jmc

Reflecting a point across the $x$-axis multiplies its $y$-coordinate by $-1$. Therefore, $D^{\prime }=(4,-1)$. To reflect $D^{\prime }$ across the line $y=x+1$, we first translate both the line and the point down one unit so that the equation of the translated line is $y=x$ and the coordinates of the translated point are $(4,-2)$. To reflect across $y=x$, we switch the $x$-coordinate and $y$-coordinate to obtain $(-2,4)$. Translating this point one unit up, we find that $D^{\prime \prime }=\boxed{(-2,5)}$.