There are 1 questions in this calculation: for each question, the 1 derivative of x is calculated.
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\[ \begin{equation}\begin{split}[1/1]Find\ the\ first\ derivative\ of\ function\ ln({(2x + 3)}^{5}{\frac{1}{({x}^{2} + 1)}}^{\frac{1}{2}})\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = ln(\frac{32x^{5}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{240x^{4}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{720x^{3}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{1080x^{2}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{810x}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{243}{(x^{2} + 1)^{\frac{1}{2}}})\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( ln(\frac{32x^{5}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{240x^{4}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{720x^{3}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{1080x^{2}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{810x}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{243}{(x^{2} + 1)^{\frac{1}{2}}})\right)}{dx}\\=&\frac{(32(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{3}{2}}})x^{5} + \frac{32*5x^{4}}{(x^{2} + 1)^{\frac{1}{2}}} + 240(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{3}{2}}})x^{4} + \frac{240*4x^{3}}{(x^{2} + 1)^{\frac{1}{2}}} + 720(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{3}{2}}})x^{3} + \frac{720*3x^{2}}{(x^{2} + 1)^{\frac{1}{2}}} + 1080(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{3}{2}}})x^{2} + \frac{1080*2x}{(x^{2} + 1)^{\frac{1}{2}}} + 810(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{3}{2}}})x + \frac{810}{(x^{2} + 1)^{\frac{1}{2}}} + 243(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{3}{2}}}))}{(\frac{32x^{5}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{240x^{4}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{720x^{3}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{1080x^{2}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{810x}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{243}{(x^{2} + 1)^{\frac{1}{2}}})}\\=&\frac{-32x^{6}}{(\frac{32x^{5}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{240x^{4}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{720x^{3}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{1080x^{2}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{810x}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{243}{(x^{2} + 1)^{\frac{1}{2}}})(x^{2} + 1)^{\frac{3}{2}}} + \frac{160x^{4}}{(x^{2} + 1)^{\frac{1}{2}}(\frac{32x^{5}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{240x^{4}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{720x^{3}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{1080x^{2}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{810x}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{243}{(x^{2} + 1)^{\frac{1}{2}}})} - \frac{240x^{5}}{(\frac{32x^{5}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{240x^{4}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{720x^{3}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{1080x^{2}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{810x}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{243}{(x^{2} + 1)^{\frac{1}{2}}})(x^{2} + 1)^{\frac{3}{2}}} + \frac{960x^{3}}{(x^{2} + 1)^{\frac{1}{2}}(\frac{32x^{5}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{240x^{4}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{720x^{3}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{1080x^{2}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{810x}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{243}{(x^{2} + 1)^{\frac{1}{2}}})} - \frac{720x^{4}}{(\frac{32x^{5}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{240x^{4}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{720x^{3}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{1080x^{2}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{810x}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{243}{(x^{2} + 1)^{\frac{1}{2}}})(x^{2} + 1)^{\frac{3}{2}}} + \frac{2160x^{2}}{(x^{2} + 1)^{\frac{1}{2}}(\frac{32x^{5}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{240x^{4}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{720x^{3}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{1080x^{2}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{810x}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{243}{(x^{2} + 1)^{\frac{1}{2}}})} - \frac{1080x^{3}}{(\frac{32x^{5}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{240x^{4}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{720x^{3}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{1080x^{2}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{810x}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{243}{(x^{2} + 1)^{\frac{1}{2}}})(x^{2} + 1)^{\frac{3}{2}}} + \frac{2160x}{(x^{2} + 1)^{\frac{1}{2}}(\frac{32x^{5}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{240x^{4}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{720x^{3}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{1080x^{2}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{810x}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{243}{(x^{2} + 1)^{\frac{1}{2}}})} - \frac{810x^{2}}{(\frac{32x^{5}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{240x^{4}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{720x^{3}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{1080x^{2}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{810x}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{243}{(x^{2} + 1)^{\frac{1}{2}}})(x^{2} + 1)^{\frac{3}{2}}} - \frac{243x}{(\frac{32x^{5}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{240x^{4}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{720x^{3}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{1080x^{2}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{810x}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{243}{(x^{2} + 1)^{\frac{1}{2}}})(x^{2} + 1)^{\frac{3}{2}}} + \frac{810}{(\frac{32x^{5}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{240x^{4}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{720x^{3}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{1080x^{2}}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{810x}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{243}{(x^{2} + 1)^{\frac{1}{2}}})(x^{2} + 1)^{\frac{1}{2}}}\\ \end{split}\end{equation} \]

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