{3}} + \frac{x^{3}{t}^{(x - 1)}}{({e}^{t} - 1)t^{3}} - \frac{3{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{2}} - \frac{6{t}^{(x - 1)}}{({e}^{t} - 1)t^{3}}\right)}{dt}\\=&-6(\frac{-4(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{5}}){e}^{(3t)}{t}^{(x - 1)} - \frac{6({e}^{(3t)}((3)ln(e) + \frac{(3t)(0)}{(e)})){t}^{(x - 1)}}{({e}^{t} - 1)^{4}} - \frac{6{e}^{(3t)}({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)^{4}} + 6(\frac{-3(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{4}}){e}^{(2t)}{t}^{(x - 1)} + \frac{6({e}^{(2t)}((2)ln(e) + \frac{(2t)(0)}{(e)})){t}^{(x - 1)}}{({e}^{t} - 1)^{3}} + \frac{6{e}^{(2t)}({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)^{3}} + \frac{2(\frac{-3(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{4}})x{t}^{(x - 1)}{e}^{(2t)}}{t} + \frac{2x*-{t}^{(x - 1)}{e}^{(2t)}}{({e}^{t} - 1)^{3}t^{2}} + \frac{2x({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)})){e}^{(2t)}}{({e}^{t} - 1)^{3}t} + \frac{2x{t}^{(x - 1)}({e}^{(2t)}((2)ln(e) + \frac{(2t)(0)}{(e)}))}{({e}^{t} - 1)^{3}t} - \frac{2(\frac{-3(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{4}}){t}^{(x - 1)}{e}^{(2t)}}{t} - \frac{2*-{t}^{(x - 1)}{e}^{(2t)}}{({e}^{t} - 1)^{3}t^{2}} - \frac{2({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)})){e}^{(2t)}}{({e}^{t} - 1)^{3}t} - \frac{2{t}^{(x - 1)}({e}^{(2t)}((2)ln(e) + \frac{(2t)(0)}{(e)}))}{({e}^{t} - 1)^{3}t} - (\frac{-2(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{3}}){e}^{t}{t}^{(x - 1)} - \frac{({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})){t}^{(x - 1)}}{({e}^{t} - 1)^{2}} - \frac{{e}^{t}({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)^{2}} - \frac{(\frac{-2(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{3}})x{t}^{(x - 1)}{e}^{t}}{t} - \frac{x*-{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t^{2}} - \frac{x({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)})){e}^{t}}{({e}^{t} - 1)^{2}t} - \frac{x{t}^{(x - 1)}({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)}))}{({e}^{t} - 1)^{2}t} + \frac{(\frac{-2(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{3}}){t}^{(x - 1)}{e}^{t}}{t} + \frac{-{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t^{2}} + \frac{({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)})){e}^{t}}{({e}^{t} - 1)^{2}t} + \frac{{t}^{(x - 1)}({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)}))}{({e}^{t} - 1)^{2}t} + \frac{4(\frac{-3(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{4}})x{e}^{(2t)}{t}^{(x - 1)}}{t} + \frac{4x*-{e}^{(2t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{3}t^{2}} + \frac{4x({e}^{(2t)}((2)ln(e) + \frac{(2t)(0)}{(e)})){t}^{(x - 1)}}{({e}^{t} - 1)^{3}t} + \frac{4x{e}^{(2t)}({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)^{3}t} + \frac{5(\frac{-2(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{3}})x{t}^{(x - 1)}{e}^{t}}{t^{2}} + \frac{5x*-2{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t^{3}} + \frac{5x({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)})){e}^{t}}{({e}^{t} - 1)^{2}t^{2}} + \frac{5x{t}^{(x - 1)}({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)}))}{({e}^{t} - 1)^{2}t^{2}} - \frac{2(\frac{-2(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{3}})x^{2}{t}^{(x - 1)}{e}^{t}}{t^{2}} - \frac{2x^{2}*-2{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t^{3}} - \frac{2x^{2}({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)})){e}^{t}}{({e}^{t} - 1)^{2}t^{2}} - \frac{2x^{2}{t}^{(x - 1)}({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)}))}{({e}^{t} - 1)^{2}t^{2}} - \frac{2(\frac{-2(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{3}})x{e}^{t}{t}^{(x - 1)}}{t} - \frac{2x*-{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{2}} - \frac{2x({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})){t}^{(x - 1)}}{({e}^{t} - 