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

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