Endpoint derivative in a negative-drift boundary layer ($y^{\prime }(1)=-2/ϵ+4+O(ϵ)$)

For $ϵy^{\prime \prime }-(1+x)y^{\prime }+y=0$, $y(0)=1$, $y(1)=1+ϵ$, the exact outer solution is $1+x$ and the mismatch is at the right endpoint. In $X=(1-x)/ϵ$, matching gives $Y_0=2-e^{-2X}$ and $Y_1=-X+(1-X-X^2/2)e^{-2X}$. Differentiation at $X=0$ gives the displayed endpoint slope. The order-one term requires the first inner correction; differentiating only the leading boundary layer loses it.