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

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