There are 1 questions in this calculation: for each question, the 1 derivative of y is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ first\ derivative\ of\ function\ ln({(2y + 3)}^{5}{\frac{1}{({y}^{2} + 1)}}^{\frac{1}{2}})\ with\ respect\ to\ y：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = 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}{The\ first\ derivative\ function：}\\&\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} \]

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