Derivative as Rate of Change of Quantities

Many real situations involve one quantity changing because another one does — distance changes with time, cost changes with production, revenue changes with production, price changes with demand. In each such pairing, we're interested not just in the two quantities themselves but in how fast one changes as the other does.

As you saw in Class XI, if $y = f(x)$ is a real function, the derivative $\dfrac{dy}{dx}$ measures exactly this: