The typical example of a rate of change is one that changes with respect to time. I would strongly suggest introducing at least one example where the independent variable is not a quantity of time.
One relatively-easy-to-visualize example is to find the rate of change of a shape's area with respect to one of its lengths. For example, a square's area $A$ is the square of the length of its side $x$, so $\frac{\mathrm{d}A}{\mathrm{d}x}=\frac{\mathrm{d}}{\mathrm{d}x}x^2=2x$. Note that it is a function of $x$. That is, the rate of change of a square's area with respect to its side length depends on what the side length currently is.
This example easily leads to the discussion on the chain rule. If a square's side length is changing at a time rate of, say, $2$ meters per second, then its area is changing at a time rate of $\frac{\mathrm{d}A}{\mathrm{d}t}=\frac{\mathrm{d}A}{\mathrm{d}x}\frac{\mathrm{d}x}{\mathrm{d}t}=2x(2)$$\frac{\mathrm{d}A}{\mathrm{d}t}=\frac{\mathrm{d}A}{\mathrm{d}x}\frac{\mathrm{d}x}{\mathrm{d}t}=2x(2$ m/s$)$ (where $x$ is a quantity having meters as the unit of measurement, so $\frac{\mathrm{d}A}{\mathrm{d}t}$ is in m$^2$/s). As the square gets bigger, its area is changing faster with respect to time. (That is, the time rate of change of the side length is constant, but the time rate of change of the area is not.)