difference quotient
Before you can talk about an instantaneous rate, you can only measure an average rate. Over a road trip, total distance divided by total time gives your average speed. The difference quotient is exactly this: change in output divided by change in input over a finite step.
For a function f and two distinct points a and a + h, the difference quotient is (f(a + h) - f(a)) / h. Geometrically it is the slope of the secant line joining the two points (a, f(a)) and (a + h, f(a + h)) on the graph. It is defined for every nonzero h for which the function values exist; no limit is involved yet.
The derivative is born from the difference quotient by letting h tend to 0. The whole drama of differentiation lives in this single passage to the limit: the difference quotient is an honest, computable ratio, while the derivative is the value it approaches — which may or may not exist. Keeping the two ideas distinct is what makes the definition rigorous rather than circular.
For f(x) = 1/x at a = 2 with h = 0.1: (1/2.1 - 1/2) / 0.1 = (0.47619 - 0.5) / 0.1 = -0.2381, close to the derivative -1/4 = -0.25.
A finite step gives an approximation; shrinking h sharpens it toward the exact slope.
A common symmetric variant is (f(a + h) - f(a - h)) / (2h), the centered difference quotient, widely used in numerical approximation; for differentiable f its limit is also f'(a), but it can converge even when the ordinary one does not.