There are 1 questions in this calculation: for each question, the 1 derivative of t is calculated.
Note that variables are case sensitive.\[ \begin{equation}\begin{split}[1/1]Find\ the\ first\ derivative\ of\ function\ (\frac{2{(1 - S{(Lt)}^{\frac{1}{2}})}^{2}}{({P}^{2})t} + \frac{2(1 - S{(Lt)}^{\frac{1}{2}})(1 - S{(\frac{L}{t})}^{\frac{1}{2}})}{P} + L{\frac{1}{(P)}}^{2})(\frac{P}{2} - arcsin({(\frac{1}{(Lt)})}^{\frac{1}{2}} - S))\ with\ respect\ to\ t:\\\end{split}\end{equation} \]
\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{4SL^{\frac{1}{2}}arcsin(\frac{1}{L^{\frac{1}{2}}t^{\frac{1}{2}}} - S)}{P^{2}t^{\frac{1}{2}}} - \frac{2S^{2}Larcsin(\frac{1}{L^{\frac{1}{2}}t^{\frac{1}{2}}} - S)}{P^{2}} - \frac{2arcsin(\frac{1}{L^{\frac{1}{2}}t^{\frac{1}{2}}} - S)}{P^{2}t} + \frac{2SL^{\frac{1}{2}}arcsin(\frac{1}{L^{\frac{1}{2}}t^{\frac{1}{2}}} - S)}{Pt^{\frac{1}{2}}} + \frac{2SL^{\frac{1}{2}}t^{\frac{1}{2}}arcsin(\frac{1}{L^{\frac{1}{2}}t^{\frac{1}{2}}} - S)}{P} + \frac{1}{Pt} - \frac{2arcsin(\frac{1}{L^{\frac{1}{2}}t^{\frac{1}{2}}} - S)}{P} - \frac{SL^{\frac{1}{2}}}{t^{\frac{1}{2}}} - \frac{2SL^{\frac{1}{2}}}{Pt^{\frac{1}{2}}} - SL^{\frac{1}{2}}t^{\frac{1}{2}} - \frac{2S^{2}Larcsin(\frac{1}{L^{\frac{1}{2}}t^{\frac{1}{2}}} - S)}{P} + S^{2}L + \frac{S^{2}L}{P} - \frac{Larcsin(\frac{1}{L^{\frac{1}{2}}t^{\frac{1}{2}}} - S)}{P^{2}} + \frac{\frac{1}{2}L}{P} + 1\\&\color{blue}{The\ first\ derivative\ function:}\\&\frac{d\left( \frac{4SL^{\frac{1}{2}}arcsin(\frac{1}{L^{\frac{1}{2}}t^{\frac{1}{2}}} - S)}{P^{2}t^{\frac{1}{2}}} - \frac{2S^{2}Larcsin(\frac{1}{L^{\frac{1}{2}}t^{\frac{1}{2}}} - S)}{P^{2}} - \frac{2arcsin(\frac{1}{L^{\frac{1}{2}}t^{\frac{1}{2}}} - S)}{P^{2}t} + \frac{2SL^{\frac{1}{2}}arcsin(\frac{1}{L^{\frac{1}{2}}t^{\frac{1}{2}}} - S)}{Pt^{\frac{1}{2}}} + \frac{2SL^{\frac{1}{2}}t^{\frac{1}{2}}arcsin(\frac{1}{L^{\frac{1}{2}}t^{\frac{1}{2}}} - S)}{P} + \frac{1}{Pt} - \frac{2arcsin(\frac{1}{L^{\frac{1}{2}}t^{\frac{1}{2}}} - S)}{P} - \frac{SL^{\frac{1}{2}}}{t^{\frac{1}{2}}} - \frac{2SL^{\frac{1}{2}}}{Pt^{\frac{1}{2}}} - SL^{\frac{1}{2}}t^{\frac{1}{2}} - \frac{2S^{2}Larcsin(\frac{1}{L^{\frac{1}{2}}t^{\frac{1}{2}}} - S)}{P} + S^{2}L + \frac{S^{2}L}{P} - \frac{Larcsin(\frac{1}{L^{\frac{1}{2}}t^{\frac{1}{2}}} - S)}{P^{2}} + \frac{\frac{1}{2}L}{P} + 1\right)}{dt}\\=&\frac{4SL^{\frac{1}{2}}*\frac{-1}{2}arcsin(\frac{1}{L^{\frac{1}{2}}t^{\frac{1}{2}}} - S)}{P^{2}t^{\frac{3}{2}}} + \frac{4SL^{\frac{1}{2}}(\frac{(\frac{\frac{-1}{2}}{L^{\frac{1}{2}}t^{\frac{3}{2}}} + 0)}{((1 - (\frac{1}{L^{\frac{1}{2}}t^{\frac{1}{2}}} - S)^{2})^{\frac{1}{2}})})}{P^{2}t^{\frac{1}{2}}} - \frac{2S^{2}L(\frac{(\frac{\frac{-1}{2}}{L^{\frac{1}{2}}t^{\frac{3}{2}}} + 0)}{((1 - (\frac{1}{L^{\frac{1}{2}}t^{\frac{1}{2}}} - S)^{2})^{\frac{1}{2}})})}{P^{2}} - \frac{2*-arcsin(\frac{1}{L^{\frac{1}{2}}t^{\frac{1}{2}}} - S)}{P^{2}t^{2}} - \frac{2(\frac{(\frac{\frac{-1}{2}}{L^{\frac{1}{2}}t^{\frac{3}{2}}} + 0)}{((1 - (\frac{1}{L^{\frac{1}{2}}t^{\frac{1}{2}}} - S)^{2})^{\frac{1}{2}})})}{P^{2}t} + \frac{2SL^{\frac{1}{2}}*\frac{-1}{2}arcsin(\frac{1}{L^{\frac{1}{2}}t^{\frac{1}{2}}} - S)}{Pt^{\frac{3}{2}}} + \frac{2SL^{\frac{1}{2}}(\frac{(\frac{\frac{-1}{2}}{L^{\frac{1}{2}}t^{\frac{3}{2}}} + 0)}{((1 - (\frac{1}{L^{\frac{1}{2}}t^{\frac{1}{2}}} - S)^{2})^{\frac{1}{2}})})}{Pt^{\frac{1}{2}}} + \frac{2SL^{\frac{1}{2}}*\frac{1}{2}arcsin(\frac{1}{L^{\frac{1}{2}}t^{\frac{1}{2}}} - S)}{Pt^{\frac{1}{2}}} + \frac{2SL^{\frac{1}{2}}t^{\frac{1}{2}}(\frac{(\frac{\frac{-1}{2}}{L^{\frac{1}{2}}t^{\frac{3}{2}}} + 0)}{((1 - (\frac{1}{L^{\frac{1}{2}}t^{\frac{1}{2}}} - S)^{2})^{\frac{1}{2}})})}{P} + \frac{-1}{Pt^{2}} - \frac{2(\frac{(\frac{\frac{-1}{2}}{L^{\frac{1}{2}}t^{\frac{3}{2}}} + 0)}{((1 - (\frac{1}{L^{\frac{1}{2}}t^{\frac{1}{2}}} - S)^{2})^{\frac{1}{2}})})}{P} - \frac{SL^{\frac{1}{2}}*\frac{-1}{2}}{t^{\frac{3}{2}}} - \frac{2SL^{\frac{1}{2}}*\frac{-1}{2}}{Pt^{\frac{3}{2}}} - \frac{SL^{\frac{1}{2}}*\frac{1}{2}}{t^{\frac{1}{2}}} - \frac{2S^{2}L(\frac{(\frac{\frac{-1}{2}}{L^{\frac{1}{2}}t^{\frac{3}{2}}} + 0)}{((1 - (\frac{1}{L^{\frac{1}{2}}t^{\frac{1}{2}}} - S)^{2})^{\frac{1}{2}})})}{P} + 0 + 0 - \frac{L(\frac{(\frac{\frac{-1}{2}}{L^{\frac{1}{2}}t^{\frac{3}{2}}} + 0)}{((1 - (\frac{1}{L^{\frac{1}{2}}t^{\frac{1}{2}}} - S)^{2})^{\frac{1}{2}})})}{P^{2}} + 0 + 0\\=& - \frac{2SL^{\frac{1}{2}}arcsin(\frac{1}{L^{\frac{1}{2}}t^{\frac{1}{2}}} - S)}{P^{2}t^{\frac{3}{2}}} - \frac{2S}{(\frac{-1}{Lt} + \frac{2S}{L^{\frac{1}{2}}t^{\frac{1}{2}}} - S^{2} + 1)^{\frac{1}{2}}P^{2}t^{2}} - \frac{S}{(\frac{-1}{Lt} + \frac{2S}{L^{\frac{1}{2}}t^{\frac{1}{2}}} - S^{2} + 1)^{\frac{1}{2}}Pt^{2}} + \frac{2arcsin(\frac{1}{L^{\frac{1}{2}}t^{\frac{1}{2}}} - S)}{P^{2}t^{2}} + \frac{1}{(\frac{-1}{Lt} + \frac{2S}{L^{\frac{1}{2}}t^{\frac{1}{2}}} - S^{2} + 1)^{\frac{1}{2}}P^{2}L^{\frac{1}{2}}t^{\frac{5}{2}}} - \frac{SL^{\frac{1}{2}}arcsin(\frac{1}{L^{\frac{1}{2}}t^{\frac{1}{2}}} - S)}{Pt^{\frac{3}{2}}} - \frac{S}{(\frac{-1}{Lt} + \frac{2S}{L^{\frac{1}{2}}t^{\frac{1}{2}}} - S^{2} + 1)^{\frac{1}{2}}Pt} + \frac{SL^{\frac{1}{2}}arcsin(\frac{1}{L^{\frac{1}{2}}t^{\frac{1}{2}}} - S)}{Pt^{\frac{1}{2}}} + \frac{S^{2}L^{\frac{1}{2}}}{(\frac{-1}{Lt} + \frac{2S}{L^{\frac{1}{2}}t^{\frac{1}{2}}} - S^{2} + 1)^{\frac{1}{2}}P^{2}t^{\frac{3}{2}}} - \frac{1}{Pt^{2}} + \frac{1}{(\frac{-1}{Lt} + \frac{2S}{L^{\frac{1}{2}}t^{\frac{1}{2}}} - S^{2} + 1)^{\frac{1}{2}}PL^{\frac{1}{2}}t^{\frac{3}{2}}} + \frac{SL^{\frac{1}{2}}}{2t^{\frac{3}{2}}} + \frac{SL^{\frac{1}{2}}}{Pt^{\frac{3}{2}}} - \frac{SL^{\frac{1}{2}}}{2t^{\frac{1}{2}}} + \frac{S^{2}L^{\frac{1}{2}}}{(\frac{-1}{Lt} + \frac{2S}{L^{\frac{1}{2}}t^{\frac{1}{2}}} - S^{2} + 1)^{\frac{1}{2}}Pt^{\frac{3}{2}}} + \frac{L^{\frac{1}{2}}}{2(\frac{-1}{Lt} + \frac{2S}{L^{\frac{1}{2}}t^{\frac{1}{2}}} - S^{2} + 1)^{\frac{1}{2}}P^{2}t^{\frac{3}{2}}}\\ \end{split}\end{equation} \]Your problem has not been solved here? 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