1. The problem asks to find the first derivative $y'$ of the function $y = x(\ln x - 1)$.\n\n2. Use the product rule for differentiation: if $y = u v$, then $y' = u' v + u v'$. Here, $u = x$ and $v = \ln x - 1$.\n\n3. Compute derivatives: $u' = 1$, $v' = \frac{1}{x}$.\n\n4. Apply the product rule:\n$$y' = 1 \cdot (\ln x - 1) + x \cdot \frac{1}{x} = \ln x - 1 + 1 = \ln x.$$\n\n5. Therefore, the first derivative is $y' = \ln x$.