1)^{2}t} - \frac{2x{e}^{t}({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)^{2}t} - \frac{4(\frac{-3(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{4}}){e}^{(2t)}{t}^{(x - 1)}}{t} - \frac{4*-{e}^{(2t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{3}t^{2}} - \frac{4({e}^{(2t)}((2)ln(e) + \frac{(2t)(0)}{(e)})){t}^{(x - 1)}}{({e}^{t} - 1)^{3}t} - \frac{4{e}^{(2t)}({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)^{3}t} - \frac{3(\frac{-2(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{3}}){t}^{(x - 1)}{e}^{t}}{t^{2}} - \frac{3*-2{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t^{3}} - \frac{3({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)})){e}^{t}}{({e}^{t} - 1)^{2}t^{2}} - \frac{3{t}^{(x - 1)}({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)}))}{({e}^{t} - 1)^{2}t^{2}} + \frac{2(\frac{-2(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{3}}){e}^{t}{t}^{(x - 1)}}{t} + \frac{2*-{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{2}} + \frac{2({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})){t}^{(x - 1)}}{({e}^{t} - 1)^{2}t} + \frac{2{e}^{t}({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)^{2}t} + \frac{4(\frac{-2(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{3}})x{e}^{t}{t}^{(x - 1)}}{t^{2}} + \frac{4x*-2{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{3}} + \frac{4x({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})){t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{2}} + \frac{4x{e}^{t}({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)^{2}t^{2}} - \frac{(\frac{-2(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{3}})x^{2}{e}^{t}{t}^{(x - 1)}}{t^{2}} - \frac{x^{2}*-2{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{3}} - \frac{x^{2}({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})){t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{2}} - \frac{x^{2}{e}^{t}({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)^{2}t^{2}} - \frac{6(\frac{-(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{2}})x^{2}{t}^{(x - 1)}}{t^{3}} - \frac{6x^{2}*-3{t}^{(x - 1)}}{({e}^{t} - 1)t^{4}} - \frac{6x^{2}({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)t^{3}} + \frac{11(\frac{-(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{2}})x{t}^{(x - 1)}}{t^{3}} + \frac{11x*-3{t}^{(x - 1)}}{({e}^{t} - 1)t^{4}} + \frac{11x({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)t^{3}} + \frac{(\frac{-(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{2}})x^{3}{t}^{(x - 1)}}{t^{3}} + \frac{x^{3}*-3{t}^{(x - 1)}}{({e}^{t} - 1)t^{4}} + \frac{x^{3}({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)t^{3}} - \frac{3(\frac{-2(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{3}}){e}^{t}{t}^{(x - 1)}}{t^{2}} - \frac{3*-2{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{3}} - \frac{3({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})){t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{2}} - \frac{3{e}^{t}({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)^{2}t^{2}} - \frac{6(\frac{-(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{2}}){t}^{(x - 1)}}{t^{3}} - \frac{6*-3{t}^{(x - 1)}}{({e}^{t} - 1)t^{4}} - \frac{6({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)t^{3}}\\=&\frac{24{e}^{(4t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{5}} - \frac{36{e}^{(3t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{4}} - \frac{6x{t}^{(x - 1)}{e}^{(3t)}}{({e}^{t} - 1)^{4}t} + \frac{6{t}^{(x - 1)}{e}^{(3t)}}{({e}^{t} - 1)^{4}t} + \frac{14{e}^{(2t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{3}} + \frac{6x{t}^{(x - 1)}{e}^{(2t)}}{({e}^{t} - 1)^{3}t} - \frac{6{t}^{(x - 1)}{e}^{(2t)}}{({e}^{t} - 1)^{3}t} - \frac{18x{e}^{(3t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{4}t} - \frac{14x{t}^{(x - 1)}{e}^{(2t)}}{({e}^{t} - 1)^{3}t^{2}} + \frac{6x^{2}{t}^{(x - 1)}{e}^{(2t)}}{({e}^{t} - 1)^{3}t^{2}} + \frac{18x{e}^{(2t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{3}t} + \frac{18{e}^{(3t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{4}t} + \frac{8{t}^{(x - 1)}{e}^{(2t)}}{({e}^{t} - 1)^{3}t^{2}} - \frac{18{e}^{(2t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{3}t} - \frac{{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}} - \frac{x{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t} + \frac{{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t} + \frac{7x{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t^{2}} - \frac{3x^{2}{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t^{2}} - \frac{4{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t^{2}} - \frac{22x{e}^{(2t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{3}t^{2}} - \frac{25x{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t^{3}} + \frac{16x^{2}{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t^{3}} + \frac{11x{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{2}} + \frac{6x^{2}{e}^{(2t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{3}t^{2}} - \frac{3x{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t} - \frac{3x^{3}{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t^{3}} - \frac{3x^{2}{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{2}} + \frac{16{e}^{(2t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{3}t^{2}} + \frac{12{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t^{3}} + \frac{3{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t} - \frac{8{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{2}} - \frac{19x{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{3}} + \frac{8x^{2}{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{3}} - \frac{x^{3}{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{3}} - \frac{10x^{3}{t}^{(x - 1)}}{({e}^{t} - 1)t^{4}} - \frac{50x{t}^{(x - 1)}}{({e}^{t} - 1)t^{4}} + \frac{35x^{2}{t}^{(x - 1)}}{({e}^{t} - 1)t^{4}} + \frac{x^{4}{t}^{(x - 1)}}{({e}^{t} - 1)t^{4}} + \frac{12{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{3}} + \frac{24{t}^{(x - 1)}}{({e}^{t} - 1)t^{4}}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{{t}^{(x - 1)}}{({e}^{t} - 1)}\ with\ respect\ to\ t：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{{t}^{(x - 1)}}{({e}^{t} - 1)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{{t}^{(x - 1)}}{({e}^{t} - 1)}\right)}{dt}\\=&(\frac{-(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{2}}){t}^{(x - 1)} + \frac{({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)}\\=&\frac{-{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}} + \frac{x{t}^{(x - 1)}}{({e}^{t} - 1)t} - \frac{{t}^{(x - 1)}}{({e}^{t} - 1)t}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}} + \frac{x{t}^{(x - 1)}}{({e}^{t} - 1)t} - \frac{{t}^{(x - 1)}}{({e}^{t} - 1)t}\right)}{dt}\\=&-(\frac{-2(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{3}}){e}^{t}{t}^{(x - 1)} - \frac{({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})){t}^{(x - 1)}}{({e}^{t} - 1)^{2}} - \frac{{e}^{t}({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)^{2}} + \frac{(\frac{-(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{2}})x{t}^{(x - 1)}}{t} + \frac{x*-{t}^{(x - 1)}}{({e}^{t} - 1)t^{2}} + \frac{x({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)t} - \frac{(\frac{-(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{2}}){t}^{(x - 1)}}{t} - \frac{-{t}^{(x - 1)}}{({e}^{t} - 1)t^{2}} - \frac{({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)t}\\=&\frac{2{e}^{(2t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{3}} - \frac{{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}} - \frac{x{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t} + \frac{{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t} - \frac{x{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t} - \frac{3x{t}^{(x - 1)}}{({e}^{t} - 1)t^{2}} + \frac{x^{2}{t}^{(x - 1)}}{({e}^{t} - 1)t^{2}} + \frac{{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t} + \frac{2{t}^{(x - 1)}}{({e}^{t} - 1)t^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{2{e}^{(2t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{3}} - \frac{{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}} - \frac{x{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t} + \frac{{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t} - \frac{x{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t} - \frac{3x{t}^{(x - 1)}}{({e}^{t} - 1)t^{2}} + \frac{x^{2}{t}^{(x - 1)}}{({e}^{t} - 1)t^{2}} + \frac{{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t} + \frac{2{t}^{(x - 1)}}{({e}^{t} - 1)t^{2}}\right)}{dt}\\=&2(\frac{-3(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{4}}){e}^{(2t)}{t}^{(x - 1)} + \frac{2({e}^{(2t)}((2)ln(e) + \frac{(2t)(0)}{(e)})){t}^{(x - 1)}}{({e}^{t} - 1)^{3}} + \frac{2{e}^{(2t)}({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)^{3}} - (\frac{-2(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{3}}){e}^{t}{t}^{(x - 1)} - \frac{({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})){t}^{(x - 1)}}{({e}^{t} - 1)^{2}} - \frac{{e}^{t}({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)^{2}} - \frac{(\frac{-2(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{3}})x{t}^{(x - 1)}{e}^{t}}{t} - \frac{x*-{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t^{2}} - \frac{x({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)})){e}^{t}}{({e}^{t} - 1)^{2}t} - \frac{x{t}^{(x - 1)}({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)}))}{({e}^{t} - 1)^{2}t} + \frac{(\frac{-2(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{3}}){t}^{(x - 1)}{e}^{t}}{t} + \frac{-{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t^{2}} + \frac{({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)})){e}^{t}}{({e}^{t} - 1)^{2}t} + \frac{{t}^{(x - 1)}({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)}))}{({e}^{t} - 1)^{2}t} - \frac{(\frac{-2(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{3}})x{e}^{t}{t}^{(x - 1)}}{t} - \frac{x*-{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{2}} - \frac{x({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})){t}^{(x - 1)}}{({e}^{t} - 1)^{2}t} - \frac{x{e}^{t}({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)^{2}t} - \frac{3(\frac{-(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{2}})x{t}^{(x - 1)}}{t^{2}} - \frac{3x*-2{t}^{(x - 1)}}{({e}^{t} - 1)t^{3}} - \frac{3x({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)t^{2}} + \frac{(\frac{-(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{2}})x^{2}{t}^{(x - 1)}}{t^{2}} + \frac{x^{2}*-2{t}^{(x - 1)}}{({e}^{t} - 1)t^{3}} + \frac{x^{2}({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)t^{2}} + \frac{(\frac{-2(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{3}}){e}^{t}{t}^{(x - 1)}}{t} + \frac{-{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{2}} + \frac{({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})){t}^{(x - 1)}}{({e}^{t} - 1)^{2}t} + \frac{{e}^{t}({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)^{2}t} + \frac{2(\frac{-(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{2}}){t}^{(x - 1)}}{t^{2}} + \frac{2*-2{t}^{(x - 1)}}{({e}^{t} - 1)t^{3}} + \frac{2({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)t^{2}}\\=&\frac{-6{e}^{(3t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{4}} + \frac{6{e}^{(2t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{3}} + \frac{2x{t}^{(x - 1)}{e}^{(2t)}}{({e}^{t} - 1)^{3}t} - \frac{2{t}^{(x - 1)}{e}^{(2t)}}{({e}^{t} - 