1. Fórmulas:
Usaremos $\frac{d}{dx}[\arcsin(u)] = \frac{u'}{\sqrt{1-u^2}}$.

2. Desarrollo:
Sea $u = x-1$, entonces $u' = 1$.
$$ \frac{dy}{dx} = \frac{1}{\sqrt{1-(x-1)^2}} $$

3. Expansión del binomio:
Desarrollamos $(x-1)^2 = x^2 - 2x + 1$:
$$ \frac{dy}{dx} = \frac{1}{\sqrt{1-(x^2 - 2x + 1)}} $$
$$ \frac{dy}{dx} = \frac{1}{\sqrt{1 - x^2 + 2x - 1}} $$
$$ \frac{dy}{dx} = \frac{1}{\sqrt{2x - x^2}} $$

Resultado:
$$ \boxed{\frac{dy}{dx} = \frac{1}{\sqrt{2x - x^2}}} $$