There are 1 questions in this calculation: for each question, the 1 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ first\ derivative\ of\ function\ ln(sqrt({e}^{(4x)}{\frac{1}{e}}^{(4x)} + 1))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( ln(sqrt({e}^{(4x)}{\frac{1}{e}}^{(4x)} + 1))\right)}{dx}\\=&\frac{(({e}^{(4x)}((4)ln(e) + \frac{(4x)(0)}{(e)})){\frac{1}{e}}^{(4x)} + {e}^{(4x)}({\frac{1}{e}}^{(4x)}((4)ln(\frac{1}{e}) + \frac{(4x)(\frac{-0}{e^{2}})}{(\frac{1}{e})})) + 0)*\frac{1}{2}}{(sqrt({e}^{(4x)}{\frac{1}{e}}^{(4x)} + 1))({e}^{(4x)}{\frac{1}{e}}^{(4x)} + 1)^{\frac{1}{2}}}\\=&\frac{2{e}^{(4x)}{\frac{1}{e}}^{(4x)}}{({e}^{(4x)}{\frac{1}{e}}^{(4x)} + 1)^{\frac{1}{2}}sqrt({e}^{(4x)}{\frac{1}{e}}^{(4x)} + 1)} - \frac{2{\frac{1}{e}}^{(4x)}{e}^{(4x)}}{({e}^{(4x)}{\frac{1}{e}}^{(4x)} + 1)^{\frac{1}{2}}sqrt({e}^{(4x)}{\frac{1}{e}}^{(4x)} + 1)}\\ \end{split}\end{equation} \]

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