1)^{3}t} - \frac{{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}} - \frac{x{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t} + \frac{{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t} + \frac{4x{e}^{(2t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{3}t} + \frac{5x{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t^{2}} - \frac{2x^{2}{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t^{2}} - \frac{2x{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t} - \frac{4{e}^{(2t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{3}t} - \frac{3{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t^{2}} + \frac{2{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t} + \frac{4x{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{2}} - \frac{x^{2}{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{2}} - \frac{6x^{2}{t}^{(x - 1)}}{({e}^{t} - 1)t^{3}} + \frac{11x{t}^{(x - 1)}}{({e}^{t} - 1)t^{3}} + \frac{x^{3}{t}^{(x - 1)}}{({e}^{t} - 1)t^{3}} - \frac{3{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{2}} - \frac{6{t}^{(x - 1)}}{({e}^{t} - 1)t^{3}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-6{e}^{(3t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{4}} + \frac{6{e}^{(2t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{3}} + \frac{2x{t}^{(x - 1)}{e}^{(2t)}}{({e}^{t} - 1)^{3}t} - \frac{2{t}^{(x - 1)}{e}^{(2t)}}{({e}^{t} - 1)^{3}t} - \frac{{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}} - \frac{x{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t} + \frac{{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t} + \frac{4x{e}^{(2t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{3}t} + \frac{5x{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t^{2}} - \frac{2x^{2}{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t^{2}} - \frac{2x{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t} - \frac{4{e}^{(2t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{3}t} - \frac{3{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t^{2}} + \frac{2{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t} + \frac{4x{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{2}} - \frac{x^{2}{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{2}} - \frac{6x^{2}{t}^{(x - 1)}}{({e}^{t} - 1)t^{3}} + \frac{11x{t}^{(x - 1)}}{({e}^{t} - 1)t^{3}} + \frac{x^{3}{t}^{(x - 1)}}{({e}^{t} - 1)t^{3}} - \frac{3{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{2}} - \frac{6{t}^{(x - 1)}}{({e}^{t} - 1)t^{3}}\right)}{dt}\\=&-6(\frac{-4(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{5}}){e}^{(3t)}{t}^{(x - 1)} - \frac{6({e}^{(3t)}((3)ln(e) + \frac{(3t)(0)}{(e)})){t}^{(x - 1)}}{({e}^{t} - 1)^{4}} - \frac{6{e}^{(3t)}({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)^{4}} + 6(\frac{-3(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{4}}){e}^{(2t)}{t}^{(x - 1)} + \frac{6({e}^{(2t)}((2)ln(e) + \frac{(2t)(0)}{(e)})){t}^{(x - 1)}}{({e}^{t} - 1)^{3}} + \frac{6{e}^{(2t)}({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)^{3}} + \frac{2(\frac{-3(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{4}})x{t}^{(x - 1)}{e}^{(2t)}}{t} + \frac{2x*-{t}^{(x - 1)}{e}^{(2t)}}{({e}^{t} - 1)^{3}t^{2}} + \frac{2x({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)})){e}^{(2t)}}{({e}^{t} - 1)^{3}t} + \frac{2x{t}^{(x - 1)}({e}^{(2t)}((2)ln(e) + \frac{(2t)(0)}{(e)}))}{({e}^{t} - 1)^{3}t} - \frac{2(\frac{-3(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{4}}){t}^{(x - 1)}{e}^{(2t)}}{t} - \frac{2*-{t}^{(x - 1)}{e}^{(2t)}}{({e}^{t} - 1)^{3}t^{2}} - \frac{2({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)})){e}^{(2t)}}{({e}^{t} - 1)^{3}t} - \frac{2{t}^{(x - 1)}({e}^{(2t)}((2)ln(e) + \frac{(2t)(0)}{(e)}))}{({e}^{t} - 1)^{3}t} - (\frac{-2(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{3}}){e}^{t}{t}^{(x - 1)} - \frac{({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})){t}^{(x - 1)}}{({e}^{t} - 1)^{2}} - \frac{{e}^{t}({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)^{2}} - \frac{(\frac{-2(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{3}})x{t}^{(x - 1)}{e}^{t}}{t} - \frac{x*-{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t^{2}} - \frac{x({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)})){e}^{t}}{({e}^{t} - 1)^{2}t} - \frac{x{t}^{(x - 1)}({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)}))}{({e}^{t} - 