Instantaneous Rate of Change
The instantaneous rate of change is the rate at which a quantity is changing at one specific moment, equal to the derivative at that point.
Formula
f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}
Definition
The instantaneous rate of change tells you exactly how fast something is changing at one precise moment in time, not over an interval; it equals the derivative $f'(a)$ at $x=a$, found by shrinking the interval of the average rate of change formula to zero: $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$. Its existence requires the function to be differentiable at $a$, a condition stronger than continuity.
Example
A car's speedometer shows its instantaneous rate of change of position: at this very second, you are traveling at $60$ mph, unlike a trip summary which only gives an average. For $f(x) = x^2$, $f'(x) = 2x$, so the instantaneous rate at $x = 3$ is $f'(3) = 6$. For exponential models like $f(t) = e^{kt}$, $f'(t) = ke^{kt}$, so in population growth or compound interest, $k$ is the instantaneous per-capita growth rate.
Key Insight
The difference between average speed and instantaneous speed is the difference between a trip summary and a live speedometer, and the instantaneous rate of change is exactly the slope of the tangent line, connecting the algebraic derivative to its geometric meaning. In physics, instantaneous velocity, acceleration, and jerk are successive instantaneous rates of change, and in thermodynamics the instantaneous rate of reaction is governed by differential rate laws.