Derivative of arcsin of a composite expression

Identify the outer function as $f(u) = \arcsin(u)$ with $f'(u) = \dfrac{1}{\sqrt{1-u^2}}$, and the inner function as $g(x) = x^2 - 1$ with $g'(x) = 2x$. By the chain rule, $h'(x) = f'(g(x)) \cdot g'(x)$.

Substituting $u = x^2 - 1$ gives $h'(x) = \dfrac{1}{\sqrt{1-(x^2-1)^2}} \cdot 2x$. This simplifies to $h'(x) = \dfrac{2x}{\sqrt{1-(x^2-1)^2}}$, which can also be written as $\dfrac{2x}{\sqrt{2x^2 - x^4}}$.