本次共计算 1 个题目:每一题对 y 求 1 阶导数。
注意,变量是区分大小写的。\[ \begin{equation}\begin{split}【1/1】求函数ln({(2y + 3)}^{5}{\frac{1}{({y}^{2} + 1)}}^{\frac{1}{2}}) 关于 y 的 1 阶导数:\\\end{split}\end{equation} \]
\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = ln(\frac{32y^{5}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{240y^{4}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{720y^{3}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{1080y^{2}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{810y}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{243}{(y^{2} + 1)^{\frac{1}{2}}})\\&\color{blue}{函数的第 1 阶导数:}\\&\frac{d\left( ln(\frac{32y^{5}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{240y^{4}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{720y^{3}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{1080y^{2}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{810y}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{243}{(y^{2} + 1)^{\frac{1}{2}}})\right)}{dy}\\=&\frac{(32(\frac{\frac{-1}{2}(2y + 0)}{(y^{2} + 1)^{\frac{3}{2}}})y^{5} + \frac{32*5y^{4}}{(y^{2} + 1)^{\frac{1}{2}}} + 240(\frac{\frac{-1}{2}(2y + 0)}{(y^{2} + 1)^{\frac{3}{2}}})y^{4} + \frac{240*4y^{3}}{(y^{2} + 1)^{\frac{1}{2}}} + 720(\frac{\frac{-1}{2}(2y + 0)}{(y^{2} + 1)^{\frac{3}{2}}})y^{3} + \frac{720*3y^{2}}{(y^{2} + 1)^{\frac{1}{2}}} + 1080(\frac{\frac{-1}{2}(2y + 0)}{(y^{2} + 1)^{\frac{3}{2}}})y^{2} + \frac{1080*2y}{(y^{2} + 1)^{\frac{1}{2}}} + 810(\frac{\frac{-1}{2}(2y + 0)}{(y^{2} + 1)^{\frac{3}{2}}})y + \frac{810}{(y^{2} + 1)^{\frac{1}{2}}} + 243(\frac{\frac{-1}{2}(2y + 0)}{(y^{2} + 1)^{\frac{3}{2}}}))}{(\frac{32y^{5}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{240y^{4}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{720y^{3}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{1080y^{2}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{810y}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{243}{(y^{2} + 1)^{\frac{1}{2}}})}\\=&\frac{-32y^{6}}{(\frac{32y^{5}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{240y^{4}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{720y^{3}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{1080y^{2}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{810y}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{243}{(y^{2} + 1)^{\frac{1}{2}}})(y^{2} + 1)^{\frac{3}{2}}} + \frac{160y^{4}}{(y^{2} + 1)^{\frac{1}{2}}(\frac{32y^{5}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{240y^{4}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{720y^{3}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{1080y^{2}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{810y}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{243}{(y^{2} + 1)^{\frac{1}{2}}})} - \frac{240y^{5}}{(\frac{32y^{5}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{240y^{4}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{720y^{3}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{1080y^{2}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{810y}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{243}{(y^{2} + 1)^{\frac{1}{2}}})(y^{2} + 1)^{\frac{3}{2}}} + \frac{960y^{3}}{(y^{2} + 1)^{\frac{1}{2}}(\frac{32y^{5}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{240y^{4}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{720y^{3}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{1080y^{2}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{810y}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{243}{(y^{2} + 1)^{\frac{1}{2}}})} - \frac{720y^{4}}{(\frac{32y^{5}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{240y^{4}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{720y^{3}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{1080y^{2}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{810y}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{243}{(y^{2} + 1)^{\frac{1}{2}}})(y^{2} + 1)^{\frac{3}{2}}} + \frac{2160y^{2}}{(y^{2} + 1)^{\frac{1}{2}}(\frac{32y^{5}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{240y^{4}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{720y^{3}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{1080y^{2}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{810y}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{243}{(y^{2} + 1)^{\frac{1}{2}}})} - \frac{1080y^{3}}{(\frac{32y^{5}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{240y^{4}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{720y^{3}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{1080y^{2}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{810y}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{243}{(y^{2} + 1)^{\frac{1}{2}}})(y^{2} + 1)^{\frac{3}{2}}} + \frac{2160y}{(y^{2} + 1)^{\frac{1}{2}}(\frac{32y^{5}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{240y^{4}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{720y^{3}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{1080y^{2}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{810y}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{243}{(y^{2} + 1)^{\frac{1}{2}}})} - \frac{810y^{2}}{(\frac{32y^{5}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{240y^{4}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{720y^{3}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{1080y^{2}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{810y}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{243}{(y^{2} + 1)^{\frac{1}{2}}})(y^{2} + 1)^{\frac{3}{2}}} - \frac{243y}{(\frac{32y^{5}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{240y^{4}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{720y^{3}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{1080y^{2}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{810y}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{243}{(y^{2} + 1)^{\frac{1}{2}}})(y^{2} + 1)^{\frac{3}{2}}} + \frac{810}{(\frac{32y^{5}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{240y^{4}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{720y^{3}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{1080y^{2}}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{810y}{(y^{2} + 1)^{\frac{1}{2}}} + \frac{243}{(y^{2} + 1)^{\frac{1}{2}}})(y^{2} + 1)^{\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!