Answer
See the explanation.
Work Step by Step
Let $y=\cos^{-1}(x)$.
Then,
$\cos y=x$ where $0\leq y\leq \pi$.
Taking the implicit differentiation,
$-\sin y\frac{dy}{dx}=1$
$\frac{dy}{dx}=-\frac{1}{\sin y}$
Since $y\in [0,\pi]$, it must be that $\sin y$ is non-negative and $\sin y=\sqrt{1-\cos^2y}=\sqrt{1-x^2}$.
So,
$\frac{dy}{dx}=-\frac{1}{\sqrt{1-x^2}}$
Thus,
$\frac{d}{dx}(\cos^{-1}(x))=-\frac{1}{\sqrt{1-x^2}}$
