Suppose f(a) = b.

Every line through the point (a,b) has an equation of the form

L(x) = y = m(x – a) + b.

If there is some open interval about the number a, for which

L(x) < f(x) whenever x is in the interval and x < a, and

L(x) > f(x) whenever x is in the interval and x > a,

then m is NOT TOO SMALL to be the value of the derivative f '(a).

In a similar way we can define "NOT TOO LARGE".

If there is only one number that is not too small and also not to large, then that is the value of the derivative.

I have heard that the textbook titled *Calculus Unlimited* takes this approach.