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

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