Exchange of $\frac{d}{dx}$ and $\lim_{n\to\infty}$ in terms of continuity of the differential operator

Question

Under certain conditions, the derivative and the limit of a real function can be exchanged.

$$\lim_{n\to\infty}\frac{df_n}{dx}=\frac{d}{dx}\lim_{n\to\infty}f_n$$

Can this be understood as the continuity of the differential operator on some metric space $\frac{d}{dx}:X\to Y$?