the (instantaneous) rate of change of f at point a. Geometrically, the derivative f ′(a) is the limit of
the slope of secant PQ as Q approaches P; that is, as h approaches zero. This limit is the slope of the
curve at P. The tangent to the curve at P is the line through P with this slope.
FIGURE N3–1a
In Figure N3–1a, PQ is the secant line through (a, f (a)) and (a + h, f (a + h)). The average rate of
change from a to a + h equals which is the slope of secant PQ.
PT is the tangent to the curve at P. As h approaches zero, point Q approaches point P along the
curve, PQ approaches PT, and the slope of PQ approaches the slope of PT, which equals f ′(a).
If we replace (a + h) by x, in (2) above, so that h = x − a, we get the equivalent expression
See Figure N3–1b.