The Derivative

The Rate of Change

The derivative measures how fast a function changes at an instant. It is defined as the 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 of the secant line through two points on the graph; as $h \to 0$ those points merge and the secant becomes the tangent line. So the derivative has two equivalent meanings: the instantaneous rate of change of $f$, and the slope of the tangent at $x$.

The sign of the derivative reads off the function's behavior:

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

Notation varies but means the same thing: $f'(x)$, $\frac{df}{dx}$, $\frac{dy}{dx}$, and $Df$. The requirement for the derivative to exist is that the defining limit exists — the function must be differentiable at $x$.

Differentiability is stronger than continuity: every differentiable function is continuous, but not conversely. A sharp corner (like $|x|$ at 0) is continuous yet not differentiable, because the left and right slopes disagree — the secant slope has no single limit. Likewise a vertical tangent or a discontinuity blocks differentiability. So smoothness (differentiability) implies no breaks (continuity), but no breaks does not imply smoothness.

Common pitfall: confusing the derivative (a rate/slope) with the function value, or reading $f'(x) = 0$ as $f(x) = 0$. $f'(x)$ is the slope, not the height — a function can be large while barely changing, or near zero while changing fast. And $f'(x) = 0$ means the graph is momentarily flat at $x$, not that its value is zero.

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

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