There are 1 questions in this calculation: for each question, the 1 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ first\ derivative\ of\ function\ {(\frac{({2}^{x} + {53}^{x} + {49069}^{x})}{3})}^{(\frac{3}{x})}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = (\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( (\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}\right)}{dx}\\=&((\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}((\frac{3*-1}{x^{2}})ln(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x}) + \frac{(\frac{3}{x})(\frac{1}{3}({2}^{x}((1)ln(2) + \frac{(x)(0)}{(2)})) + \frac{1}{3}({53}^{x}((1)ln(53) + \frac{(x)(0)}{(53)})) + \frac{1}{3}({49069}^{x}((1)ln(49069) + \frac{(x)(0)}{(49069)})))}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})}))\\=&\frac{-3(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})}{x^{2}} + \frac{{2}^{x}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln(2)}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})x} + \frac{{53}^{x}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln(53)}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})x} + \frac{{49069}^{x}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln(49069)}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})x}\\ \end{split}\end{equation} \]

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