Calculus 1: Inverse and Hyperbolic Trig Derivatives

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

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

Show that for $y = \cos^{-1}(x)$ the first derivative, $\frac{dy}{dx} = \frac{1}{x^2 + 1}$

For the following function, find the first derivative

$\theta = \frac{\tan^{-1}(2r)}{\pi{r}}$

Find the derivative of the following hyperbolic function

$f(x) = \sin{(\sinh{(x)})}$