Derivatives of Inverse Trigs via Implicit Differentiation

We can use implicit differentiation to find derivatives of inverse
functions. Recall that the equation $$y = f^{-1}(x)$$ means the same things as
$$x = f(y).$$
Taking derivatives of both sides gives
$$1 = f'(y) \frac{dy}{dx}.$$
Dividing both sides by $f'(y)$ (and swapping sides) gives
$$\frac{dy}{dx} = \frac{1}{f'(y)}.$$
Once we rewrite $f'(y)$ in terms of
$x$, we have the derivative of $f^{-1}(x)$.

In the following video, we use this trick to differentiate
the inverse trig functions $\sin^{-1}$, $\cos^{-1}$ and $\tan^{-1}$.