Question

Solve:
$\frac{d\sqrt{a^{{\sin }^{-1}t}}}{dt}$

Answer 1

$\frac{d}{dt}\sqrt{a^{{\sin }^{-1}t}}=\frac{1}{2\sqrt{a^{{\sin }^{-1}t}}}\cdot \frac{d}{dt}\left(a^{{\sin }^{-1}t}\right)$

$=\frac{1}{2\sqrt{a^{{\sin }^{-1}t}}}\cdot a^{{\sin }^{-1}t}\cdot \frac{d}{dt}\left(si{n}^{-1}t\right)lna$

$=\frac{1\sqrt{a^{{\sin }^{-1}t}}}{2}\cdot \frac{1}{\sqrt{1-t^2}}lna$

$\frac{d}{dt}\sqrt{a^{{\sin }^{-1}t}}=\frac{\sqrt{a^{{\sin }^{-1}t}}\times lna}{2\sqrt{1-t^2}}$.