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

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