Rate of change and derived number

The rate of change

For a function f and two real numbers a and a+h (where h ≠ 0), the rate of change of f between a and a+h is the quotient: (f(a+h) - f(a)) / h

This quotient represents the slope of the straight line connecting the points A(a; f(a)) and B(a+h; f(a+h)) on the graph of f. This straight line is called a secant (or chord).

Example: f(x) = x², a = 2, h = 1. Rate = (f(3) - f(2)) / 1 = (9 - 4) / 1 = 5

The derivative

As h approaches 0, the rate of change may tend towards a limit value: this is the derivative of f at a, denoted by f'(a): f'(a) = lim (as h approaches 0) (f(a+h) - f(a)) / h

Geometrically, f'(a) is the slope of the tangent line to the curve of f at the point with abscissa a. We then say that f is differentiable at a.

Example: for f(x) = x², we show that f'(a) = 2a. Therefore, f'(2) = 4: the slope of the tangent line at x = 2 is 4.

Common pitfall

Do not confuse the rate of change (which depends on h and gives the slope of a secant) with the derivative (a precise limit, the slope of the tangent at a single point). Another pitfall: a continuous function is not necessarily differentiable everywhere (for example, f(x) = |x| is not differentiable at 0, as the limit of the rate of change differs to the left and to the right).

Summary table

Concept	Definition	Graphical interpretation
Rate of change	(f(a+h) – f(a))/h	Slope of the secant (AB)
Derivative f'(a)	Limit of the rate as h → 0	Slope of the tangent at a