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

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