There are 1 questions in this calculation: for each question, the 3 derivative of y is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ third\ derivative\ of\ function\ \frac{(ln(x + y + 1))}{(1 + {y}^{2})}\ with\ respect\ to\ y：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{ln(x + y + 1)}{(y^{2} + 1)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{ln(x + y + 1)}{(y^{2} + 1)}\right)}{dy}\\=&(\frac{-(2y + 0)}{(y^{2} + 1)^{2}})ln(x + y + 1) + \frac{(0 + 1 + 0)}{(y^{2} + 1)(x + y + 1)}\\=&\frac{-2yln(x + y + 1)}{(y^{2} + 1)^{2}} + \frac{1}{(x + y + 1)(y^{2} + 1)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2yln(x + y + 1)}{(y^{2} + 1)^{2}} + \frac{1}{(x + y + 1)(y^{2} + 1)}\right)}{dy}\\=&-2(\frac{-2(2y + 0)}{(y^{2} + 1)^{3}})yln(x + y + 1) - \frac{2ln(x + y + 1)}{(y^{2} + 1)^{2}} - \frac{2y(0 + 1 + 0)}{(y^{2} + 1)^{2}(x + y + 1)} + \frac{(\frac{-(0 + 1 + 0)}{(x + y + 1)^{2}})}{(y^{2} + 1)} + \frac{(\frac{-(2y + 0)}{(y^{2} + 1)^{2}})}{(x + y + 1)}\\=&\frac{8y^{2}ln(x + y + 1)}{(y^{2} + 1)^{3}} - \frac{2ln(x + y + 1)}{(y^{2} + 1)^{2}} - \frac{2y}{(x + y + 1)(y^{2} + 1)^{2}} - \frac{2y}{(y^{2} + 1)^{2}(x + y + 1)} - \frac{1}{(x + y + 1)^{2}(y^{2} + 1)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{8y^{2}ln(x + y + 1)}{(y^{2} + 1)^{3}} - \frac{2ln(x + y + 1)}{(y^{2} + 1)^{2}} - \frac{2y}{(x + y + 1)(y^{2} + 1)^{2}} - \frac{2y}{(y^{2} + 1)^{2}(x + y + 1)} - \frac{1}{(x + y + 1)^{2}(y^{2} + 1)}\right)}{dy}\\=&8(\frac{-3(2y + 0)}{(y^{2} + 1)^{4}})y^{2}ln(x + y + 1) + \frac{8*2yln(x + y + 1)}{(y^{2} + 1)^{3}} + \frac{8y^{2}(0 + 1 + 0)}{(y^{2} + 1)^{3}(x + y + 1)} - 2(\frac{-2(2y + 0)}{(y^{2} + 1)^{3}})ln(x + y + 1) - \frac{2(0 + 1 + 0)}{(y^{2} + 1)^{2}(x + y + 1)} - \frac{2(\frac{-(0 + 1 + 0)}{(x + y + 1)^{2}})y}{(y^{2} + 1)^{2}} - \frac{2(\frac{-2(2y + 0)}{(y^{2} + 1)^{3}})y}{(x + y + 1)} - \frac{2}{(x + y + 1)(y^{2} + 1)^{2}} - \frac{2(\frac{-2(2y + 0)}{(y^{2} + 1)^{3}})y}{(x + y + 1)} - \frac{2(\frac{-(0 + 1 + 0)}{(x + y + 1)^{2}})y}{(y^{2} + 1)^{2}} - \frac{2}{(y^{2} + 1)^{2}(x + y + 1)} - \frac{(\frac{-2(0 + 1 + 0)}{(x + y + 1)^{3}})}{(y^{2} + 1)} - \frac{(\frac{-(2y + 0)}{(y^{2} + 1)^{2}})}{(x + y + 1)^{2}}\\=&\frac{-48y^{3}ln(x + y + 1)}{(y^{2} + 1)^{4}} + \frac{24yln(x + y + 1)}{(y^{2} + 1)^{3}} + \frac{8y^{2}}{(x + y + 1)(y^{2} + 1)^{3}} + \frac{16y^{2}}{(y^{2} + 1)^{3}(x + y + 1)} + \frac{4y}{(x + y + 1)^{2}(y^{2} + 1)^{2}} + \frac{2y}{(y^{2} + 1)^{2}(x + y + 1)^{2}} - \frac{4}{(x + y + 1)(y^{2} + 1)^{2}} + \frac{2}{(x + y + 1)^{3}(y^{2} + 1)} - \frac{2}{(y^{2} + 1)^{2}(x + y + 1)}\\ \end{split}\end{equation} \]

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