There are 1 questions in this calculation: for each question, the 2 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ second\ derivative\ of\ function\ \frac{arctan(x + y)}{(1 + ({x}^{3} + {y}^{3})sin(xy))}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{arctan(x + y)}{(x^{3}sin(yx) + y^{3}sin(yx) + 1)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{arctan(x + y)}{(x^{3}sin(yx) + y^{3}sin(yx) + 1)}\right)}{dx}\\=&(\frac{-(3x^{2}sin(yx) + x^{3}cos(yx)y + y^{3}cos(yx)y + 0)}{(x^{3}sin(yx) + y^{3}sin(yx) + 1)^{2}})arctan(x + y) + \frac{(\frac{(1 + 0)}{(1 + (x + y)^{2})})}{(x^{3}sin(yx) + y^{3}sin(yx) + 1)}\\=&\frac{-3x^{2}sin(yx)arctan(x + y)}{(x^{3}sin(yx) + y^{3}sin(yx) + 1)^{2}} - \frac{yx^{3}cos(yx)arctan(x + y)}{(x^{3}sin(yx) + y^{3}sin(yx) + 1)^{2}} - \frac{y^{4}cos(yx)arctan(x + y)}{(x^{3}sin(yx) + y^{3}sin(yx) + 1)^{2}} + \frac{1}{(x^{3}sin(yx) + y^{3}sin(yx) + 1)(x^{2} + 2yx + y^{2} + 1)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-3x^{2}sin(yx)arctan(x + y)}{(x^{3}sin(yx) + y^{3}sin(yx) + 1)^{2}} - \frac{yx^{3}cos(yx)arctan(x + y)}{(x^{3}sin(yx) + y^{3}sin(yx) + 1)^{2}} - \frac{y^{4}cos(yx)arctan(x + y)}{(x^{3}sin(yx) + y^{3}sin(yx) + 1)^{2}} + \frac{1}{(x^{3}sin(yx) + y^{3}sin(yx) + 1)(x^{2} + 2yx + y^{2} + 1)}\right)}{dx}\\=&-3(\frac{-2(3x^{2}sin(yx) + x^{3}cos(yx)y + y^{3}cos(yx)y + 0)}{(x^{3}sin(yx) + y^{3}sin(yx) + 1)^{3}})x^{2}sin(yx)arctan(x + y) - \frac{3*2xsin(yx)arctan(x + y)}{(x^{3}sin(yx) + y^{3}sin(yx) + 1)^{2}} - \frac{3x^{2}cos(yx)yarctan(x + y)}{(x^{3}sin(yx) + y^{3}sin(yx) + 1)^{2}} - \frac{3x^{2}sin(yx)(\frac{(1 + 0)}{(1 + (x + y)^{2})})}{(x^{3}sin(yx) + y^{3}sin(yx) + 1)^{2}} - (\frac{-2(3x^{2}sin(yx) + x^{3}cos(yx)y + y^{3}cos(yx)y + 0)}{(x^{3}sin(yx) + y^{3}sin(yx) + 1)^{3}})yx^{3}cos(yx)arctan(x + y) - \frac{y*3x^{2}cos(yx)arctan(x + y)}{(x^{3}sin(yx) + y^{3}sin(yx) + 1)^{2}} - \frac{yx^{3}*-sin(yx)yarctan(x + y)}{(x^{3}sin(yx) + y^{3}sin(yx) + 1)^{2}} - \frac{yx^{3}cos(yx)(\frac{(1 + 0)}{(1 + (x + y)^{2})})}{(x^{3}sin(yx) + y^{3}sin(yx) + 1)^{2}} - (\frac{-2(3x^{2}sin(yx) + x^{3}cos(yx)y + y^{3}cos(yx)y + 0)}{(x^{3}sin(yx) + y^{3}sin(yx) + 1)^{3}})y^{4}cos(yx)arctan(x + y) - \frac{y^{4}*-sin(yx)yarctan(x + y)}{(x^{3}sin(yx) + y^{3}sin(yx) + 1)^{2}} - \frac{y^{4}cos(yx)(\frac{(1 + 0)}{(1 + (x + y)^{2})})}{(x^{3}sin(yx) + y^{3}sin(yx) + 1)^{2}} + \frac{(\frac{-(3x^{2}sin(yx) + x^{3}cos(yx)y + y^{3}cos(yx)y + 0)}{(x^{3}sin(yx) + y^{3}sin(yx) + 1)^{2}})}{(x^{2} + 2yx + y^{2} + 1)} + \frac{(\frac{-(2x + 2y + 0 + 0)}{(x^{2} + 2yx + y^{2} + 1)^{2}})}{(x^{3}sin(yx) + y^{3}sin(yx) + 1)}\\=&\frac{18x^{4}sin^{2}(yx)arctan(x + y)}{(x^{3}sin(yx) + y^{3}sin(yx) + 1)^{3}} + \frac{12yx^{5}sin(yx)cos(yx)arctan(x + y)}{(x^{3}sin(yx) + y^{3}sin(yx) + 1)^{3}} + \frac{12y^{4}x^{2}sin(yx)cos(yx)arctan(x + y)}{(x^{3}sin(yx) + y^{3}sin(yx) + 1)^{3}} - \frac{6xsin(yx)arctan(x + y)}{(x^{3}sin(yx) + y^{3}sin(yx) + 1)^{2}} - \frac{6yx^{2}cos(yx)arctan(x + y)}{(x^{3}sin(yx) + y^{3}sin(yx) + 1)^{2}} - \frac{6x^{2}sin(yx)}{(x^{3}sin(yx) + y^{3}sin(yx) + 1)^{2}(x^{2} + 2yx + y^{2} + 1)} + \frac{2y^{2}x^{6}cos^{2}(yx)arctan(x + y)}{(x^{3}sin(yx) + y^{3}sin(yx) + 1)^{3}} + \frac{4y^{5}x^{3}cos^{2}(yx)arctan(x + y)}{(x^{3}sin(yx) + y^{3}sin(yx) + 1)^{3}} + \frac{y^{2}x^{3}sin(yx)arctan(x + y)}{(x^{3}sin(yx) + y^{3}sin(yx) + 1)^{2}} - \frac{2yx^{3}cos(yx)}{(x^{3}sin(yx) + y^{3}sin(yx) + 1)^{2}(x^{2} + 2yx + y^{2} + 1)} + \frac{2y^{8}cos^{2}(yx)arctan(x + y)}{(x^{3}sin(yx) + y^{3}sin(yx) + 1)^{3}} + \frac{y^{5}sin(yx)arctan(x + y)}{(x^{3}sin(yx) + y^{3}sin(yx) + 1)^{2}} - \frac{2y^{4}cos(yx)}{(x^{3}sin(yx) + y^{3}sin(yx) + 1)^{2}(x^{2} + 2yx + y^{2} + 1)} - \frac{2x}{(x^{2} + 2yx + y^{2} + 1)^{2}(x^{3}sin(yx) + y^{3}sin(yx) + 1)} - \frac{2y}{(x^{2} + 2yx + y^{2} + 1)^{2}(x^{3}sin(yx) + y^{3}sin(yx) + 1)}\\ \end{split}\end{equation} \]

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