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

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