Rate of change and derived number

Why talk about the rate of change?

When a function f describes a position, a temperature or a cost, we often want to know how fast it changes at a specific moment. That is precisely what the derivative is for.

The rate of change

Between two points a and a+h (where h is not equal to 0), the rate of change of f is:

rate = (f(a+h) - f(a)) / h

Geometrically, this is the slope of the line connecting the points (a, f(a)) and (a+h, f(a+h)) on the curve: this is known as the secant line.

The derivative

If, as h approaches 0, this rate of change approaches a fixed number L, we say that f is differentiable at a and that L is the derivative of f at a, denoted by f'(a):

f'(a) = the limit as h approaches 0 of (f(a+h) - f(a)) / h

Practical example

Let f(x) = x² and a = 3. Let’s calculate the rate of change:

(f(3+h) - f(3)) / h = ((3+h)² - 9) / h = (9 + 6h + h² - 9) / h = (6h + h²) / h = 6 + h

As h approaches 0, this rate tends towards 6. Therefore, f'(3) = 6: in the vicinity of x = 3, the function x² varies with an instantaneous slope of 6.

Summary table

Concept	Formula	Interpretation
Rate of change	(f(a+h)-f(a))/h	Slope of a secant
Derivative	limit of the rate as h → 0	Instantaneous slope at a

Common pitfall

Do not confuse f'(a), a NUMBER (the value of the slope at a specific point), with the derivative function f', which associates the derivative f'(x) with each x. Also note: a function may not be differentiable at a point (for example, |x| at 0), even if it is continuous there.