Hyperbolic Trig Functions

Find the derivative, $y^{\prime}$ of the following implicit function

$y = \tan^{-1}(xy)$

Show that $\frac{d}{dx}(\arcsin{x}) = \frac{x^{\prime}}{\sqrt{1 - x^2}}$

Determine the derivative of the following inverse trig function

$f(x) = \arctan{(\sqrt{x})}$

Determine the derivative of the inverse trigonometric function

$f(x) = \sec^{-1}{(5x)}$