There are 1 questions in this calculation: for each question, the 1 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ first\ derivative\ of\ function\ sqrt(\frac{((1 + 2x)(3 + 4x))}{((5 + 6x)(7 + 8x))})\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = sqrt(\frac{8x^{2}}{(48x^{2} + 82x + 35)} + \frac{10x}{(48x^{2} + 82x + 35)} + \frac{3}{(48x^{2} + 82x + 35)})\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( sqrt(\frac{8x^{2}}{(48x^{2} + 82x + 35)} + \frac{10x}{(48x^{2} + 82x + 35)} + \frac{3}{(48x^{2} + 82x + 35)})\right)}{dx}\\=&\frac{(8(\frac{-(48*2x + 82 + 0)}{(48x^{2} + 82x + 35)^{2}})x^{2} + \frac{8*2x}{(48x^{2} + 82x + 35)} + 10(\frac{-(48*2x + 82 + 0)}{(48x^{2} + 82x + 35)^{2}})x + \frac{10}{(48x^{2} + 82x + 35)} + 3(\frac{-(48*2x + 82 + 0)}{(48x^{2} + 82x + 35)^{2}}))*\frac{1}{2}}{(\frac{8x^{2}}{(48x^{2} + 82x + 35)} + \frac{10x}{(48x^{2} + 82x + 35)} + \frac{3}{(48x^{2} + 82x + 35)})^{\frac{1}{2}}}\\=&\frac{-384x^{3}}{(48x^{2} + 82x + 35)^{2}(\frac{8x^{2}}{(48x^{2} + 82x + 35)} + \frac{10x}{(48x^{2} + 82x + 35)} + \frac{3}{(48x^{2} + 82x + 35)})^{\frac{1}{2}}} - \frac{808x^{2}}{(48x^{2} + 82x + 35)^{2}(\frac{8x^{2}}{(48x^{2} + 82x + 35)} + \frac{10x}{(48x^{2} + 82x + 35)} + \frac{3}{(48x^{2} + 82x + 35)})^{\frac{1}{2}}} + \frac{8x}{(48x^{2} + 82x + 35)(\frac{8x^{2}}{(48x^{2} + 82x + 35)} + \frac{10x}{(48x^{2} + 82x + 35)} + \frac{3}{(48x^{2} + 82x + 35)})^{\frac{1}{2}}} - \frac{554x}{(48x^{2} + 82x + 35)^{2}(\frac{8x^{2}}{(48x^{2} + 82x + 35)} + \frac{10x}{(48x^{2} + 82x + 35)} + \frac{3}{(48x^{2} + 82x + 35)})^{\frac{1}{2}}} + \frac{5}{(48x^{2} + 82x + 35)(\frac{8x^{2}}{(48x^{2} + 82x + 35)} + \frac{10x}{(48x^{2} + 82x + 35)} + \frac{3}{(48x^{2} + 82x + 35)})^{\frac{1}{2}}} - \frac{123}{(48x^{2} + 82x + 35)^{2}(\frac{8x^{2}}{(48x^{2} + 82x + 35)} + \frac{10x}{(48x^{2} + 82x + 35)} + \frac{3}{(48x^{2} + 82x + 35)})^{\frac{1}{2}}}\\ \end{split}\end{equation} \]

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