本次共计算 1 个题目:每一题对 t 求 3 阶导数。
注意,变量是区分大小写的。\[ \begin{equation}\begin{split}【1/1】求函数\frac{{t}^{\frac{5}{2}}}{({e}^{t} - 1)} 关于 t 的 3 阶导数:\\\end{split}\end{equation} \]
\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{t^{\frac{5}{2}}}{({e}^{t} - 1)}\\&\color{blue}{函数的第 1 阶导数:}\\&\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}{函数的第 2 阶导数:} \\&\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}{函数的第 3 阶导数:} \\&\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} \]你的问题在这里没有得到解决?请到 热门难题 里面看看吧!