The derivative

Average rate of change is easy: pick two points, divide the change in output by the change in input. Instantaneous rate of change is the hard case, because at a single instant both changes are zero and 0/0 says nothing. The derivative solves this by taking the average rate over an interval and then letting the interval shrink to nothing — a limit, not a very small subtraction.

Geometrically that limit is the slope of the tangent line. A secant through two nearby points has a definite slope; as the second point slides towards the first, those slopes settle on a value, and that value is f′(a). This is why the derivative answers "how fast, right now" for velocity, for reaction rate, for marginal cost — anything where the average over an interval is not the answer wanted.

Two boundaries are worth knowing. Differentiability is strictly stronger than continuity: |x| is continuous at 0 but has no derivative there, because the slope approaches −1 from one side and +1 from the other and no single limit exists. A corner is enough to break it. And a zero derivative means only that the tangent is horizontal — a maximum, a minimum or an inflection like x³ at the origin. Finding f′(x) = 0 locates the candidates; deciding which kind they are takes further work.