Derivatives and Rates of Change

The Instantaneous Rate

The derivative measures how fast a function changes at one exact instant. It is the limit of average rates of change over shrinking intervals:

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

This fraction is the slope between two points. As $h \to 0$, the points merge into the slope of the tangent line.

Notation varies: $f'(x)$, $\frac{df}{dx}$, and $\frac{dy}{dx}$ all mean the same thing.

Limits and Pitfalls

A function with a sharp corner or a jump is not differentiable there because the slope has no single value.

Every differentiable function is continuous, but not vice versa. A corner is continuous yet not differentiable.

Common pitfall: Confusing the derivative (slope) with the function value (height).

$f'(x)$ tells you how fast it changes, not where it sits. $f'(x) = 0$ means the function is momentarily flat, not that $f(x) = 0$.