There are 1 questions in this calculation: for each question, the 1 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ first\ derivative\ of\ function\ ln(sqrt(\frac{({e}^{3}x)}{({e}^{3}x + 1)}))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = ln(sqrt(\frac{xe^{3}}{(xe^{3} + 1)}))\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( ln(sqrt(\frac{xe^{3}}{(xe^{3} + 1)}))\right)}{dx}\\=&\frac{((\frac{-(e^{3} + x*3e^{2}*0 + 0)}{(xe^{3} + 1)^{2}})xe^{3} + \frac{e^{3}}{(xe^{3} + 1)} + \frac{x*3e^{2}*0}{(xe^{3} + 1)})*\frac{1}{2}}{(sqrt(\frac{xe^{3}}{(xe^{3} + 1)}))(\frac{xe^{3}}{(xe^{3} + 1)})^{\frac{1}{2}}}\\=&\frac{-x^{\frac{1}{2}}e^{\frac{9}{2}}}{2(xe^{3} + 1)^{\frac{3}{2}}sqrt(\frac{xe^{3}}{(xe^{3} + 1)})} + \frac{e^{\frac{3}{2}}}{2(xe^{3} + 1)^{\frac{1}{2}}x^{\frac{1}{2}}sqrt(\frac{xe^{3}}{(xe^{3} + 1)})}\\ \end{split}\end{equation} \]

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