1)^{2}t} + \frac{(\frac{-2(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{3}}){t}^{(x - 1)}{e}^{t}}{t} + \frac{-{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t^{2}} + \frac{({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)})){e}^{t}}{({e}^{t} - 1)^{2}t} + \frac{{t}^{(x - 1)}({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)}))}{({e}^{t} - 1)^{2}t} + \frac{4(\frac{-3(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{4}})x{e}^{(2t)}{t}^{(x - 1)}}{t} + \frac{4x*-{e}^{(2t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{3}t^{2}} + \frac{4x({e}^{(2t)}((2)ln(e) + \frac{(2t)(0)}{(e)})){t}^{(x - 1)}}{({e}^{t} - 1)^{3}t} + \frac{4x{e}^{(2t)}({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)^{3}t} + \frac{5(\frac{-2(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{3}})x{t}^{(x - 1)}{e}^{t}}{t^{2}} + \frac{5x*-2{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t^{3}} + \frac{5x({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)})){e}^{t}}{({e}^{t} - 1)^{2}t^{2}} + \frac{5x{t}^{(x - 1)}({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)}))}{({e}^{t} - 1)^{2}t^{2}} - \frac{2(\frac{-2(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{3}})x^{2}{t}^{(x - 1)}{e}^{t}}{t^{2}} - \frac{2x^{2}*-2{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t^{3}} - \frac{2x^{2}({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)})){e}^{t}}{({e}^{t} - 1)^{2}t^{2}} - \frac{2x^{2}{t}^{(x - 1)}({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)}))}{({e}^{t} - 1)^{2}t^{2}} - \frac{2(\frac{-2(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{3}})x{e}^{t}{t}^{(x - 1)}}{t} - \frac{2x*-{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{2}} - \frac{2x({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})){t}^{(x - 1)}}{({e}^{t} - 1)^{2}t} - \frac{2x{e}^{t}({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)^{2}t} - \frac{4(\frac{-3(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{4}}){e}^{(2t)}{t}^{(x - 1)}}{t} - \frac{4*-{e}^{(2t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{3}t^{2}} - \frac{4({e}^{(2t)}((2)ln(e) + \frac{(2t)(0)}{(e)})){t}^{(x - 1)}}{({e}^{t} - 1)^{3}t} - \frac{4{e}^{(2t)}({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)^{3}t} - \frac{3(\frac{-2(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{3}}){t}^{(x - 1)}{e}^{t}}{t^{2}} - \frac{3*-2{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t^{3}} - \frac{3({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)})){e}^{t}}{({e}^{t} - 1)^{2}t^{2}} - \frac{3{t}^{(x - 1)}({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)}))}{({e}^{t} - 1)^{2}t^{2}} + \frac{2(\frac{-2(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{3}}){e}^{t}{t}^{(x - 1)}}{t} + \frac{2*-{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{2}} + \frac{2({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})){t}^{(x - 1)}}{({e}^{t} - 1)^{2}t} + \frac{2{e}^{t}({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)^{2}t} + \frac{4(\frac{-2(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{3}})x{e}^{t}{t}^{(x - 1)}}{t^{2}} + \frac{4x*-2{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{3}} + \frac{4x({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})){t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{2}} + \frac{4x{e}^{t}({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)^{2}t^{2}} - \frac{(\frac{-2(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{3}})x^{2}{e}^{t}{t}^{(x - 1)}}{t^{2}} - \frac{x^{2}*-2{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{3}} - \frac{x^{2}({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})){t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{2}} - \frac{x^{2}{e}^{t}({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)^{2}t^{2}} - \frac{6(\frac{-(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{2}})x^{2}{t}^{(x - 1)}}{t^{3}} - \frac{6x^{2}*-3{t}^{(x - 1)}}{({e}^{t} - 1)t^{4}} - \frac{6x^{2}({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)t^{3}} + \frac{11(\frac{-(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{2}})x{t}^{(x - 1)}}{t^{3}} + \frac{11x*-3{t}^{(x - 1)}}{({e}^{t} - 1)t^{4}} + \frac{11x({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)t^{3}} + \frac{(\frac{-(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{2}})x^{3}{t}^{(x - 1)}}{t^{3}} + \frac{x^{3}*-3{t}^{(x - 1)}}{({e}^{t} - 1)t^{4}} + \frac{x^{3}({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)t^{3}} - \frac{3(\frac{-2(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{3}}){e}^{t}{t}^{(x - 1)}}{t^{2}} - \frac{3*-2{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{3}} - \frac{3({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})){t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{2}} - \frac{3{e}^{t}({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)^{2}t^{2}} - \frac{6(\frac{-(({e}^{t}((1)ln(e) + \frac{(t)(0)}{(e)})) + 0)}{({e}^{t} - 1)^{2}}){t}^{(x - 1)}}{t^{3}} - \frac{6*-3{t}^{(x - 1)}}{({e}^{t} - 1)t^{4}} - \frac{6({t}^{(x - 1)}((0 + 0)ln(t) + \frac{(x - 1)(1)}{(t)}))}{({e}^{t} - 1)t^{3}}\\=&\frac{24{e}^{(4t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{5}} - \frac{36{e}^{(3t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{4}} - \frac{6x{t}^{(x - 1)}{e}^{(3t)}}{({e}^{t} - 1)^{4}t} + \frac{6{t}^{(x - 1)}{e}^{(3t)}}{({e}^{t} - 1)^{4}t} + \frac{14{e}^{(2t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{3}} + \frac{6x{t}^{(x - 1)}{e}^{(2t)}}{({e}^{t} - 1)^{3}t} - \frac{6{t}^{(x - 1)}{e}^{(2t)}}{({e}^{t} - 1)^{3}t} - \frac{18x{e}^{(3t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{4}t} - \frac{14x{t}^{(x - 1)}{e}^{(2t)}}{({e}^{t} - 1)^{3}t^{2}} + \frac{6x^{2}{t}^{(x - 1)}{e}^{(2t)}}{({e}^{t} - 1)^{3}t^{2}} + \frac{18x{e}^{(2t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{3}t} + \frac{18{e}^{(3t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{4}t} + \frac{8{t}^{(x - 1)}{e}^{(2t)}}{({e}^{t} - 1)^{3}t^{2}} - \frac{18{e}^{(2t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{3}t} - \frac{{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}} - \frac{x{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t} + \frac{{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t} + \frac{7x{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t^{2}} - \frac{3x^{2}{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t^{2}} - \frac{4{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t^{2}} - \frac{22x{e}^{(2t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{3}t^{2}} - \frac{25x{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t^{3}} + \frac{16x^{2}{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t^{3}} + \frac{11x{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{2}} + \frac{6x^{2}{e}^{(2t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{3}t^{2}} - \frac{3x{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t} - \frac{3x^{3}{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t^{3}} - \frac{3x^{2}{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{2}} + \frac{16{e}^{(2t)}{t}^{(x - 1)}}{({e}^{t} - 1)^{3}t^{2}} + \frac{12{t}^{(x - 1)}{e}^{t}}{({e}^{t} - 1)^{2}t^{3}} + \frac{3{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t} - \frac{8{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{2}} - \frac{19x{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{3}} + \frac{8x^{2}{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{3}} - \frac{x^{3}{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{3}} - \frac{10x^{3}{t}^{(x - 1)}}{({e}^{t} - 1)t^{4}} - \frac{50x{t}^{(x - 1)}}{({e}^{t} - 1)t^{4}} + \frac{35x^{2}{t}^{(x - 1)}}{({e}^{t} - 1)t^{4}} + \frac{x^{4}{t}^{(x - 1)}}{({e}^{t} - 1)t^{4}} + \frac{12{e}^{t}{t}^{(x - 1)}}{({e}^{t} - 1)^{2}t^{3}} + \frac{24{t}^{(x - 1)}}{({e}^{t} - 1)t^{4}}\\ \end{split}\end{equation} \]