ABSTRACT
Let f be a function with domain an open interval I. If x ∈ I then the quantity f(t)− f(x)
t− x measures the slope of the chord of the graph of f that connects the points (x, f(x)) and (t, f(t)). See Figure 6.1. If we let t → x then the limit of the quantity represented by this “Newton quotient” should represent the slope of the graph at the point x. These considerations motivate the definition of the derivative:
Definition 6.1 If f is a function with domain an open interval I and if x ∈ I then the limit
lim t→x
f(t)− f(x) t− x ,
Figure 6.1: The Newton quotient.
