Derivatives and Rates of Change

The Instantaneous Rate

The derivative measures how fast a function is changing at a single instant. It is defined as a limit of average rates of change over shrinking intervals: $$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}.$$ The fraction is the slope between two points on the curve; as $h \to 0$ the two points merge and the slope becomes the slope of the tangent line at $x$. So the derivative has two equivalent meanings: an instantaneous rate of change and a tangent slope.

The intuition is everywhere. If $f(t)$ is position, $f'(t)$ is velocity — the rate position changes. If $f$ is cost, $f'$ is marginal cost. If $f$ is a smooth curve, $f'(x)$ tells you how steeply it rises or falls at each point:

• $f'(x) > 0$ — the function is increasing there.
• $f'(x) < 0$ — the function is decreasing there.
• $f'(x) = 0$ — the tangent is horizontal (a peak, valley, or flat spot).

The notation varies: $f'(x)$, $\frac{df}{dx}$, and $\frac{dy}{dx}$ all mean the same derivative. The requirement is that the defining limit exists — a function with a sharp corner or a jump is not differentiable there, because the slope has no single value. Differentiability is stronger than continuity: every differentiable function is continuous, but not vice versa (a corner is continuous yet not differentiable).

Common pitfall: confusing the derivative (a rate/slope) with the function value, and misreading the sign. $f'(x)$ is not $f(x)$ — a function can be large while barely changing (small derivative) or near zero while changing fast (large derivative). And $f'(x) = 0$ means the function is momentarily flat, not that $f(x) = 0$; the derivative being zero is about the slope, not the height.

A curve with a secant line through two points; the second point slides toward the first, the accent secant rotating into the tangent line as h shrinks to zero — the derivative as a limit of slopes.

$$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$