Determining whether the evaluation mapping $p \mapsto \frac{d}{dx}\bigg|_p$ is smooth

Question

For some reason, this quetion gives me a mental block: Suppose that we have the mapping $f(a, p) = a\cdot \frac{d}{dx}\bigg|_p$ for $a, p \in \mathbb{R}$, and we'd like to argue why this is a smooth mapping. Evidently the first partial derivative exists and is continuous for all $a\in \mathbb{R}$. But what about the partial derivative w.r.t. $p$? In this case, what would it even mean to take the partial derivative of the evaluation value?