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

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