Differentiate the function y = (x−1)sqrt(x2−2x+2) and simplify the result.

The function $y = (x-1)\sqrt{x^2-2x+2}$ is a product of two functions, $u = x-1$ and $v = \sqrt{x^2-2x+2}$. We will use the product rule $\frac{d}{dx}(uv) = u'v + uv'$. For $v$, we will also need the chain rule.

Rewrite $v$ as $v = (x^2-2x+2)^{1/2}$.
Using the chain rule, $\frac{d}{dx}(f(g(x))) = f'(g(x))g'(x)$, where $f(z) = z^{1/2}$ and $g(x) = x^2-2x+2$.

So,

To combine the terms, find a common denominator.

Expand $(x-1)^2$:

Substitute this back into the numerator:

The final answer is $\frac{2x^2-4x+3}{\sqrt{x^2-2x+2}}$.