If a car drives 100km in 2 hours, its average speed is 50km/h. But that's just an "average" — in reality, it probably stopped at red lights and also hit 100km/h on the highway at times. So how do you find "the speed right at this instant"? Answering exactly this question is what differentiation is all about.
It becomes clearer when you think in terms of a graph. The line joining two points A=(a, f(a)) and B=(a+h, f(a+h)) on a curve f(x) is called a secant. This secant's slope, (f(a+h)−f(a))÷h, is the average rate of change from A to B (the exact same calculation as "average speed" in the car example). Now slowly move B closer and closer to A (shrink h toward 0). The secant keeps rotating, gradually approaching a tangent — a line that just barely touches the curve at point A.
If you make h exactly 0, A and B overlap and you can't calculate a slope at all (it becomes dividing by 0). So instead of substituting h=0, you find the limit value the slope approaches as h gets infinitely close to 0. This limit value is called the instantaneous rate of change or the derivative, defined as f'(a) = lim(h→0) [f(a+h)−f(a)]÷h. In the car example, this is exactly "the speed right at this instant."
Actually calculate this with the function f(x)=x², and a curious pattern appears. The instantaneous rate of change is 2 at x=1, 4 at x=2, and 6 at x=3. Notice it always comes out to double the original x value? Turn this rule into a function, and you get the derivative f'(x)=2x. The derivative is a new function that tells you all at once "how steep the tangent is at each point of the original function" — instead of calculating a limit every single time, you can just plug whatever x you want into f'(x), and the instantaneous rate of change at that point comes right out.
Because differentiation deals with "the instantaneous speed of change," it's used in a truly wide range of fields. In physics, velocity is the derivative of position, and acceleration is the derivative of velocity. In economics, differentiating a cost function gives you "the extra cost of producing one more unit" (marginal cost). On a graph, finding the point where the derivative becomes 0 lets you locate a function's maximum or minimum (the peak or the bottom of the graph), which is why differentiation is also a core tool for finding an optimal choice.
When studying this with kids, it helps to start with the question, "what happens if you keep cutting the distance between two points in half, forever?" Actually using a ruler and protractor to move two points on a curve closer and closer together, and measuring how the line's slope changes, helps build a physical sense of "approaching a tangent in the limit." On our activity page, you can move h with a slider toward 0 and watch directly as the secant becomes the tangent, then compare the original function and derivative graphs side by side and confirm that the tangent's slope and the derivative's value always match.