The derivative and the tangent

Letting b tend towards a

The rate of change gives the average velocity between two points. To obtain the instantaneous velocity at a point a, we let b tend towards a (or, equivalently, we set h = b – a and let h tend towards 0).

If the rate of change (f(a+h) - f(a)) / h has a finite limit as h approaches 0, this limit is called 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

Detailed example

Let us return to f(x) = x² and calculate f'(2):

(f(2+h) - f(2)) / h = ((2+h)^2 - 4) / h = (4 + 4h + h^2 - 4) / h = (4h + h^2) / h = 4 + h

As h approaches 0, the expression 4 + h approaches 4. Therefore, f'(2) = 4.

Graphical interpretation: the tangent line

Geometrically, as b approaches a, the secant line (AB) rotates about A and approaches a limit position: the tangent line to the curve at point A(a; f(a)). The derivative f'(a) is therefore the slope of this tangent line.

The equation of the tangent line at the point with abscissa a is:

y = f'(a) (x - a) + f(a)

For f(x) = x² at a = 2: f(2) = 4 and f'(2) = 4, so the equation of the tangent is y = 4(x - 2) + 4 = 4x - 4.

Common pitfall

If the limit of the rate of change does not exist (or is infinite), the function is not differentiable at a and there is no tangent line in the usual sense at that point (this is the case, for example, for sqrt(x) at 0).