The average rate of change can be modeled as a function: $f:D^2 \to \mathbb{R}$ where $D$ is the domain of the primary function in consideration. It maps two variables - the ends of the interval- to the average rate of change of that particular interval.
The derivative is $f$ restricted to the set: $\{(x,x): x \in D\}$, essentially making it a "slice".
Is this a valid interpretation?