Mathematics

The key is recognizing that $p'(x)$ exists only where $p$ is differentiable, which requires continuity and a well-defined derivative from both sides. First, simplify the left piece—it factors nicely and isn't actually the discontinuous mess it appears. Then check if $p$ is continuous at $x=3$ by comparing the limit from each side with $p(3)$. Finally, compute one-sided derivatives at $x=3$ to see if they match; if they don't, that point is excluded from the domain of $p'$.

The key is recognizing that $p'(x)$ exists only where $p$ is differentiable, which requires continuity and a well-defined derivative from both sides. First, simplify the left piece—it factors nicely and isn't actually the discontinuous mess it appears. Then check if $p$ is continuous at $x=3$ by comparing the limit from each side with $p(3)$. Finally, compute one-sided derivatives at $x=3$ to see if they match; if they don't, that point is excluded from the domain of $p'$.